ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Technical Report THE INTERACTION OF PLANE AND CYLINDRICAL SOUND WAVES WITH A STATIONARY SHOCK WAVE W. R. Johnson 00 Laporte Project Supervisor Project 2539 DEPARTMENT OF THE NAVY OFFICE OF NAVAL RESEARCH WASHINGTON, D. C. CONTRACT NO. Nonr-1224 (18) June 1957

This report has also been submitted as a dissertation in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1957.

ACKNOWLEDGMENTS The author wishes to express his appreciation to the staff of The University of Michigan for the. assistance given him during the course of this research. Professor Otto Laporte deserves special thanks for suggesting the problem presented herein, for directing the research to its conclusion, and for criticizing the final manuscript. Thanks are also due to Professor R. C. F. Bartels and Doctor W. C. Meecham for their help and encouragement during the course of this research. The author also wishes to express his appreciation to the Office of Naval Research for support of this work under Contract No. Nonr1224(18). ii

TABLE OF CONTENTS Page LIST OF FIGURES v LIST OF SYMBOLS vii ABSTRACT x OBJECTIVE x Io INTRODUCTION 1 Ao DESCRIPTION OF THE PROBLEM 1 Bo PREVIOUS INVESTIGATIONS 8 IIo THE LINEAR DIFFERENTIAL EQUATIONS AND SHOCK CONDITIONS 11 Ao THE DIFFERENTIAL EQUATIONS 11 Bo THE SHOCK CONDITIONS 15 IIIo PLANE WAVES AND THEIR INTERACTION WITH A STATIONARY SHOCK WAVE 19 Ao PLANE WAVES IN A MOVING FLUID 19 Bo INCIDENCE FROM THE SUBSONIC SIDE 24 1. Reflection Laws 26 ao Reflected Sound Waves 26 bo The Reflected Entropy-Vorticity Waves 28 2o "Fresnel" Formulae for Reflection 29 Co INCIDENCE FROM THE SUPERSONIC SIDE 33 lo Refraction Laws 34 ao Refracted Sound Waves 35 bo Refracted Entropy-Vorticity Waves 38 2, "Fresnel" Formulae for Refraction 40 IV CYLINDRICAL SOUND WAVES IN A MOVING GAS 44 Ao THE CONVECTIVE WAVE EQUATION 44 Bo LINE SOURCE IN A SUBSONIC GAS 45 Co LINE SOURCE IN A SUPERSONIC GAS 52 Vo THE INTERACTION OF CYLINDRICAL SOUND WAVES WITH A STATIONARY SHOCK WAVE 65 Ao SUBSONIC INCIDENCE 63 lo The Interaction Integrals 65 2o -The Shape Function: flt)66 35 The Entropy Function: A,(7['Ai)69 4. The Vorticity Potential:C( jt)jt) 71 5o The Sound Potential: ~ ( it 72 Bo SUPERSONIC INCIDENCE 74 lo The Interaction Integrals 74 2o The Shape Function: fl[Yt) 77 ao Inside the Mach Wedge: Xo > \\ 78 b o Outside the Mach Wedge: Hi IM'1 81 iii

TABLE OF CONTENTS (Concluded) Page 3, The Entropy Wave:Ail,( t) 82 a. Inside the Edge Streamlines: Xo > 1' 83 b. Outside the Edge Streamlines: XA' C<1 83 4. The Vorticity Potentialo A )!t) 83 ao Inside the Edge Streamlines: X< >l'| 86 b. Outside the Edge Streamlines: )(<'l 86 5. The Sound Potential: l x,y,t) 87 VI. EXTENSIONS AND CONCLUSIONS 93 A. POINT SOURCE 93 Bo MOVING SHOCKS 97 lo Reflection of Plane Waves from a Moving Shock 98 2. Refraction of Plane Waves from a Moving Shock 100 35 Reflection and Refraction of Cylindrical Sound Waves by a Moving Shock 103 C. CONCLUSIONS 104 APPENDICES 106 APPENDIX A - THE DIFFERENTIAL EQUATIONS AND SHOCK CONDITIONS 107 APPENDIX B - CHARACTERISTICS OF THE LINEAR FLOW EQUATIONS 118 APPENDIX C - SIMPLIFICATION OF THE LINEAR SHOCK CONDITIONS 121 APPENDIX D - TRANSFORMATION OF THE "FRESNEL" COEFFICIENTS FOR THE REFLECTED WAVES 124 APPENDIX E - TRANSFORMATION OF THE "FRESNEL" COEFFICIENTS FOR THE REFRACTED WAVES 127 REFERENCES 132 iv

LIST OF FIGURES Figure Page 1 Undisturbed state of the gaso 1 2 Reflection and refraction of plane waveso 5 3 Cylindrical waves in a moving gas. 6 4 Interaction of cylindrical waveso 7 5 The disturbed state of the gas 16 6 Plane waves in a moving gas, 21 7 Energy transport for sound wavesO 23 8 Reflection of plane sound waveso 28 9 Angle reflections for reflectiono 29 10 Refraction of plane sound waveso 39 11 Angle relations for refractiono 40 12 Path of integration for'l(x' ) subsonic caseo 48 13 Path of integration for 0(x,y') subsonic caseo 49 14 Mapping of 8 to 0o; subsonic case 53 15 Path of integration forX(x' ) supersonic caseo 55 16 Path of integration for O(x',y ), supersonic case, inside the Mach wedgeo 57 17 Path of integration for 0(x',y' ) supersonic case, outside the Mach wedgeo 58 18 Mapping of 9 to GO; supersonic caseo 62 19 Path of integration for F(y' ) subsonic caseo 67 20 Path of integration for 0l(x',y'); subsonic case, 73 21 Reflection of cylindrical waveso 75 22 Paths of integration for F(y' ) supersonic incidence inside the Mach wedgeo 79 v

LIST OF FIGURES (Concluded) Figure Page 25 Path of integration for F(y'); supersonic case outside the Mach wedgeo 81 24 Path of integration for X(y'); supersonic case outside the Mach wedge, 85 25 Paths of integration for l(x',y')' supersonic case. 89 26 Refraction of cylindrical waveso 92 27 Paths of integration for O(x',y,z), 96 28 Undisturbed state of a moving shocko 97 vi

LIST OF SYMBOLS Symbol Explanation 0o Amplitude of distortion N AAmplitude of sound wave 5b Amplitude of vorticity wave Cp Cv Specific heats AC~ Undisturbed sound velocity, amplitude of the entropy wave G ^Disturbed sound velocity ad ciPerturbation to density D Undisturbed density 01U Disturbed density (rt) Shape of shock F (X) Reduced shape function G (2j Reduced vorticity function h(.) Unit step function H (c,$) Integrand of entropy integral (supersonic) 3T(0o) Integrand of vorticity integral with poles removed (supersonic) e AnAmplitude of wave vector \K Pole of sonic integral %k<i 6Integrand of vorticity integral Xa^ Phase constant ~LpiC Integrand of sonic wave (supersonic) Y1 Perturbation to Mach number K -Undisturbed Mach number iH Disturbed Mach number ^,p PPerturbation to pressure vii

LIST OF SYMBOLS (Continued) Symbol Explanation eP Undisturbed pressure yGi Disturbed pressure fS Distance Q4%S Perturbation to entropy S Undisturbed entropy %c ) Reduced entropy function (supersonic) "Poynting" vector t oDisturbed entropy t Time %U, U Perturbations to x-velocity U Undisturbed x-velocity CU% Disturbed x-velocity V, V Perturbation to y-velocity 6- ~ Disturbed y-velocity (C Y,) Coordinates t ^Cos Sin Ratio of specific heats, "/C Fr\ Polynomial in o( Dirac's delta function v ~ Gradient operator 6t'Strength of incident disturbance fQ -Angle h Ratio of sound speeds C,/C /\ Polynomial in o(I viii

LIST OF SYMBOLS (Concluded) Symbol Explanation eP Transformed distance Angle )(D^ XReduced velocity potential ~2XV,8t) Velocity potential (xi8t) Vorticity potential I' Y) Reduced vorticity potential ~r_ Angle (CO FFrequency ix

ABSTRACT This investigation treats the problem of the interaction of plane and cylindrical sound waves with a stationary shock wave theoretically. The linearized Euler differential equations and the corresponding linearized shock conditions serve as the fundamental laws for this study. Plane-wave solutions to the differential equations are found, and the analogues of Snell's laws of reflection and refraction are determined, together-with the "Fresnel" formulae. Integral solutions to the differential equations corresponding to incident cylindrical waves are then found from the planewave solutions by a method devised by H. Weyl. These integrals are investigated to determine the reflected and refracted field of a line source. Generalizations of the theory for moving shocks and point.sources are set up but not investigated in detail. OBJECTIVE The purpose of this study is to determine how sound waves interact with a stationary shock wave in a compressible, inviscid gas. Two types of incident sound waves are considered, plane and cylindrical. For incident plane waves the analoques of the reflection and refraction laws of optics are desired. One may use these plane-wave laws in a manner devised by H. Weyl to solve the cylindrical wave problem. x

I. INTRODUCTION A. DESCRIPTION OF THE PROBLEM This introductory chapter will be devoted to a purely qualitative discussion of the interaction of sound waves with a stationary shock waveo The fluid medium which is to serve as a carrier for the shock wave is assumed to be a compressible inviscid gas. When no sound waves are present, we assume that the shock wave is a plane at rest normal to the flow of this gas. Such a plane normal shock wave constitutes a rigorous solution of the equations of hydrodynamics in which the physical properties of the gas assume constant values. A situation of this type is illustrated in Fig. lo It should be noticed that the flow enters the shock supersonically and leaves subsonically. The pressure, density, and entropy on the supersonic side are smaller than the corresponding quantities on the subsonic side. SUPERSONIC SIDE SUBSONIC SIDE Velocity, U> C U < C Pressure, p PI Density, D D Entropy S S, Stationary Shock Wave Figo 1I Undisturbed state of the gaso We now introduce a sound wave into this uniform but discontinuous flowo This sound wave disturbs the shock wave and thus the flow, presumably on 1

2 both sides. Let us suppose that the incident wave is a plane waveo Since the situation seems analogous to the reflection and refraction of plane light waves in a medium of discontinuous refractive index, one would expect to find reflected and refracted plane waves present in the flowo It is therefore natural to inquire after the analogues of Snell's law and of the Fresnel coefficients~ In other words, we wish to find the relations between the angle of incidence and the angles of reflection or refraction as well as the amplitudes of the reflected and refracted waveso However, the optical analogy is only superficial, as the two problems differ in certain important respects, Perhaps the most important way in which the problems differ is in the types of waves present~* In the optical case, there is only one type of wave present; this is referred to as a light wave. In the problem presented here, we may have two types of waveso One of these is an irrotational, isentropic wave, referred to as the sound wave; the other, a solenoidal, isobaric wave, referred to-as the entropy-vorticity waveo The sound waves here are not identical with the sound waves of classical acoustics since the medium carrying these waves is in motiono But they do behave like ordinary sound waves if they are observed from a frame of reference moving with the fluide.On the other hand, the entropy-vorticity waves are waves moving with the fluido they would appear to be stationary to an observer moving with the flow. These waves transport entropy and *In optics we have only a transverse wave to consider, while here we have both transverse and longitudinal waveso We should, therefore, compare our theory to th theory of propagation of waves in an elastic medium, but since this subject is apt to be less familiar to the general reader, we shall be content with the optical analogueo

5 vorticity changes into the flow, but do not transport pressure changeso In the optical case, the light waves do not distort the reflecting surface, but it can readily be seen that in the present problem the shock itself changes shape, while at the same time an entropy wave is thrown off. Another novel feature is illustrated in Figo 2. It is seen that when the sound wave is incident from the supersonic side, only refracted waves appear, while waves incident from the subsonic side give rise to reflected waves onlyo In other words, the disturbance in the flow created at the shock front is apparent only on the subsonic sideo The reason for this behavior in the case of the entropy-vorticity waves is obviouso These waves move with the fluid which in turn moves into the subsonic flow, away from the shocko Thus entropy-vorticity waves created at the shock will appear only on the subsonic sideo The sound wave, on the other hand, moves relative to the flow with the sound speedO On the left, the flow speed is greater than the sound speeds thus, sound waves created at the shock wave will appear only on the subsonic sideo SUPERSONIC SIDE SUBSONIC SIDE SUPERSONIC SIDE SUBSONIC SIDE Uo U1 Uo U1 Incident Sound Incident / \ / Wave Sound // Refracted r\ ^\\. Wave / Entropy-Vorticity Wave Reflected V Entropy -Vorticity Wave Reflected Refracted S/Sound Wove Sound Wove a) SUBSONIC INCIDENCE b) SUPERSONIC INCIDENCE Figo 20 Reflection and refraction of plane waveso Let us consider the boundary conditions nexto Small disturbances on one side of the shock wave are related to small disturbances on the other

4 side by means of the linearized shock conditionso These conditions determine the disturbance in pressure, velocity, and entropy on the subsonic side as linear functions of the disturbances in pressure, velocity, and entropy on the supersonic side and the distortion of the shock waveo Since the disturbances are plane waves with amplitudes independent of the coordinates, it is necessary that the exponential factors match at the shock surface. This fact gives us the analogue of Snell s laws of reflection and refraction, as in opticso The shock conditions may be written as four linear equations relating the amplitudes of the reflected or refracted waves to the amplitude of the incident wave. The four amplitudes appearing in the linear equations are those of the sound wave, entropy wave, vorticity wave, and the distortion of the shocko This latter amplitude has no optical analogueo These equations may be solved to give the amplitudes of the four waves, which are the required solutions to the plane-wave problem. We now turn to the interaction of cylindrical sound waves with a shock wave. A three-dimensional electromagnetic analogue to this problem is the propagation of radio waves from a transmitter above a plane earth~ Let us first consider the properties of cylindrical sound waves in a moving medium with no shock wave present If the medium were not in motion, one would be able to find the properties of such waves by solving the wave equation for an oscillating point sourceo Thus one is inclined to view this problem from a frame of reference moving with the fluid, with the result that the equation governing the velocity potential of the disturbance becomes the wave equation, but that the source is now in motiono To transform the source back to rest and yet to preserve the form of the wave equation, an appropriate Lorentz transformation* on the moving*The Lorentz transformation referred to here is based on the speed of sound rather than the speed of lighto

5 source equation is used. The resulting equation is still the wave equation but now with a stationary source, Thus from this peculiar Lorentz-Galileo frame of reference the waves should look like sound waves from a stationary source in a fluid at resto One merely transforms the known solution back to the original frame of reference to find out what the waves actually look likeo This procedure gives a reasonable result when the fluid speed is subsonic. Further, upon using the same method with a "supersonic" Lorentz transformation, a perfectly acceptable solution is obtained. However, since it is essentially an elementary function of the transformed distance from the source [ x2 - (M2 - l)y2], it will be real not only in the downstream Mach "wedge" but also in the unphysical upstream Mach wedgeo It is therefore clear that the desired solution cannot be written in closed form, but will have to be an integral or series which, when evaluated in various regions, will be represented by different analytical expressionso The method adopted to overcome this difficulty is a technique commonly employed in wave mechanicso2 The solution is expressed as a Fourier integral and the integrand is foundo Various solutions to the problem may be obtained by choosing different paths of integrationo Two of the solutions are of the form discarded above, but another is a physically reasonable result which is chosen to be the solution of our problemo The solutions, which are finally obtained are illustrated in Fig. 3o When the flow is subsonic, The lines of constant phase are circles which are blown downstream with the flow (Fig, 3a)0 The supersonic case -

6 U,<C Uo< co Souurce Sr 900 Phase Lead Source. (Ut)2+22 to (x Uot)2+y2C2t2 x-U^+y f C1t= Mach Wedge a) SUBSONIC FLOW b) SUPERSONIC FLOW Fig. 3. Cylindrical waves in a moving gas. j0i X X ( >.-0- ma ) -" otherwise -is more interesting since the circles are confined to the interior of a wedge, the Mach wedge. In Fig. 3b the part of the circle to the right of the intersection with the Mach "wedge" leads the remainder of the circle by 90~. This phase shift is analogous to the 90~ phase shift, discovered by Debye,'4 of a wave passing through a focal line. It should be noted that the fact that these circles cross one another brings about destructive interference along certain curves. The existence of the stationary zeros in the flow gives these waves the appearance of standing waves, although they are, in reality, progressing downstream, and preserving their nodal line. The incidence of cylindrical waves from both the subsonic and supersonic side is treated in full, and rigorous closed form solutions for all flow variables of both sonic and entropy-vorticity waves are obtained. The results are qualitatively pictured in Fig. 4. In the case of subsonic incidence the reflected waves are obtained using the saddle-point method. The sonic waves are cylindrical waves whose centers move downstream with

7 SUPERSONIC SIDE SUBSONIC SIDE SUPERSONIC SUBSONIC SIDE Uo> Co U, < C, SIDE U, < Cg ~ —--- ----- U > Co Reflected I -" Sound Entropy- A, - S Vorticity Sound -// WavSound/Wave wave Incident Wove a) SUBSONIC INCIDENCE b) SUPERSONIC INCIDENCE Fig. 4. Interaction of cylindrical waves. the flow, while the entropy-vorticity waves are segments of hyperbolas. As is to be expected, the case of incidence from the supersonic side is more complex. The transmitted sonic waves are no longer circles, although the deviation from circularity is of the order xo/p, xo being the distance of the source from the shock. The entropy-vorticity waves are confined to the strip - x < < x_,' and are swept downstream within it, s/Mo - 1 //M2 - 1 The detailed presentation of the theory just outlined is to be found in Sections II to VI. Section II is devoted to a study of the linear differential equations and linear shock conditions governing the propagation of small disturbances. In Section III, plane-wave solutions to these differential equations are studied. Parts B and C of Section III are devoted to the interaction problem for the subsonic and supersonic cases, respectivelyo Each of these parts is again divided into to t subsections, the first dealing with te analogue of nell law and the analogue of Snells law and he second with the analogue of the Fresnel coefficients.

8 In Section IV the problem of cylindrical waves in a medium without shocks is studied and the decomposition into plane waves is made. Section V is devoted to a detailed study of the integrals describing the interaction of the waves of Section IV with a plane normal shock wave. In Section VI we discuss the extension of this theory to the spherical waves generated by a point source, as well as the necessary modifications for the case of a moving shock wave interacting with a stationary source. This case could lend itself to observation in the shock tube, Because of their relative complexity, certain computations pertaining to the shock conditions and Fresnel coefficients have been referred to Appendices A to Eo B. PREVIOUS INVESTIGATIONS Several authors have interested themselves in problems involving the interaction of sound waves and shock waves.5 The first of these was V. Bargmann, who gave the solution to the problem of the diffraction of a shock wave by a thin wedge. In this problem a moving shock wave strikes a thin wedge normally. The tip of the wedge acts as a source of cylindrical sound waves. These sound waves interact with the shock wave and cause it to be slightly distorted. Bargmannts solution to the problem involved the assumption that the shock wave was so weak that the flow behind the shock was irrotational and isentropic. Lighthill7 examined this diffraction problem for strong shocks and thin wedges. He was able to eliminate consideration of the entropy and vorticity and find the pressure field and resulting distortion of the shock wave. Lud8 loff was able to extend the results of Lighthill to the case of thin bodies of various shapes~ The diffraction problem, although it involves the interaction of the sound field and the shock wave in a manner different from that of our con

9 cern here, is of fundamental importance since it was the first in which the linear shock conditions were used as boundary conditionso A problem which is closer to the one solved, here was treated by Car9 riero This problem is that of interaction of a plane sound wave with a stationary shock wave inclined to the flowo Carrier introduces the entropyvorticity wave as well as the sound waveo He assumes that a'plane sound wave in the flow behind the inclined shock interacts with the shock, and he calculates the resulting distortion of the shock waveo No treatment of sound waves incident from the front of the shock is giveno Ribner1 considers the problem of the convection of a plane vorticity wave of a given profile through a moving shock wave, and calculates the resulting sound, vorticity, and entropy waveso Fo Ko Moorell calculates the interaction of sound waves of a given shape incident from either side of a moving shocko This problem is quite closely related to our plane-wave problem0 The problem of the cylindrical waves in a supersonic flow with no shock wave present may be compared with I. G. Tammrsl2 theory of Cerenkov radiationo In fact, we may formally compare our source with a body, whose charge.oscillates in time, moving faster than the speed. of light in a nondispersive medium0 The interaction of electromagnetic dipole radiation incident upon a plane earth has received a great deal of attention013 It is from this source that we are able to obtain the greatest amount of guidance for the present work Two related methods have been proposed for solving the radio problem, 14 The first of these is due to Sommerfeldo He writes the reflected and refracted waves as integrals which have integrands of a form suggested by the incident wave, except for an undetermined function, which is then determined

10 by the boundary conditions. The solution so obtained is in integral form, and may be expanded asymptotically to give the far fieldo Weyl 5 writes the incident wave as a superposition of plane waveso He then writes the reflected and refracted waves as superpositions of plane waves of the same formo He applies the boundary conditions to the integrals and finds the amplitudes of the reflected and refracted waves, which are (as one would expect) given by the Fresnel coefficientso Thus he also has the solution in the form of an integral which may be expanded asymptotically to give the far fieldo Our method of treatment of the cylindrical wave problem should be considered as the analogue of Weyl s method for the radio wave problemo *It may be demonstrated that the integrals obtained by Sommerfeld are equivalent to those obtained by Weyl; however, the results arising from the discussion of the different formulations are in striking disagreement0 This gave rise to a controversy which continued from 1919 until 19439 when it was resolved by Ott who used a method originally devised by Pauli for a different purpose0 The difficulty occurs because poles and saddlepoints may be near one another in the plane of integration. In Sommerfeldts treatment, this difficulty is not as obvious as in Weyl~s method, and in fact was completely overlooked, giving rise to the disagreement0 Van der Waerden has subjected the problem of evaluating integrals of this type to a rigorous mathematical treatment which has done much to clear up the problemo (See Refso 16 and 17o)

12 We let the flow on either side of this plane normal shock be slightly disturbed. The dimensionless perturbations,,)S,, and are introduced by means of: Pressure: = P + D x-velocity: I = U + - U y-velocity:'J = (2 2 (2.2) Density: + Entropy: - S + A Cp Upon substituting Eqs. (2.2) and neglecting the squares of the small perturbing terms, Eqs. (2.1) become: Continuity: +U + U( s) = Momentum x-component:.A+U'C s = 0 (2-3) y-component: t+ -'A +C Entropy: a U 0 j Equation of State: U = C( -h) The entropy equation and equation of state in (2o3) may be used to eliminate c from the continuity equation. If this is done, we may rewrite the system as four linear differential equations involving, f,ir, and A, and one linear equation defining a in terms of $ and A.o The resulting equations, which shall serve as the fundamental equations of this study, are:

13 a x C (a+ U ) +2 (2~ /~__d.t ~___t) +~0 - C at co X w-A- In the following, only the first four of Eqso (2~4) shall be consideredo The fifth equation of (2.4) may be used to find the density after the first four have been solvedo Subscripts 0 or 1 should be used on all the constants and dependent variables in Eqso (204), 0 referring to the supersonic side of the shock, and 1 referring to the subsonic side, In Appendix B the characteristic curves of Eqso (2~4) are studiedo The results found there lead us to the conclusion that the characteristic curves either move relative to the flow with the sound speed, or move with the flowo In the former case there is a jump in pressure and a jump in velocity which is normal to the curveo There is no jump in entropyo These isentropic, longitudinal waves are called sound waveso In the latter case there is no jump in pressure across the characteristics and the velocity jump is tangential to the curve, There is a jump in entropy which has an amplitude independent of the amplitude of the jump in velocity0 These transverse, isobaric waves are called the entropyvorticity waveso

14 We see here why the density was eliminated from Eqs. (253). If the density had been used, we would have had jumps in density across both families of characteristics, whereas pressure and entropy have jumps across one or the other of the characteristic families but not both. Let us now derive the energy transport equation associated with Eqs. (2o4) found above. The equation governing the transport of energy is found by multiplying the first three of (2.4) by1p,Uj, and r, respectively, and adding the resulting equationso This gives the following expressions C \U r- ~) +) + Q } ~ (2.5) which may be rewritten in the form of a conservation equation as: wat rutpX L 2t~Z )+9 (Cg1Pu++ C+, (2.6)?t)c z z / The term = in (2.6) is the "dimensionless" energy density of the wave, while the vector S —(Ctups~U t (>C spa), which has the dimensions of velocity, represents the energy flux, or "Poynting," vector. Equation (206) then expresses the conservation of energy as:.* |&F+ V * = 0. (2.7) The vector.consists of two parts, % = C'qlv) and z U )) 0^ is the "Poynting" vector in a frame of reference moving with the flow, while S2 is the contribution to the flux arising from the motion of the gas, *This derivation of the energy-conservation law (as well as the related momentum-conservation law) is essentially due to 00o Laporteo

15 B. THE SHOCK CONDITIONS We have disturbed the flow on both sides of the shock wave, and since we have in no way constrained the shock wave, it will be disturbed also. Equations (2.4) describe the behavior of the disturbance on either side of the shock, and we shall now find the relations connecting these disturbances across the shock front, The shock wave is assumed to be only slightly disturbed from a state of rest at -= O. Let this disturbance be described by =- f (,t) The normal and tangent vectors to the shock wave are then given to first order by: (2.8) The shock velocity is given by: -ts = C X ) (2.9) The relative velocities of the flow to the shock are given by: i^ = U,, X - u, o U- iJ - i 1 -J) (2.10) > - u- _ ^'"' ^ -^. ".-^ I+ t t i The relative normal and tangential velocities of the flow are obtained by forming the scalar product of (2.8) and (2.10): UTh = Uto+ U)f n w ith h e isto Ue(UItda- it); Ukrin 3 Yw\ + oll Uot — ~ U~o+v tq -so\ot -Uu i (~ + t 0+; U,. =U,0\j0S,, -QB+U-l^^3').To A A The situation with which we wish to deal is shown in Fig. 5.

16 y SUPERSONIC SIDE SUBSONIC SIDE o: P +oDoCUo., >- PR + 4, C,U, U^o- U o kUo ID U U^^.U. Uo = fUn ^ t <= _, U, o o -- o c _ 5= t, Cp __x = f(y,t) Fig. 5. The disturbed state of the gas. 20 The conditions governing a general curved, moving shock wave are: Continuity:'- $O Momentum = + Normal component: U + kI Av - _ + <o4o (2.12) Tangential component: Ult U oRt Energy: -'-ao l _ o In Appendix A we show that these may be written simply as: 4, =9t G(Tn) v (2. 13) where= o where

17.' D t F Cm) = ^^^'') ) SAak )= ) __ ___One may find the relations between the first order disturbances by differentiating the relations (2.13)o The first order relations are: t %1 = %FI~o +>< ~M~-~i,~t-t-'C(MO); (21l4) At\ o LGV TV < F ^ ~ where f\is defined by: xnb = X.C - These relations may be rewritten to give I- p r j,j and L as linear functions of _I 0k ) and A, o This rather lengthy manipulation is carried out in Appendix Co We summarize the linear shock conditions as follows: Th = c\\oeffcitos 1Aisld.\its it rs = NvZPo + K22Ulo+ 4 j i (2 ol5) oTh = cA33yeod pi iS ^ = R4\ to A\42WUo C A so+ P\^S ft The coefficients Aij are displayed explicitly in Appendix C.

18 Although this system of linear relations decomposes into four relations connecting the quantities A ~jU A and one relation between'\ and i the actual boundary conditions between plane-wave amplitudes do not decompose in this mannero In this section we have found the differential equations (204) which small disturbances satisfy, and the conditions (2l.5) which connect the variables describing these small disturbances across a distorted shock wave In the next section we shall investigate the solutions to these equations in the case of plane-wave disturbances,

III. PLANE WAVES AND THEIR INTERACTION WITH A STATIONARY SHOCK WAVE A. PLANE WAVES IN A MOVING FLUID In this section we wish to consider solutions of the differential equations (2,4) which are harmonic in time and constant on lines perpendicular to a certain fixed unit vector = 0 A), ( and f being the cosine and sine of the angle Y makes with the positive x axis. From the discussion of characteristics in Section II, we expect the planes of constant phase to be of two types, those which move downstream with the fluid and those which move relative to the fluid with the sound speed. That this is so will be verified below. Let us consider plane-wave solutions of (2.4) in the form: _- u ~'~el^ x+A - L~t 1 A = U eS (~ks b - L St Equations (2,4) become: Wdok P+ t- + ( +,d) 4 X4 0 ) (352) T P+ k_ +MWA) V = O (- ~ +,)S = O. with k = - These equations have a solution if the coefficient deterC minant vanishes, This gives the following equation for ~: (-\Q+tI^d^ ltt R+>o%0() >] = o.) The roots of this equation are given by: 19

20 A_\Q~~~ fe(3.4) Al A = d- w^^w~ ~(305) * - _ (356) The solutions to Eqs. (3.2) which correspond to the first eigenvalue?\ = s\ are: ~- = O ) v.= ~ L. B' ~ -'~.; pU ieetviX^^Pd-txt (307).^,dX~M la L Wt,= Ce Ni, with B and C arbitrary constants. This wave corresponds to the entropyvorticity wave of Section IIo There are two important properties of this wave which we wish to mention here. These ares 1) The wave moves with the fluido Proof: The planes of constant phase are described by - WOt constant. The x-component of the phase velocity is given by = = flow velocity (Figo 6a), 2) The velocity may be derived from a vector potential. Proof: Po - O from (3~7), hence: u= |9'- }=- a_ ) (538)

21 Flow Velocity U Velocity U a) ENTROPY-VORTICITY WAVE X b) SOUND WAVES Fig. 6. Plane waves in a moving gas. where =-'L i C _ _ 1. (3.9) Now we turn to the second eigenvalue, vL =.: _ ^ XstL - Got L gSe=" IPX^' vl^ t+ ) (3.10) O with A an arbitrary constant. This wave corresponds to the sound wave of Section II. There are two important properties of this wave analogous to, but differing from, those for the entropy-vorticity wave. These are: 1) The wave moves with the sound speed relative to the fluid.

22 Proof: consider the surface of constant phase \ + U o V~'___ —__-+-__. — Lkt - constant. Hence, & lOX ) = v+Wol) C = ( C Uol) (Fig. 6b)o 2) The velocity may be described by means of a scalar potential. Proof: M^~L - a- - 0, O from (3 10); hence: a-^ @. -^(3 11 where whr Q' ( l W$\ ~) t;M 0~ X I) _ L, t (3 12) Notice that, may be found also in terms of ~ as: (3513) Finally for the third eigenvalue, k = 3 we find from Eqso (352): %~ =_ ~ % ~~x- ( L _ -^t Lot. ) - t - _- (3f14) A= O This wave is identical with wave (3o10) if (d,g) is replaced by (-O. -p),

23 i.e., if the normal vector is reversed. This is a sound wave also, but moving in the direction of - rather thanvt. Each of the waves considered above transports energy into the gas. It is of interest to consider the direction of the energy flux associated with sound waves and with entropy-vorticity waves. Energy is transported through the gas in the direction of g given in (2.6). The time average of 5for the plane sound waves (3.10) is: XL A 3 \^\ y > >\(5-15) S =, C ^,) = (cy+OGl ), where A,=_()0), and s indicates the time average. Formula (3.15) is the generalization of the ordinary "Poynting" theorem for plane waves, = c n. The direction of propagation of energy for the sound waves is found by the velocity addition law, as illustrated in Fig. 7a. Poynting Vector Wave Fronts Sound circles Direction of Wove Normal Wave i Normal CP ^ Direction of 8i9~ ~ / Poynting gcSM 8P / Vector 9 \ Un' Flow Direction UnL Flow Direction a) POYNTING VECTOR FOR SOUND b) GRAZING ANGLE Fig. 7- Energy transport for sound waves. If U C the vector S will always lie interior to the forward Mach "wedge" (and will vanish on this surface); thus the energy is always blown downstream. On the other hand, if U<C, the vector S will have a component in the direction of - ~, for angles in the range lT 6- COS -,. For incidence upon a shock the angle % =-C05-O'-M plays the same role as

24 the 90~ angle of grazing incidence in the "Fresnel" problem of electromagnetic theory (see Fig. 7b)o In particular we must consider waves with angles in the range 0 < e, as downstream waves, and waves with angles in the range - < ~ TT as upstream waves. The entropy-vorticity waves also have an associated "Poynting" vector, but since - = for these waves, the term, which gives the energy flux relative to the flow, vanishes. Therefore the "Poynting" vector for the entropy-vorticity waves consists of the convective term onlyo From (357) we find the time average of; to be~ C -Sun, =UE, v* (5.16) The energy of the entropy-vorticity wave is blown downstream with the flow. B. INCIDENCE FROM THE SUBSONIC SIDE Now that we have studied the behavior of plane waves in a moving fluid we are in a position to describe the interaction problem. First we shall describe the interaction of plane waves incident from the subsonic side of the shock wave (Fig. 2a, p.3). The incident plane wave is a sound wave moving in the subsonic flow toward the shock and is described by Eqs. (3510), Let us choose the following notation.:. -L = E e.. i+., x -t, ^-iY Lqt. a; = -ox o e \e 1.^o X a ^o XXDi i t;- (5,17) j, = ^fr^^h~~; O l 0 C

25 where 6 is the amplitude of the wave and is supposed much less than 1 to make the use of the linear equations reasonable, We shall see that there is, in conjunction with this incident sound wave, a reflected sound wave as well as a reflected entropy-vorticity waveo These are described by: Reflected sound wave: -~LL~t (5318) v,=-, Reik' #(,A x Lt AA= 0 Reflected entropy-vorticity wave: w = o (3519),5*= o. t lofr ^ i I. At x = 0, all the quantities, CU,, AJ, and A are related linearly by the shock conditions (2,15)o Thus we must require that the exponentials

26 match at x = 0. This matching may be accomplished by requiring that the coefficients of y in the exponents all be the same. From this we conclude: \+ tA ~\o ) (3520) and (3,521) Equations (3.20) and (3521) represent the analogues to Snell's law of reflection in optics. 1. Reflection Laws In this section we wish to examine in detail the angle relations for the case of subsonic incidence. a. Reflected Sound Waves — The relation governing the angle of reflection for the sound wave was just found to beo — ~-1 — *3, -4^2 —- (35.20) \ \+,, \ l+ tdo Squaring, we find: 0r-~ _,1+. = Wo t\ +zM m\ y1 ^ iv or This equation may be solved for b,; we find: o( = l do 0 - %jo+Z= The first of these corresponds to (3 _ S, whereas the second corresponds to:

27 The solution o(, = 0(,' = is trivial and will not be considered further. The remaining nontrivial solution, which corresponds to the reflected sound wave, is: (_ _- i+ )do + 2W: 0( (\0 \ I) 0 M+ i (3522) I = " -M2) I (|+ +2) +2 M, ioD It should be noted that the angle of incidence Q equals NT for normal incidence, since the waves move toward the shock from the subsonic side, X > O As we have seen in Part A of this Section, only waves in the angle range'TF- ~ > e, where 4g is the grazing angle, transport energy upstream. We therefore limit our incident angles to this range, It can be seen easily from Eqso (3.22) that as the incident wave normal goes from TT to G, the reflected wave normal goes from 0 to 49 o The fact that Bg t ~ is evidently due to aberration. It is evident from Figo 7a that Qp, the angle of the "Poynting" vector, is related to, the angle of the associated wave normal, by the formula:' M + CO's. (23) Thus for the reflected sound waves treated above and illustrated in Fig. 8 we find: tonO- =4- - Misd C oL (3.24) This equation has as a solution: 4P, *-8 _ (3525)

28 SUPERSONIC SIDE SUBSONIC SIDE SUPERSONIC SIDE SUBSONIC SIDE Uo U U0 U _ Reflected -- Wove Normal Reflected Poynting Vector eflected \ \^ \ \^ ^ ^^\^' Poynting -VX -^ -jV^'T- \ Vector Reflected / \^^ / /~ t^ Wave Normal Incident Wave Normal\ Incident/ \ Shock ---- Poynting ~ \ Incident Poynting Vector a) NEAR NORMAL INCIDENCE b) NEAR GRAZING INCIDENCE Fig. 8. Reflection of plane sound waves. In other words, the ordinary form of Snell's law of reflection holds for the "Poynting" vectors. b. The Reflected Entropy-Vorticity Waves.-The reflection governing the angle of reflection for the entropy-vorticity wave is: 4AO 1 0 io (3.21) Squaring this, we find: or ( t\+~M?) 42to)2 = (\2 + -M oo, which may be solved for od to give: 0(_ = 1. ^M>00 {2 =+. %' ~ \+M W +, o 0 + + If we choose the positive sign in both of these expressions, then',= TT corresponds to ~2=0. This gives:

29 ~(Mi d \ -- + 6 (__a) 3__, (3.26) l+W14) -YAi- o it\+ Mh + e, The maximum angle of the entropy-vorticity wave corresponds to Go = a 9 = COS-I- I. For this angle oz = - H or $4 ~ -- C0 S-l ~ \ oM As $0 decreases from TT to ~*, Q increases from 0 to z o The relations expressed by Eqso (3.22) and (3.26) are illustrated in Fig. 9. 2 Reflected M/ a2 Entropy-Vorticity 2.0 t 1~ E \< 0,-g 09 lo I +M~ia / / ~'?..... y -2.0 1+MMal S of the incident sound waveo -2.0 Fig~ 9o Angle relations for reflection~ 2. "Fresnel" Formulae for Reflection The next step, after having explored the relations between the various

30 In the introduction we mentioned how this calculation was to be performed. We use the linear shock conditions (21l5) to relate the amplitudes of the waves on the subsonic side of the shock to the amplitude of the distortion of the shocko This gives four linear equations for the determination of the amplitudes A,, C of the waves and the amplitude a of the distortiono The pressure, velocity, and entropy on the subsonic side of the shock are given by (3516), (3017), and (3518): =^ ~ ~ ~~~ {'* ee^',. j }M-U L =,' - i Wt ~a de, 2o.e,5 e~', ~ (3.27) a = C e'",^ e-'t We now carry out the transition to the other side using the linear shock conditions (2.15)o Since all the quantities 0' d,, Ut, and C0are zero, these conditions reduce too MONO VA-) (3.28c-^ )) R'\ C —^ )' ^- ^4(-L^ >

31 where p, qU, (L3, and A are to be evaluated at x = 0O We now make the assumption that f(y,t) depend upon its variables in the same way as j, 9, 9 and.AS 0-= el +i^l i (3.29) t (ftl J -k, U. \ e: Ms~wT;; (7330) ^^^=tt _ t Bi ab ikSw _ (3031) Substituting (5330) and (3531) in Eqs. (3 28), dropping the common. exponential factors, and noting that o CC, F =' MO C\ and that D...,- -.. -, we findo ) Ve. V CL 3l 2 U. = I te- b j, ) where bx z - ~ ~-~T- _,. _ llv t;7_ _ +(3)33) b _ LT /dan o r t To determine a, A, B, and C, consider the relations (3527) at x = Oo Let us then remove the common exponential factors [by virtue of the reflection laws (3520) and (3521)] to finds

32 P, = X + e6 = i,b,; 0\ = o(^ tdor - 6 = C_ tib2 (3534) V, = p,(\ ^eo~i +-^ dL =k ab3; &^ =. QC = tkita b04, Since the unknown quantities to be determined in terms of 6 are A, B, C, and a, these equations may be written in the form: ('L, - A = ei ih\b2)oi - c, R + krB = 0o@, (3)35) (i~,b/ a-.A 3 - = o The determinant of the system (3535) is: /i =^ i, u' b - (n, An d b-), (3.36) where the vector 0 = k.bzb ) is parallel to the velocity. The solution to the system is: (3537) B= b'tttv rpo^ -~ (d,-0i^b3 + (dir- ^b )b.^i; C = i4 (t ^ ) - (R. )]Th4 / The last three expressions (3537) are the analogues of the Fresnel coefficients in optics; the first has no optical analogueo It should be noted that a, which is the amplitude of the shock-wave distortion is

33 imaginary, and thus 90~ out of phase with all the waves. In Appendix. D these formulae are written entirely in terms of the incident wave-direction cosines o( and Q. The resulting expressions will be used in Section V to calculate the amplitudes of the reflected cylindrical waves. C. INCIDENCE FROM THE SUPERSONIC SIDE In this section we shall describe the interaction of plane waves incident from the supersonic side of the shock wave (Fig. 2b, p. 5). Although many of these considerations are identical with those of the preceding section, the fact that complex angles occur here make it necessary to study this case in detail. The incident plane wave is a sound wave moving in the supersonic flow. In this case we must consider angles of incidence ranging from 0 to1T since the energy, even of waves whose normals do not point toward the shock, is nevertheless swept downstream, the flow being supersonic. The incident wave is described by:;~k +^ ~ -;Ltf ~i =E0 a; =d e \v Moo Ad + -^ (~S0Y _ ot- (3558) V ere = th l where e is the amplitude of the incident wave. Together with this incident sound wave we consider a refracted sound wave as well as a refracted entropy-vorticity wave. These waves are described by:

34 Refracted Sound Wave:. gd\W,, -c13^ _ - t i 4iv>_8 _ (3539) Q,= O, Refracted Entropy-Vorticity Wave:;_ Wt (_(3.40o) ^'P= ab^^^ Ja.*= c^ e i^' Since at x = 0 the quantities d, U, r, and A are related linearly by means of the shock conditions (2.15), it is again necessary that the exponentials match at x =0 with the result that: s, = \,W 130 o 4\M, d, \ + -0o d0 (3.41) and _ Nd \c+MoAoo (35.42) Equations (35.41) and (35.42) are the analogues of Snell's laws of refraction in optics. 1. Refraction Laws In this subsection the angle relations for the case of supersonic incidence will be studied in detail;,

35 a. Refracted Sound Waves —The relation governing the angle of refraction for the sound wave is: k,'_ _ i_,_ \+tA A\ X v o D (5.41) If we let Z' = i and square both sides of (3.41), we find: x'W' (\ 2 OMt + maZ\d,) = (\-i )( H\+% o ) or ( 9 +t++ o)2l)olIt+2 M, d%2( % ^-(\ Mo^)= 0 We may solve this equation to find: M y AF + M% Moo)2 (3543) R M\Xpt j 2r~[W- t- )Ndt For the physically realized wave a definite choice of sign in Eqs. (3043) has to be made. For this purpose let us consider first the angular range described by (I + t"o oav 0 (( — e)~ (2 ~ In this case the functions d\ and Hare both realo Waves which are normally incident upon the shock wave must correspond to waves transmitted at an angle Sty_ 0, so both - ) = 0 and 0& = -T must correspond to O -- ) o Thus we are led to choose the positive sign in both Eqs. (5.43) for (+t o0C) 2 il_ ~ o, and to choose the negative sign in both equations for In the range (4. - o0( < - the functions a and In the range, the functions it, and Ad are complex and thus do not correspond to the cosine and sine of the angle refraction. We find instead that:* *The subscripts R and I will be used henceforth to denote the real and imaginary parts of a complex number.

36 1+sm' i tioX+3,)= - kt>d +'L kk (+ M l^'d\t A + = - u, + i',t, (Xfo<) + where and \ \ + M\, dO \V \+ Mo0o The factor V is to be interpreted as a damping factor, while i is the wave number of the damped wave. This wave moves in the direction j = (Q, ) (3,). In particular we findo ^ ^ +;2 2\ 4 ^ - \ + M'4 d " (^tro^)(\-^) ) (3.44) where the positive or negative sign arises from the ambiguity in (3.43). To achieve damping, we must require ),> O 0 Thus we choose the positive sign in (3.43) for the range \- > \+- mo(&.) > O, and we choose the negative sign for the range - i\_ t Mc +(6)0 o Combining this condition with the condition determined for the case when the waves are undamped, we arrive at the following choice of signs: choose the positive signs in (3.43) for + MIAo() > o0 choose the negative signs in (3.43) for ( \ + AMoLo) 0. Let us return to the case of damped waves and note that we have~ Since _ _ _ -- _, w _ hae de 1 M$,\ 1- ) (3.45) Since \ + wo ha o Since a ^ W >t = \, we have:

37 = - +.'_.' t-: o_ (-_M, — _ (3.46) - \- IM H) \+Mo Co) where the positive sign is to be chosen for (l+C M0(o) >, and the negative sign for ( + Moa) O Hence for t\+m 0>)) 0 O ( _( -^ M Ri^o0). _t _ E__o_ _-_MI _ (3.47) and for O\+MNeo^)j: \= \ (t#Moo i) _; tt= ltit\- @M ) (3.48) The relations (3.47) and (3.48) are then the appropriate angle relations in the region where the wave is dampedo In the subsonic case a simple relation connected the incident and reflected "Poynting" vectors, but in the supersonic case no simple relation of this type (ordinary Snell's law for refraction) exists, One may easily show, though, that for the undamped waves region, (\- 0io 0>(I'\- X A the refracted "Ponyting" vector is related to the incident wave angles by means of the formulae: o( (\t'o ao).- e =. ^;l. (3.49) A\m t At At + ku + Mo^ where the positive sign is chosen for \+Mo(o) > O and the negative sign is chosen for (\tAo(d < O At the critical angles, ( \+Modo- = l-M?-)? o ) these formulae predict that the "Poynting" vector will be directed along the shock.

38 In the damped region (\+ M.o)Oo \-,) the formula (3.15) relating t to 0 is no longer valid but must be replaced by: t, =.e... (Ci..-R -u 2, CR) (3.50) which is the obvious generalization of (3515) for complex angles. Substituting the formulae (3543) for o(, and t one finds: l^ ^ er VIA C o -p ^ Mo0 + ) or G _ t r L^^I^ J ^ L Lor g \ o J X n (3.51) where. ( lo,) o This formula shows that the "Poynting" vector is directed along the shock in the entire damped region, directed upward for (I+M0oo0)>0 and downward for (lt- oto, < 0, as is to be expected, It should also be noted that this vector field is damped in the direction normal to the shock. The relations discussed here are illustrated in Fig. 10o b. Refracted Entropy-Vorticity Waveso-The relation governing the angle of refraction for the entropy-vorticity wave is: Js> =, Qv H- (3.42) M'$t 1+ Moo Setting 3 = -2 C and squaring both sides, we find: ik Co (\- \ <Ao o2 = M: X or [ tl+ MoCo)0^ + 0 h\ 2 n = (\+ M<^f Thus M ) (5 0(2 - (3-52) andA 2= t- \,',. A t 2 at+ 1\ +voVa)Z Again we choose the positive sign for (\tMOO) > 0, and choose the negative sign for (\+,N\CoO) o. This choice of signs corresponds to

39 Uo,SUPERSONIC U, SUBSONIC Uo, SUPERSONIC U, SUBSONIC k reft. n.- s rt nc. Sefr. SrSt~~~efryfr. sinc. nrefr. a) NEAR NORMAL INCIDENCE b) NEAR FIRST CRITICAL ANGLE U0, SUPERSONIC U,SUBSONIC_ _ refr. nnc. /X~// <~ Amplitude Damping Factor s1nc. _.~'Vn rfr. c) CRITICAL REGION U, SUPERSONIC U,,SUBSONIC UO,SUPERSONIC U,,SUBSONIC n. "refr.; refr. s S inc~~.. Srefr. nrefr. d) PAST SECOND CRITICAL ANGLE e) NEAR ANTI- NORMAL INCIDENCE Fig. 10. Refraction of plane sound waves.

40 a choice of o^ = 0 for %O= 0 or. = 1 The relations expressed in Eqs. (53.43) and (3.52) are illustrated in Fig. 11. 2k | )9 2 | Jncident Sound Wave M 1 — 2 - koo Refracted Entropy- I, +MOao Vorticity Wave Incident / Refracted Sound Wave Sound / k/ 1 Wove I/ I+MI a Fg 1/. Angle li First Critical Angle % I It 9 Mximum / I -n O r "2 / / V Wave 2.~ trenel Formua fr R c eotioSecond Knowing the^^ "nell~s"law of refraction we mCritical n \'"/ Angle v^I~~~ ^^ ^ / ~~~...Incident \ ^J' I Sound. ^^ r ~ I_! Wave ko/ o Incide nt Refracted I + Mo ao Sound / Entropy-Vorticity Wave / Wove Fig. 11. Angle relations for refraction. 2. "Fresnel" Formulae for Refraction Knowing the "Snell's" law of refraction we may now calculate the amplitudes A, B, and C of the refracted sound, vorticity, and entropy waves, in terms of 6, the amplitude of the incident wave. On the supersonic side of the shock wave the pressure, velocity, and entropy are given by: ^o = e e i d Uo - dvo 6 ~tl- i tf; (3-53),0., = O )

41 where 6 is the amplitude of the incident wave. On the subsonic side of the shock wave the pressure, velocity, and entropy are given by: = idA Re (X ~t) Lk --- _____ (3.54) U {AAC ^ X do e tit^ VQ At x = 0 all the exponential factors are equal; thus we may drop this common factor from the Eqs. (35.53) and (35.54) and find: Uo =c~o E - Uo = oe ) So = < ) and uO = a v > -i a V (3.56) 5s - C. At x = 0 the quantities ji,, %k, and.4on the subsonic side of the shock are related to the quantities-,,, 1, and A on the supersonic side of the shock by means of the linearized shock conditions (2.15). Let us suppose again that r (t) depends upon its variables in the same fashion as the remaining quantities in these equations:

42 4. W.\ lulMooo; (3.57) f~^,~:) = -'I^, a? b~ ~,~; oo io (3558) ~^ = ^5^ a- Mvwfo = 059) Using (3.58) and (3.59) for sand f and (3555) for P0, U0, o,, and S,, the linearized shock conditions (2.15) reduce to: UQ = + L C z oL) =.6& + b.oUk)o ) o O- E +.00C) O 5, = ag6 + b i oOL), where P t^o #- 14V-\; b, -= zw); kMIN M. =-ftA^F(K L ))o> = sF~; (O These shock conditions may be written as four linear equations for the four unknowns a, A, B, and C, by utilizing Eqso. (3556):'LA F F (MO (5.62) -i obt( + dz A i)L = A1 ~b Gi L= S

43 The determinant of this system is: \ =- k, ^btu n- (n, ), (3.63) where ~ = (bb3 The solution to the system is: it = ratio - yia et sh ) t = h ( -^oizbp(ao -iv)+>larb,-b la b,-hcs^a (3 64) C= fl>,^la 4b-^bA4Y- ^ A2 b4(AYY with ( =O(A,3 These equations are the analogues of the "Fresnel" equations for the supersonic case. In Section V we shall use these equations in determining the interaction of a cylindrical wave with a shocko

IV. CYLINDRICAL SOUND WAVES IN A MOVING GAS A. THE CONVECTIVE WAVE EQUATION This entire section will be devoted to a study of the properties of sound waves in a moving gas. In treating this problem quantitatively, it is convenient to replace the system of first-order equations used in Section III by a single second-order equatiqn. Let us define a sound wave as an irrotational, isentropic disturbance in a compressible inviscid gas.* From Section II, Eqs. (2.4), we find that the differential equations governing such disturbances are: C'd~P ( 9 U M )~ azL -t- E 23 = Ol I~ (t Cat X ^; _3 - X O. It should be noticed that these equations are not identical with the system (2.4), as the irrotationality condition in (4.1) replaces the entropy equation in (2.4). The last three equations in (4.1) imply the existence of a scalar function ([y such that: *We are here using the properties described in Section III, Part A, for plane sound waves as defining properties for sound waves in general. 44

45 = se ) (4.2) + U This function (Xijt) is called the velocity potential. The first equation in (4o.1) becomes by virtue of (4.2): ^a a9y - C2 3t+a- t ax ( ~ If we subject this equation to the transformation X= -X-Vt I ='j ) tl = t, we find: X' aX CX 7t V) dx'o. = (4 4) In particular, if we move with the flow, by choosing V= 0 the equation reduces to the ordinary wave equation. The single second-order equation, (403), determines the behavior of the sound field when no sources are present in the flow. If there are sources, one must modify this equation as in electrostatics by adding the source term to the right sideo In particular for an oscillating line source located at QX 0), we have: a 2 a M c(.1t aUx).m =4r4S -E%)-3 j ) 1)e (4.5) This "convective" wave equation will serve as the fundamental equation for cylindrical waveso Bo LINE SOURCE IN A SUBSONIC GAS Let us now find the solution to Eq. (4,5) when the speed of the flow is subsonic. We place the source at (0,0) for convenience, and find:

46' - ( UI ) = 41 i (X e, (4.6) If we set |lM~t_ ec e-~ W ty (4-7) then Eq. (4.6) becomes: t -m ) +CAL t+p T m = 4n 6(x6$6) IC ( (4.8) Let us now simplify Eq. (4.8) by the following transformations: ^ = — =:> ^,)~=..... —6 =b —A-'-(~9) Equation (4.8) becomes the inhomogeneous Helmholtz equation:*,22P +, +'t = 4W6 & 0<1) S' 3') (4.10) Equation (4.10) arises frequently in optics and wave mechanics, where? is interpreted as a Green's function. We are therefore led to use the common technique employed in these fields to solve the equation.1 Let us assume that the solution to (4.10) may be written as a Fourier Integral: *toO kK& (4..1) We may write lTr:' TwS C S.Ly' as: O4fTt% 6C = y \ eem &^^6k. (4.12) Equation (4.10) for (P is thus equivalent to the following equation for^: *We have used the property of the S -function: L ~ ( tcLx) =c %Ls

47 i L — _ it- - Lei y''i ),D -~, (4.13) - 09 which may be satisfied by choosing ir^^ 3 (isT^^ ^ Ir~.z- il-T \b )(4.14) Choosing this value for j(Q\ ARE, Eq. (4,11) becomes: (-1 jg S (4.15) This integral is formally the solution to our problem. It is possible to carry out the integrations in (4.13) in the following manner. Define XL (>c) by: % <$ 4 - - K4( ) (4,16) with Then e. rt = 6^ rd \ s. (4.17) The integrand of <(~) has poles at.y - + o When K is imaginary, A ~< \,, the integral is well defined, but when K is real, V kx, the question arises as to how the Fourier Integral is to be interpreted in the vicinity of the poles. One may extend the path of integration into the complex plane near the poles, and by choosing various paths arrive at dif22 ferent solutions to Eq. (4.10). Sommerfeld shows that, in the one-dimensional case, the choice of path shown in Fig. 12 is equivalent to his radiation condition, o - O as X-w o We shall use this path in our subsonic case since the resulting integral in the limit m (_ must satisfy the Sommerfeld condition.

48.-~ t > kx - Plane / PF / ()K(for k2 > ky2) \ / \ r~1~~~~ ~ ~Path of Integration K (fork12>ky2) I \K ^ / )-K >/ \ / \ __ s / Fig 12. Path of integration for;(x'); subsonic case. The integral (4.16) may then be converted into a contour integral by adding the semi-circular segment P, to the path of integration for' > o 9 and P to the path when X IC 0 The contour integrals over paths p\ and Ep - O as their radii -- oO, and thus we find: r 0AT, RQ5.' t for X O O7WLx' <a) 9L t-, for. f xor or C ) = or T fr all x'. (4.18) The integral (4.17) then reduces to: Hankel function by the transformation: in = W1 S\a\er Q IZ = AVCOSTe &@; (4.20) with the result with the result:

49 9') = 6'.4 I5. o + i 5iLc0.l (4.21) where p is the path in the G -plane shown in Fig. 13. 8 Plane Allowed Regions Allowed Regions Fig. 135 Path of integration for /5(x,y'), subsonic case. Now let \I\ = p CDS and i = p n+v\, with I- 7,H+, and - / 4 / <?T Then integral (4.21) becomes: " S 6S' e'L coSe- a. (4.22) We may translate the imaginary axis to the right by an amount a\: [\_ _-_) and find: X 1e ^ ^teo5+ by (4.23) The integral (4.23) is Sommerfeld's integral for the Hankel function, and the 23 path ~p is the well-known path of integration. Thus we find for (: @~(pl,^ -= irH"'( ) (4.24)

50 and the solution to our problem is given by: (Xxt^ ) =-*F H t i' X isL+(\_m ) t ^^\_^-^ (4.25) Notice that as V,- O this solution gives: Gtr ( _______ (4.26) 24 which is the well-known Green's function for an oscillating line source. The solution (4'25) could have been found directly from the differential equation (406) by first applying a Galileog transformation to a frame of reference moving with the flow, and then a Lorentz transformation to put the source at rest. The solution to the resulting wave equation could be written down inmediately, and is in fact identical with (4.25). We use the above method, though, for comparison with the supersonic case, which we cannot solve by the transformation method. Let us examine the "far field" due to this source. We know the asymptotic behavior of H, ),and thus of f: Tk w l- gl -. C (4.27) for The lines of constant phase are given by (\ -w) -W M )- It -T = i, +(4.28) Setting tt - + 4. ) this becomes \3j )2N XY\-N\+ = \- t \_ )C.t ) (4.29) which is equivalent to: Qex- t l= o c (4.50) The lines of constant phase are therefore circles which are blown down

51 stream with the flow. [See Figo (3a), po 6.] Thus the solution described by this method is entirely in accord with the results one would expect on physical grounds, Although formula (4025) represents a "closed form" expression for the cylindrical wave, it is not of a form appropriate, for reasons stated in the introduction, to the solution of the boundary value problem which interests us A "Weyl-type" expression of (4.25) will therefore be developedo The expression (4o21) represents the function (, and hence 5, as a superposition of plane waves of constant amplitude and wave number. It is for X( 0 (the region of interest for us, since our subsonic source is to be located at o > O ), ~(~eP. ) A) (4.21) which becomes in terms of X3~t: ~- eLut _... 0C5te. l (43 1) However, these plane waves are different from those used in Section III. But the connection is established by means of the aberration relations: -_ l ~oS = ~C05B-.. gin e _ ~tan L o /___(4-32) \-MIA t,CO o > i -r \ + M, A C+ 5 4o These equations may be solved to give: co^= -: C>s. -e - _h-^ SOSea I. \ + N\ ~cOSo ) I \ wt - BOS&o ) (4.33) CO ^=- COSTaMh \2Mg _!-% 5w Wg \+Ccose ) O, 40 The at first surpising occurrence of the relativistic aberration formulae is explained by the fact that the wave equation (4,4) is Lorentz invariant.

52 Differentiating the equation for sin 3 in (4353), we find: &' = T- c+ Oao' (4~34) Expressions (4531) and (4,34) may now be substituted into Eqo (4031) to give the desired superposition: (,K (4,3S) where do= Co0S. o ),-o -,Go and the path P" is the image of P under the transformations (4,33)o The mapping of the path P -e? is illustrated in detail in Fig, 140 The appearance of two essential singularities on the lines yeo _ylyT may be notedo We may find the pressure and velocity of the flow by differentiating (4,35) as in (4,2), Comparing the results of this differentiation with Eq, (3510), we see that we may interpret the integral as a superposition of plane sound waves, each of amplitude fc*^ - it____. (4036) (\+ Mo4^ The integral (4~35) will serve as the fundamental expression in the diffraction problem of Section Vo C, LINE SOURCE IN A SUPERSONIC GAS Let us now consider the more complicated case of an oscillating line source in a supersonic flowo The fundamental equation (4~5) is for this case: 7'- h ^ ^i+ 0o;x) ~ 6S C6Y,) S e; (4~37) wherein we have placed the source at (0,0)o If we make the transformation analogous to (4o7): i^(.^ -^ = Mo"l (4.58)

ossao osuosqns foe oq e jo SuTcddeTW'-[ ~'T d d0 9uDld- & euo8d-0 t~~8 ~ -— ~ 0 i

54 then Eq. (4.37) becomes: r 8\2z _ E\ - - _ f e( 0-' l - y - w Sls^ * (4.39) We now simplify Eq. (4.39) by means of XL ~2' _X =,;, ) 1= X, be1 = f * ) A' = - ) (4.40) and find the analogue of the Helmholtz equation: tw2 X-4 t' 6? i 5 - -6 (X ). (4.41) Let us seek solutions of (4.41) which may be expanded into a Fourier Integral as in the subsonic case: Btr',y'l- Lt>XW X L4 & kW &C ) (4.42) With the aid of (4.12) we may write Eq, (4,42) as: S| l(k2-; -^' -j ]Q k^^']:)3 -& & O a (4435) This equation may be satisfied by choosing: ~t(R 5 7 (I TuJ2.~ b -'r -k) e (4.44) whereupon the integral (4.41) becomes: %'Mj) = T i| X-h -he &istir. (4.45) This integral again is formally the solution to the problem. As in the subsonic case, it is possible to reduce (4.45) to a cylinder function. Let us define: ( > -' __ kx - (4.46) where K = \\ \2" h"; then:

55 -oo (4.47) When choosing a path of integration, we cannot apply the reasoning of Part B, since A>l. We are thus forced to impose some other physical condition on the problem to replace the Sommerfeld radiation condition. We shall require instead that there be no radiation upstream of the source. This condition may be satisfied by requiring that ) ( ) = 0 for X' 0 Referring to Fig, 15, we see that if this condition is to be satisfied, we must choose the path of integration as shown. kx - Plane // / \ I -K K Path of Integration / \ / \ f \ \/ Fig ~ 1 Pt fn ri f X sps ce/ / Fig. 15. Path of integration forj(x'); supersonic case. The integral (4I46) may then be converted into a contour integral precisely as in the subsonic case. The path for X1 < 0 encloses no poles, and hence our radiation condition is satisfied, as:

56 (^l^ 2 (ge9 Res-0), X'>0 or KX -Ot (, =0 for 1 < 0 8 We may reduce this integral to Sommerfeld integrals for the two Hankel functionso To do this, let: ~'~1 =;'s -S) _____ in the first term in (4o49), and in the second let: tU'Skm =-i'~OSW w T=_' 4 * (05l) The integral for ( tX'), 7), (449) becomes: q* tsqj L 4) (49 bcoms9 ~~x',Y`) ~ ~ P (,' where ~ and uc are th e paths shown in Fig t 16 This integral will be seen to be represented by quite different mathematical expressions according as wow ftm Q.0 P where ~~, and Pzare the~ Patssonin'g 6 Ti nera il

57 ^Path e e-Plane Equivalent To R-e i Allowed P2 Regions __!^.a~ Path Equivalent To P2 - e -w- 0 mO 27 e P Fig. 16. Path of integration for 0(x',y'); supersonic case, inside the Mach wedge. Kx > \%t inside the Mach wedge, ^%' < ^\M' outside the Mach wedge. Inside the Mach wedge we set A' = b4 tLO 4 ) i M - i;, with i = i Sew then we find OR C O and X = chlr'. This substitution gives ~,is tW v/^ v and thus we must have < X Hence we find for al > \\\: Aip^, l) t (cto - (4.53)?,t PL If we let & = G-t', the entire plane is translated along the imaginary axis, and we find: I1y-b )k L eL?.'P' C. 0 e,Q (5.54) P. + PI The paths P. and Pmnay be distorted into the allowed region as is shown in

58 Fig. 16 to give the standard Sommerfeld integrals: Inside the Mach wedge, X> > 1'1, W c'')= v i\, S - t- ) J - t 67t 30 ti Up),(4.55) On the other hand, outside the Mach wedge,' > >' > 0 we let - Kx 1d p cOS ) Y =\ p 5 A.' I where ~%- - _\. Setting XR we find X'- C 5 I, \ = O\ Chtt o Thus. "^ %t~,, and: "?G+i, =_) scO5v) V (4.56) These paths are illustrated in Figo 17. P3 I 0- Plane Path P Equivalent To P2 -____r_?_0 2ir 4 P2 Fig 17 Path of int egion for (x supersonic case, Fig. 17o Path of integration for (x' supersonic case, outside the Mach wedgeo We may replace the path Pz by the equivalent path shown in Fig. 17, and close the paths with the segments FP and P 4 The integrals over P3 and _P_ _ 0 as the segments — t I o, and since the integrand has no poles in the allowed region, we conclude that:

59 For t > O, Similarly, for <' <-) O < 0, cpl' )= O, Thus we find the solution to our problem: 2r11^Te' - t -- (4.57) O ) otherwise The somewhat surprising occurrence of the Bessel function was noted in the introduction. The calculation of the surfaces of constant plane follows. When the argument of the Bessel function is large, we may again carry out an asymptotic expansion, the first term being: which gives the asymptotic formula for ~ ~ts ~;t) _,ZE -., C OS __-___-___0 t-H.9 _ nt Here we have a wave which is composed to two components. There are two lines of constant phase to examine. Let them be denoted by: _____- _ 4_ iN =le - -- t; - (^-\^ _ ^ ) M - _t: (4.:9) Now let:

6o t, =- t + i> ), -tand then As2_ —w -n^M = [(mt-Q^oX 3, - (4.60) and XM = tt -n\C(ot,-A x 1 (4.61) Squaring either we find~ x - ot t) + = C t, (4.62) Let us relate the phases 9, and 9lby Q2z =, + j then t, = t- and both waves correspond to the same circle. Thus each circle (4~62) is composed of two parts, The part corresponding to phase. leads the remaining part,,, by 90~ The expression' ~' - (t^. -y /0, the equality sign applying on the Mach wedgeo Thus for wave 1 ( ot and for wave 2\ V4?' — C. ot, A M.,a From this we.may conclude that the part of the circle convex to the source is wave 1 and the part concave to the source is wave 2. (see Figo, 3b of page 6. We may now inquire about the zeros of the cosineo These occur when -,== -(L2. (2n-., - or o \ X4 _63 = 4 X, >.3) —- (4.63) The intersection of Jj = Q T and fJ =. -3 > for example, corresponds to the zero VY\.\ of the cosineo This zero then moves with the flow as t increases, giving rise to an hyperbola (asymptotic to the Mach wedge) along which = O We have here an example of an interference phenomenon

6l occurring for traveling waves. Let us proceed now to the "Weyl-type" expansion of 3 in terms of the plane waves of Section III, Part Ao Consider Eqo (4~52): q (*tM) =&& 6, 1( titcoatLt ( 452) P\+ t from which it follows that: _-:'~o~ ~- t i v b ~'3 Xi. A(4o63) - (wt MO Y Qx.O%;t) =' e - A e _' C M -1r A)(4.6) Let cos - co _ ct6-eo MOe= 0 Li o \ + ot ^Qeo C\+) oC0s o C ) These are the abberation formulae corresponding to a "supersonic" Lorentz transformation, Differentiating the equation for L'fnlO in (4.64), we find: hi@-_ = - to-o (4I65) \ -t+ to (O O We may now write our potential (as: 7a (bat 4. s S T — t, cit(4.66) Ps o MO- PS where p, and'P are the images of P, and Plunder the transformation (4163), as shown in Figo 180 By reference to (3o10), we see that we may interpret (4.66) as a superposition of plane waves of amplitude a e"su = (e Lrn o The integral (4t66) will serve as the fundamental expression for the supersonic interaction problem of Section V.

-se3o os-uosasdns f ~e oq e Jo BU-ddIedW'81'*T -c d~_ I'd lZd 08_ auold - 0 ad d —'C, d - 9' aUold -9 39

V. THE INTERACTION OF CYLINDRICAL SOUND WAVES WITH A STATIONARY SHOCK WAVE A. SUBSONIC INCIDENCE 1. The Interaction Integrals Let us consider a cylindrical sound wave generated in the uniform subsonic flow to the right of our stationary shock (at Xo, O )o In Part A of Section IV we found that such a sound wave can be decomposed into a weighted superposition of plane waves of the type considered in Section III. Since we have solved the interaction problem explicitly for each plane wave occurring in the superposition, we may thus obtain the solution of the interaction problem for incident cylindrical waveso We shall write down the resulting integrals for the reflected waves below, and, in the remainder of Part A, evaluate these integrals approximately to find expressions for the far field of the reflected waves. From Eq. (4.35) of Section IV we know that the incident wave potential from a source located at ) > 0 may be written as: Lk, _I z~2Lj3\ _ LU.t 4? t6 &+ MOCo - (5.1) for Xo > X, where P1' is the path shown in Fig. 14. Knowing', we may calculate;, %(u, q/4 and 0 by (4.2). In particular: =,~ ~ i A (1o2X i wt (5.2) where 65

64 The incident pressure wave is thus described as a superposition of planepressure waves, and likewise we may show that the incident velocities are described as superpositions of plane-velocity waves, the pressure and velocities all having the same amplitude (or weighting)factor, 6 Q(e0) An incident wave of the form: ( Dt -X'.t- b i,t.; = E0 o) e ^* g gives rise to a reflected wave field: L,.o7 ) e. - M 1 and the corresponding distortion (5.5) [i Cje^= a ^> 1 e-^if -e0 and the corresponding distortion S-CY^)= ^'___ -jlo t (5.5) The functions OC (C),' C o,' }(, and C*'(6) are the amplitudes a., A,, and C of Section III, Part B (and Appendix D) if is replaced by

65 Thus: W -^-o) _A = - -— ^ ) (5.6) "(<} = _ -" d, C- o. j As in Section III, Part A, we introduce the scalar potential, to describe the sound field, and the vector potential C, to describe the vorticity wave. The sound and vorticity potentials are.given by (30.12) and (359) of Section III, Part A: - w I\+ t4 tart 1+ I, (5-7) fi= 0(4 ),(a.xN) where _^~- L;-tt M=v^io5X (5.8) ~'B' - -\ _ oL2 The entire reflected field may be described in terms of these two potentials and the entropy function. Thus the solution to the reflection problem is given by the four integrals: {-',-,, W"~x~.,. _^. -, Ja -i ->f ^\ — >^ \+ MAodo ml\e0 ( C-.( t i M%.l OL (~+ tcL d k -dX+&Hr~ s~

66 where P' is the path in the &O plane described in Section IV, Part A (Fig. 14), and A,, C, and Q are the "Fresnel" coefficients for reflection calculated in Section III (and Appendix D). -These integrals give the exact behavior of the reflected sound wave. In the remainder of this section we shall study the approximate behavior of these integrals in detail. 2. The Shape Function: -, jt) The shape of the shock wave was just shown to be: &(3,tz = 11^ ^ 5|K^ - t\ X (5.10) where OX is the amplitude of the shock distortion for plane waves, given in Section III, Part B. To simplify this integral, and thus to facilitate the approximations, we transform back to the G-plane of Section IV, by means of the formulae (4.33). From (4,34) we find: __i = __ _A - (5.11) Thus in the G-plane the integral for the shape is found to be: n^ - t SiCLtv \^^ ) e A A (5.12) where p is the standard path for HQ, shown in Fig. 14 of Section IV, and _ > M\ = X > A = lipas in Section III, Part B. In Appendix D we show that the expression for (i.,a.) is:

67 0 \)+L { ^o C L + 1}' (D.11) The formula for C(,t) in the G-plane reduces to: t1=,6 eE' - e F ) (5-13) fcl.-fc^ =~ h. ^ M.^ eA ^-) c(1 where F t3'5 = \ <^a + >o____ k lood.+1 (5.14) 5 o(%(+j -NAwoC) - p + 3 and P is the path illustrated in Fig. 19. Region I I.. Where Poles Region Fig. 19, Path of integration for F(y'); subsonic case. We now obtain the first term in the asymptotic series for F(3'), in terms of yA by means of the saddle-point method The saddle May Lieof the integrand is located at: LrK ^....cj=....- ( Saddle Region Where. — Saddle Point May Lie I Fig. 19o Path of integration for F(y' );. subsonic case. We now obtain the first term in the asymptotic series for R11), in terms of k 4X[Z-+~'~', by means of the saddle-point method. The saddle

68 Thus:,gp- CoS, i, S, (S (5o16) with e- = and -2 < a o Near the saddle point the exponent becomes ~t\R',1 ^.ltokYo ) = v~kR' (- t ~^ \ ----;~ ~ ) ) (5o17) which indicates that this is an ordinary saddle pointo Let us now examine the poles of the integrando These poles occur at: t2 o-Z tM,^o(+ \ = O, or, = - M ~ i- 1 (5.18) The values of ~,2 are always real, since MMt>J and negative, since -2 -Z M<oI.c<0 o Thus the poles lie outside the strip -i S $ < I, and are isolated from the saddle point * We may therefore apply the ordinary saddle-point formula, \(*)e;9'c"g.'xoWt 9^ 4.) j (5.19) to find Ft- 8 4 *) so ( ~c M,> (ae 4 - LE (5.20) l A\ WAXe2%,o + p )where *If this were not so, we would have the possibility of a pole being near the saddle point and thus the additional complications, discussed by Ott and Van der Waerden,^ referred to in the introductiono

69 The asymptotic formula for kt,) is found by using the asymptotic expression for F(), (5.20), in the formula for i(,3t), (5o13)o Let us now discuss briefly a few conclusions which may be drawn from the asymptotic formula (5.20). We see that the points of constant phase is given by: (+ MX) -t\- M. C.t =, (5.21) Differentiating this expression we find that the velocity of these points is given by: (Qt 7Z- " (5.22) On the axis A =0, we have p s, X, and the distortion formula reduces to: (Ot)= - IAAMI^4t (5025) \k (e U-5t\ _, ll Xo'+tU^, % w) Near the axis, p, x C + M- (I_>)M' ), the formula for t (Y t) becomes: %,t^ -= ( \ + t <) i0o~,t) (5.24) We may also obtain an expression for I3,t) when ~ >> Xo )jt4= )ZA ^)(AZTrE i %-1T)z-, 6- (5.25) L )-V4^ % nl/2 This latter formula shows in particular that \ (l))\ -, as o The shape function is illustrated in Figo 21. 35. The Entropy Function: A ( tt) The entropy function is given in formula (5~9) as:

70 C O.* XO + Oa 3i - T7 (N e 0 (5.26) where od and z, are given by formula (3.26). We again transform from the Go-plane to the Q-plane of Section IV by means of the formulae (4o33), and find that o(, and 2z become:.{ = I\-^ Ah, )a $2 n (5027) r' l+M,d ) \, c Thus in the G-plane the integral for the entropy is: -A M (5.28) C (d is shown in Appendix D to be: C(r = -4 - tL-Ml(y q (Do.l) and p is the path illustrated in Fig. 19. Thus we write the entropy integral as:,. tA- M;'/"~ -''.l: ^, d ^ 4 iv l0^-t ti 1, -L t\ P t4 (5.29) where F () is precisely the same integral as occurred in (5o13) for the shape function. The asymptotic formulae for the entropy function is found by using the asymptotic expression for Fl), (5.20), in Eq. (5.29). Two facts are immediately apparent. First we see that the entropy function at (Xl i t) is proportional to the shape function at the same value of ~ but at time t -.. Thus the entropy field is given as the image of the shape blown downstream with the flow. Second, we see

71 that the surfaces of constant phase are, choosing the constant to be-' ^r Zt MXo) + ^V( -. = 0, (5.30) which may be written as: \ T^? ^ 4 \, nn o-\ t- 2,G * (5031) Thus the surfaces of constant entropy are hyperbolas, blown downstream with the flow. These surfaces are illustrated in Fig, 21. 4. The Vorticity Potential;: \(C,^,t) Let us now turn to the vorticity potential which is shown in (5~9) to be: tv~~h M 5+ (5.32) We again transform from the go-plane to the G-plane to simplify the integration. Using (5.27), we find: and(W is t he path illustrated i-n \ Fig 1 9 \Thou (5e33) where (~WO~,~.' r a~, +z'~.w,~. [-1'~ E 2a (D.1l) ji \ _ t \++ a ] o+ and P is the path illustrated in Figo 19. Thus we write:,,y.,t, = Z G 6() * (5.34) where rW)~ - l [ (" t(^ 1 d' \+Flro ^ ^\ WV*J/ j\+m~i) L-nD d+o<+2^^A^ 3 e e 4 (5~35

72 The saddle point of G lfu) is the same as that for F l'); however, G(i') has poles at 03 -_ — _ in addition to those of F(')o These additional poles are located on the lines (>x- ~2.A4 )r and thus are again isolated from the saddle point. We may therefore immediately write down the first term in the asymptotic series for Gr(')from the general saddle-point formula (5.19).t (1 )h-t'Th(M?+rC^ jK+ o M~l ^^; (55036) J? e Cko X( P + ZM^^ l a ^r i M2 % } where e = AXX + (IA)M We find the asymptotic formula for y X, jt) by substituting (5.36) into (5.34). Formula (5.36) shows us that the vorticity vanishes on the axis = 0, and that the lines of constant phase for the vorticity wave are the same as those for the entropy wave (see Fig. 21). 5. The Sound Potential:', i),,t) This is given in (5.9) as: $ (",tJ le-^-PI ^1^ ii, aIX L iWt A (5-37) 1p" Let us again transform this integral to the G-plane by means of Eqs. (4.33). Since C, and ~, are given in terms of o< and o by (3o22), we find in the G-plane: d, = =i - i (5-38) -M,Mt C k -~ M,& 1 Thus in the Q-plane the integral for the sound potential becomes: ~> t~i-rA,

75 where: __ __ _ _ m I_ ___,C M (D.11) 2ZvOk 1.4-MNI'and P is the path illustrated in Fig. 20. Ie ^-^^ ~I — d' - Plane Fig. 20. Path of integration for ($(x', y'); subsonic case. The integral for the sound potential may thus be written: l~,~],-t =xs where ~~, ~ is given by: ~'~' = I (- h'l ). p The saddle point of the integrand is at: i sp' COS + = X+XQ ) pg - e t ^^ = i (5.42) with - =..+K, +~' and, i ~ The poles of the integrand are located at the points shown in Fig. C-0 5 1+ =~~~~~~~~~~~\

74 19, and in addition at o( _ ~ j M. These latter poles occur on the imaginary axis, but even in this case are sufficiently distant from the saddle point. Applying formula (5ol9) to the integral (5.41) we find: Q)l~~y~ 2r(l-MF) ^l+x+Or l8+XDU pxiy) LT.(Ec 1 a S^ e ) (r1" I'nzR dpi L^ -t^bWlFMo)]MOlt A^+TI tleaxo)p +X0 )a where The asymptotic formula for 9 (>X Jt) is found by substituting (5.43) into Eq. (5.40). The lines of constant phase are given by (choosing the constant to be - a ): O = l-Xo L\- I,) Ct ) (5.44) which reduces to: ^ -^ -*-^t^[^ W}.M (5.45) These are circles moving downstream with the flow. The shape function, as well as the entropy, vorticity, and sound waves are illustrated in Fig. 21. B. SUPERSONIC INCIDENCE 1. The Interaction Integrals As in the preceding section, we shall develop the integrals which describe the refracted sound field. We have seen in Section IV, Part B, that the incident sound wave may be described as a superposition of plane sound waves of amplitude E'(a) = (\ M

75 SUPERSONIC SIDE SUBSONIC SIDE U>Co U< C U < Entrop -V/ ort/ci tyIncident ( I / -^^ ^^ Sound Wvve HrSe\/^~~ e> ~/ S ectiSee Section ReflectedAEntropy-Vortici ty Wove See Section A-3,4 See Section I Source A-2 Reflected Sound \ Wave See Section -—'' A-5 Fig. 21. Reflection of cylindrical waveso Consider an incident plane sound wave: -B;6 o t —x)^ o^' -_Lwt. (5.46) The resulting refracted wave will be described by: f _ e M t ksX+flt^ 1,teo0^ -uo'st I l j W,0 a. 4tj'-ci:o'1Yex ei^~ ~ Q-z Id BrU 1 ol —-- -eJ uto A^ ^ L ^ ^~o(^ i q~~~~~ ~I4oo(4o

76 and the shape of the distortion will be: K ^y,t) C,'iSL l+~-^lat i \ d o t, (5.48) where ~ <.\+NMoiv ) 0 0+\;W y (5.49) Ck = M~ ) C" = ~C \+Mowdo o'\+ Foio We shall again introduce the scalar and vector potentials. These potentials are given by: I t + td0^ d _-1 d (5.50) i = B' e~,+MOoO M+W ) where A' = - L + _) _ ~ o 5, (5521) ^8= i~ 7 -^ ^"' i o?;''l oAf. * P and p are the paths illustrated in FigXo 18 in the Q plane, and and P are the path"Fresnel" corefficients for r efraction described

77 in Section III, Part C, and Appendix E. 2. The Shape Function: -,'. t) We have just seen in formula (5.52) that the shape of the distorted shock is given by: ia..45 ) _ iCt 6'r~t).~~ P~ tgOIodo~i o (5,53) Let us again, for the convenience of calculation, transform this integral to the Q-plane- of Section IV, Part B, by means of the.supersonic aberration formulae (4.64). The integral (5.53) becomes: f^^yM ~ ='_I)^|^\^ H oa) J i~tMw4e I\'' -_ dab (5.54) where P. and Pz. are the paths in the @-plane illustrated in Fig.. 18. We may rewrite integral (5.54) as: ~lt =,,' (5'55) where FA) = ( l\+(am.4 Sl^ e1 ^"bY le, (5.56) with 0= ^.,, and -- From Appendix E we find that: (+tM.^lh o) _ eO[- r" r VX, v-ew (Eo7) t^-I)Lr, t rX eQ^^_ v- + Q - 4' -v*^-' where the P are polynomials. in. d( of degree not higher tnan 2. To make a saddle-point expansion of F t3, we must first study the poles and branch. points of the integrand. The branch points occur at:

78 tee t h) + ( 1)*> Using (I- L o b l)] typ q we find: o_ =_UO <_\ J hence t =~lJ -Mo = +il o l (5~57) The poles of the integrand are located at: r, -r- tt[w - +\- (,,^ ^] =0) or ~(382^ + \ 7 t") =) ^ 1s? [ (8+ O, which we may solve for A': CL 2^^ (^ Z ^ l tA r (558) It is easy to verify that both of these poles belong to the negative branch of the square root,, - rz t,^- t- ~)P^ = = o a. Inside the Mach Wedge, yo\ >\\. —In Fig. 22 we illustrate the paths of integration together with the associated branch points and branch lines for X0 < M The choice of branch lines depends on the location of the saddle point and is always made in such a way as to keep the path of integration on the positive branch of Thus the poles of the integrand need never be considered, For case i), tpl~ 1 i, we may carry out a saddle-point expansion. If we set A = o ~OS ~o, i;.-= inl.o, then'2-= - Xo'-.1am, and to is imaginary; om T-\ /.1. The integral (5o56) may be written:

79 - Plane B.L. Equivalent Segment For Case aii a= thf y/Xb -w- 0' _____2r B.L. B.L. Fig. 22. Paths of integration for F(y' ); supersonic incidence inside the Mach wedge Ft= -r r w_ M," -,- {"-+'-'i;' e &Pcoso, (5-59) On P, the saddle point is located at t-= o -,. On?P the saddle point is located at Q- 1T+ iL Tox If we set GL~-, r- -- L-+,.....-^(A G ~~~"k- = + qr -'t*- q' —(,x"A% (5.60) we find from formula (5.19)'Ffe = t' (5.61) + G -cos s in, r e-''t'-) The final form for the shape function is: -t' = o-T_ lt'- P )e,^t0"-1t (5.62) with

80 Now the criterion i l >) i implies o + This inequality may be satisfied by choosing first )z >> and then?el Xo( - Tm. Thus the approximation is valid in the vicinity of the X axis and then only for situations wherein the source is located sufficiently far from the shock wave. For case ii), i i <, a n wo, and we are dealing with the immediate vicinity of the intersection of the Mach wedge with the shock wave. Let us replace the part of the path P? above the saddle point in Fig. 22 by the equivalent path P. Then we let = a- L 4 z, and expand the exponential in powers of W p? to find: F y' G = (>CostV v'B X n+~ I +' i1'c'co s^ t - ) 14(5 6) G itself may be expanded in powers of i A using: cos0' <*0X) =, cosv-L, sine'; (5.64) Now we find that the segments of paths off the x axis cancel and we are left with:* w = X'GoT )) + -- \ 0 (5o65) = X G ) a \ (G,\t)+ l G, ) co )' t + - - It is easily seen that Go Le) and G@C|() are constants and thus: *We use G(e() instead of G& COSt0% t +', sivl (' ^4o0 ), for the sake of brevity.

81 F (/^ = T~T( Cr,-(2 Gr. -^Ol^Y^ (5.66) The first term gives for f t): b. Outsidette Mach Wedge, \'\ S'. If we assume'>, then: F ^t - \ O, x e ^ " + L'' 0se ) (5.68) Letting L - PI =CoY' r Lv p Iv^1, then t = t' V (P3 =~ -T^^^ i, and the paths are as illustrated in Figo 23. P4 _ B.L. B. L. _-Pl ne Figo 253 Path of integration for F(yl); supersonic case outside the Mach wedgeo Since we have no poles in the allowed region, we may close the path and find Fl) =O for By > > ~ y similar considerations we find c (t\ = O for 1 <'- 4t Therefore we conclude:

82 t j^t) = 0, for'\~ > o ~ (5.69) The results of these considerations are illustrated in Fig. 26. 3. The Entropy Wave;,, xyt From formula (5.52) we see that the refracted entropy wave is given by: = - X\ — ^ = i0 C -^ 6ro "l WIADr;,i (5.70) This integral is transformed to the G-plane by means of the aberration formulae (4.64) to give: = K, -. Ta (5 71) Thus we write: ____) = L, iot - tA- Sv)t (5.72) where and From Appendix E we find From Appendix E we find: (+^+od=C Jz4Q-\5 Fr1 - Fa ^ lI~??Ire (E.ll) ML4Ft(4 Tt'(o),+ r a The integrand of 5(1y has poles and branch points at precisely the same points as the shape integral F') o Therefore we may treat tQ%') by the method used for F [~').

83 a. Inside the Edge Streamlines, X) > \1 o — For case i),' ~ 1, let \ (4,A) = \4 k i C; then by the saddle-point method we find: sb= I.H("o.,s <)g + H ogI^ nt\ e1 0 (5.74) and therefore the formula for the entropy wave becomes: (Mo'^ ^ -o liwith An examination of the surfaces of constant phase conducted in the manner of Section IV, Part C, reveals that the waves have an elliptical shape. These ellipses are all tangent to the edge streamlines, and are blown downstream with no change in shape. There is a phase difference of 9 0 between the front and back surfaces again, just as in the case of the interaction-free sonic waves. For case ii), t H < ~, we carry out a series expansion and find that the leading term in the series for H is Therefore, since _Adz) is a constant tli) 2 2T ~_, and the entropy wave has a singularity on the streamlines through the intersection of the Mach wedge and the shock. b. Outside the Edge Streamlines, \,\ > - Y. - ) = 0 by the same reasoning as in Subsection 2b. Therefore the entropy wave vanishes outside the edge streamlines. The entropy wave is illustrated in Fig. 26. 4. The Vorticity Potential: i Clx,),t) The refracted vorticity wave is shown in formula (5.52) to be:

84, \5- * Ioo ~ ^ t, (.76), Let us again transform this integral to the G-plane of Section IV. Then: where ^X,..t — - _. =& %' \'- -.e, (5~77) Hence: Using formula (Eo9) of Appendix E for B we find: This expression has the same branch points as the previous amplitudes but in addition has poles at: Equation (5o81) has two solutions for Corresponding to each of these two root there are two possible values for Thus in each strip of width z-M in the -plane we have 4 poles79) with Using fornula (Eo9) of Appendix E for B we findo M,^j\^^~d)B= is l&^^^aiTLT1^ (M.-L^O^^^I~X fe (5.80) This expression has the same branch points as the previous amplitudes but in addition has poles at: [iM: - t^Y l =0.2PZI O (5.81) Equation (5o8l) has two solutions for fi. Corresponding to each of these two roots. there are two possible values forc ~ Thus in each strip of width 2?-tT in the Q-plane, we have 4 poles:

85 [~l,~ =3 / ^- 4~xz: 9' h-i ( i_.\) ); t - a- ( MD-\- 7-) / \ AnX / (5.82) These poles are illustrated in Fig. 24. 3 o FOR case outside the Mach wedge LAgain we consider two cases namely, nside and outsid-Plane -iTT -7r/21 Q 71/2 iT 27 ^ t^84 ^f ^^^!^:x: POLES FOR BB.L..L. p P -BL I3;I <1 o: POLES FOR /' s;^ ^:i i,>1 Figo 24. Path of integration for X(y'); supersonic case outside the Mach wedge. Again, we consider two cases, namely, inside and outside the streamlines emanating from the intersection of the Mach wedge with the shock. In the first case (inside the edge streamlines) the situation is identical with 2a and 5a since the poles are clearly isolated from the saddle point, In the other case (outside the edge streamlines), an interesting and different result arises. The shape function and entropy wave both vanished in the region, > \\\ but as we shall see presently, the vorticity potential does not vanish, due to the presence of the poles described aboveo

86 a. Inside the Edge Streamlines, X. > \> -n For case i), p' >, we carry out a saddle-point expansion as before. Using the notation: K(dit )= V \ \ d^)I +Mo ) (5.83) we find: (5.84) <(.' SE ) -~.aA, p-c>o>+'W1 l^.^-U>t) with The surfaces of constant phase are identical to those of the entropy wave. For case ii), p < i, we carry out a series expansion for. }K(o') = \<oti) = h 1 8(4') c -a -, (5.85) Therefore XKc~ 2 zTT o (5.86) and t,,(Y.',t} stays finite on the edge streamlines b. Outside the Edge Streamlines, Xo\ < \.\ eFor case i), suppose A y X; then the poles are either on the real or A+ ri axis, depending on whether or not (N \-) < %%L Thus referring to Fig. 24, we find -X(I by means of Cauchyts theorem as: (5.87) w-^{5 ^^S^}

87 Let 1j,,)_= _ \(J-!~ A't,'') ] tA, A,) (5.88) Therefore we have: Kt~,~ -- ) --,,7 (5.89) where wand.^ are defined by (5.82). Both ~ and 4Q lie in the allowed region. Therefore:,RQS (K!,XO:, ),,, =g T',,, Res ( -K(Al,ta- J 2 dT, (; - )"*~~~~~~~ (5.90) and (5.91) L A + i \.,-, MJ,X y For case ii), ~1 < -xo, since T Co, ) is a function of, the effect of replacing 31, by,tis to leave T unaltered. Thus the result of integration may be introduced by changing Vt to \l\ in the exponential of (5.91). Thus we find: (5.92) ^Jtf', { )F Xl- i, tt,Cx-u^t)- bll which represents a damped vorticity wave outside the edge streamlines. 5. The Sound Potential: g (^,,t) From Eqs. (5.52) we see that the formula for the refracted sound

88 potential is: gt~ bt=;5o; od i+'' & (5 93) Because of the complicated relationship between the functions i,, (, and the aberrated angles functions O,., transformation to the G-plane is of no great help in simplifying this integral. To carry out a transformation which will simplify the integral we first transform g0 to G as before, and then transform 9 to G' by means of the relations: (5.94) 402( _M)tol = t (\3, With the aid of this transformation, the integral (5.93) becomes: tt%)-3,t)*^~c = X -1i3 m(i - O _t, _4- < t,^)l, (5.95) where i s,ly) is defined by:.~r-O ___ Akk Y,+ (5.96) - l'^^K''' ^ ) e h\o \Ai' Pa and p1 are the paths illustrated in Fig. 25, and L are given by: 1 ~,_ ~C h \+ =t'"'op ~d) (5-97 Using the expression (E.14) of Appendix E for A, the integrands of (5.96) become:

89 "" ^eI I I Figo 25o Paths of integration for iS(x 2 ye); supersonic caseO Before we can proceed with an approximation of ( we must ex amine the branch points and oles of the ntegrandi The branch points occur at: or (5.99) + L~I Thus these points are located on the lines as illustrated 0n = >' (5Ploo) B.I_

90 which occur on the lines & R; and \R = 0O or Zt X c t+2 ModQ+oI -+ O 0 Solving we find: (,2, = - t 2 -- ii (5.101) Thus - Z. ~ * 4 O o The expressions (5.100) and (5.101) show that the poles are again isolated from the saddle points of the integrands. Therefore, we proceed with the ordinary saddle-point expansion, using formula (5.19). We may introduce CO,,.and find: w'- p\ | tof ) ^^w~fY-) ~ R l^, (5.102) We may now approximate (5.102) as in subsection A5. The saddlepoint expansion for p> j gives:,cos W, MsIM eL M 1\, V 1- L& -I 4. _?__( -M_ (5.105)...sF-" k.X I 1+ t (C0o Y\snr) e* v s i \ p\ * \-~) P.l Thus: )^-^- ]^ {^tr i'l f^)k ^^ L'* t&3 t) t ^ | p+ )(5.104) \ W p ) \ iX r i- M %,F + h@'' t It is of interest to examine the formula (5.104) as - > Q o In this limit the sound potential becomes:

91 These are cylindrical waves blown downstream with the flow. To determine the deviation from cylindricality, we examine the surfaces of constant phase. Let t* = t -. MJYa, and we find the Co' (~t-O surfaces of constant phase from (5.104): or as before: (,><x-uu~ty^^^Y =.~t'' ~,(5.108) Thus at sufficiently large distances from the origin (in units of Xo), the sound waves become circular. The deviations from circularity as we get nearer to the shock may be calculated to first order. Squaring (5.105) and keeping only first order terms in Xo, we find: 2_3 i (M^ ^ = K-,^ -., o ) If we set; = U_,t + Ct' Co05- t. Cos (5.110) i n Ct' s;-e 5 tve

92 then g measures the deviation from circularity. In particular: tt\ ^ ~X1 ^H 4 \ c os4)t+t% -\ s2i ) (50111) and hence the distorted circles are described by: R (e =,t't Xc ^ -r m \+ coDSv^ ^ ^ (\;Fl e (5.112) Figure 26 illustrates this wave together with those described in the preceding subsections Bl-B-4 SUPERSONIC SIDE SUBSONIC SIDE U0 > Co U, < C, Mach c\ _j^f_~~./ \ / R fracted -_ ^"g| / / Entropy -Vorticity Source Wae S \ \ / /See Sct / \ a-5 / See Section \\ \rA-2 \ -ig/ 2 Rfato ofcl Edge ^^^^\ ~~~/ / Streamline / Inciden Re racted / Incident ^^^_^^ Sound Wave / Sound Wave j See Sectio \ See Section A-5 / / See Section A-2 F~1o' 26~ Refraction of cylindrical waves.

VI. EXTENSIONS AND CONCLUSIONS Although the theory presented in the preceding sections can in principle be checked, experiments dealing with the interaction of sound and shock waves can most easily be performed in the shock tubeo However, for this purpose the theory will have to be generalized in several respects, On the one hand, tube shocks are not stationary but proceed into gas at rest. This causes the sound source which of course will be at rest also, to be moving supersonically with respect to the shocko Further, the sound source, which for reasons of simplicity, was assumed to be two dimensional will in practice be a point source. In Part A the point source interacting with a stationary shock is briefly discussed and in Part B the ground work is laid for a treatment of the problem of the moving shocko A. POINT SOURCE Let us first consider the generalization of the expressions for a line source as presented in Section IV to the case of a three dimensional point source. The differential equation governing the propagation of spherical sound waves in a moving gas is found to be: 7 - U -C) 41TE 8(x) 8,()8Q() - (6.1) The substitution tC* fi^,t)t) = (p^,(X, ) ge- -'"^^", -'9t7 93

94 with I~ = l and k = reduces Eq. (6.1) to: C C -4-) - *4t ) _ 4We1CE (6.e) If -\= v\ <, we use the similarity transformation X' = - 2' = Z' -- ) - -. (6.4) to reduce (6.3) to:'p -Vr $ =~ 4rr El &i(') S') (SG) (6.5) On the other hand, if V\ = \ >, we use the transformation x'-= *,ei -~ - (6.6)r to reduce (6o3) to:' ^ \ - -I^'1 = 1- ~Tr C = x'- FC') C 3), (6.7) We may treat Eqs. (6.5) and (6.7), as in Section IV, by means of a Fourier Integral. Carrying out one integration, we find the solution to (6.5):, l(x1^ ~%)= e —\\, (6.8) and the solution to (6.7); GI SA' -, + ) x_ 4____ -. ) (0= rO 00 A/+',^ -"\0 (6.9) for (' >O j Q O ) otherwise.

95 After introducing the variables t and. ( into (6.8) by means of?.n -= i coc ^itnvO ) (6.10) i = - t'' c te ) (6.8) becomes: d(', ^l) = 27 & siYA t3clp (6.11) O o Since (6.5) is rotationally invariant, (6.11) reduces to cn (>^ \ ^ ^ ^\ | \& zosi,ne& Dt, (6.12) * T p Jo where' _x -'2z'4- _z,, and P' i isllustrated in Fig. 27a. Equation (6.12) may be integrated immediately to give )(X\,\) = - e (6513) Thus for tA, C we find: )S4L - 4? -.L..'',?-.~ -., r,') twm~~~;sk l-y~t~t)- x)-MI (6.14) We reduce (6.9) in a similar fashion. Let -= i -, COS% L-6 9' i (6.15) then (6.9) reduces to: wher'e (.an ~ \ he osv ix'+Lstrt-Gecs sbn F i.b) @(P.^-Z.) = J Qe 9bl(6.16) where P\ and p? are the paths illustrated in Fig. 27b.

96 8. 8-Plane 0-Plane ok!v So 2rr iT 0 g 2 l a) SUBSONIC FLOW b) SUPERSONIC FLOW Fig. 27. Paths of integration for 0 (xi, yI, z). Since (6 7) is rotationally invariant in terms of the variables ( X ) L \,.'), we find that (6.16) may be written: ( VC 6) ^ =.T 2 s> nog 4 (6.17) We integrate (6.17) to find: Qtl^ )-< J1C1Y ) ) (6.18) otherwiseO Therefore for > I we conclude: ( 2ecoS (^T^ ^ -I>C -) )' (6.19) for: > L\A-.; v L O \ otherwise. It should be noted that Eq. (6.19) reduces to the well-known result of Pi'I I. G. Tamm21 for Cerenkov radiation in the limit LO =. Equations (6.14) and (6.19) give the exact behavior of sound waves generated by a point souree in a moving medium when no shock wave is present, When a shock wave is present in the flow, we use Eqs. (6.11) and (6.16) as the fundamental expression for our treatment. These equations

97 are expressed in terms of aberrated angles as in Section IV, and the resulting integrals are interpreted as superpositions of plane waves. The corresponding reflected or refracted plane waves are found from Section III, and the resulting integrals give the appropriate behavior of the induced disturbance. B. MOVING SHOCKS The theory presented in Sections II, III, and V applies strictly to stationary shocks and thus must be modified if it is to be checked experimentally in the shock tube, In this part we shall consider the nature of the modifications to be made and the results which one would expect for the case of a moving shock. Let us assume that the shock moves to the left into a gas at rest. The gas on the right then moves to the left as illustrated in Fig. 28. GAS AT REST GAS MOVING TO THE LEFT UOREL = O- (-UO)= U UIREL =U + UO= UI Shock Velocity -U0 -- Gas Velocity U -- Pressure, < Pi Density, D < DI Entropy, SO < S -_ Moving Shock Wave Fig. 28,o Undisturbed state of a moving shock. The situation illustrated in Fig. 28 may be arrived at by observing a stationary shock wave from a frame of reference moving to the right with the speed Uo. In this way we find that the speed of the gas on the supersonic side is Uo-\o)O, while the speed on the subsonic side is [,-~oaT and the speed of the shock is - o0=-U0.

98 The shock conditions, discussed in Appendix A, and used throughout the text, apply here without change since the relative velocities U,!and ] are the same as before. Let us choose our coordinate system so that at t O the shock is located at X O. The undisturbed motion of the shock is described by X - - ot; and a transformation of coordinates "'= X+Uot, 20= M t) =t ) (6 20) gives us precisely the situation treated in the text. The plane-wave problem may be solved immediately by reference to Section III. Let us indicate how this solution is obtained, first for incidence from the right X > - Uot o 1. Reflection of Plane Waves from a Moving Shock Consider a plane-wave incident from the right of the shock: \\ i e A -'CP ^I tud =0 R _ -V, - = \ and i If we apply C> If we apply the transformations (6.20) to Eqs. (6.21) we find the result

99 ing incident waves of the form discussed in Section III, Part Bo The frequency of.these waves, however, will contain a Doppler factor due to the motion of the observer. The appropriate frequency in this moving frame of reference is given by: O' = I_-~M,,. M, (6.22) Now as mentioned above, the problem in the moving coordinate system is precisely the same as that treated in Section IIIo Thus we may apply the results of Section III to find the reflected sound field, and the reflected entropy-vorticity field, as well as the shape of the shock. If we transform these reflected fields back to the original coordinates, the reflected sound wave becomes:, 4. (ivx+,l _ t i (6.25) _ ED Hi'e~ ( t c(S8)-i u0~-t; vA- ^O ) ~.=O where atv -i _ 1-,o (6.24) \ +,fi\ \ t+Mgdo and _ C, (6.25) The entropy-vorticity wave reduces to:

100 P=o o VI - t&e, 0(2- i (6.26),nit t e' J t )- where H. = {L l 9 )v\ 0 0, c);(6o27) \+AM Co and b~ = C~ (6o28) The shape function is found to be - ^ t + _ - I+tA \( Tio / C (6.29) y' - Uo1 t \+ MIo In Eqs. (6.23), (6.26), and (6.29), we use the values of A, 5, C, 0, and (d,>\ ), (ozl ~ ) given in Section III, Part B, (3o37), (3522), and (5o26)o These relations completely solve the problem of reflection of a plane sound wave from the back of a moving shock wave. 2. Refraction of Plane Waves from a Moving Shock. Let us now turn to the problem of a plane wave incident from the left of the shock:

101 doTLL lb +t -- t t (6.530) with, = - Co We may apply the transformations (6.20) to the Eqs. (6.30) to find the incident sound field in a system of coordinates moving with the shock. We then find that the system (6.30) transforms to an incident wave of the type considered in Section III, Part C if we use the frequency X3 = ( \+ MAo o) L (6.31) We find the refracted wave field corresponding to the incident field (6.16) by reference to Section III, Part C. Transforming the refracted wave field to the original coordinate system, the refracted sound wave becomes: (6.32).4=.,,.> \ S -; O A= 0 0)

102 where \ N\ t i\t-, I ^OdL).) (6,33) and v C ( (6.34) C\ The refracted entropy-vorticity wave is given by: -u = O )?^ -" (6)55) with (6~36) (. = -( 1+ Moo') ( (6.36) and i-^ \ _ 2(6.537) The shape function is found to be: x - U* (- ~ e\+ ^' (\^ -t) (68) Equations (6.32), (6.35), and (6o38) solve the problem of the refraction of a plane sound wave incident from the front of a moving shock. The coefficients A, b,C, and 0. and the angle functions ( o( \ ), ( ( 1) Z) used herein are given by (3.64), (3o43), and (3.52) of Section III, Part C.

105 5. Reflection and Refraction of Cylindrical Sound Waves by a Moving Shock Since the plane wave interaction may be reduced to the problem treated in Section III, it might be expected that the cylindrical wave interaction would reduce to the problem treated in Section V. This is not the case though, since the shock wave is moving relative to the source here, and this motion causes the interaction integrals to be more complicated than those of Section V. To make this fact apparent, let us discuss the appropriate interaction integrals. For incidence from the right we have according to Section IV: For 0 > W\ > -1 (subsonic flow behind the moving shock), i*^ -& oiX- us ) do &(6.39) Pl being the path shown in Fig. 14, p. 53. For \<(- (supersonic flow behind the moving shock), L W_. -LI -Jl-f-.L.P.t ep. I J^ wo^. (6.4O) P. and P are the paths shown in Fig. 18. Both of these integrals may be described as superpositions of plane waves of amplitudes E(e-o=) = E ( 6.41) The corresponding integral for incidence from the left side is: ( ^>>,5st = e S Q~ktdt>(X+Xhoo3)-Wk (AKo 4 (6.42)' being the Sommerfeld path of Fig. 14, p. 53. The corresponding plane wave amplitudes are * tIo) ^ ~.-_ c k, (6.43)

1o4 The reflected waves with amplitudes given by (6.41), and the refracted waves corresponding to (6.43) may be written as integrals of the form considered in Section V. It should be remembered that now the time factor may not be removed from these integrals as in the cases considered in Section V, and thus the saddle points of the integrands are time-dependent. This gives rise to time-dependent wave amplitides, which is to be expected because of the motion of the shock. These interaction integrals will be considered in detail in a forthcoming paper. An observer at rest in front of a moving shock will notice the following sequence of events if he is originally located between the source and the shock. First, he will see only the field of the sound source since there is no reflection into the gas at rest. Then he will notice the refracted field as the shock travels between him and the sourceo Finally, he will see a combination of the incident and reflected fields when the shock has passed the sourceo If the observer were originally on the side of the source away from the approaching shock he would see first the incident field until the shock passed the source. After that time he would see no field as the shock moved between the source and him, and finally he would see a combination of the incident and reflected field as beforeo C. CONCLUSIONS To test this theory experimentally with the shock tube, one would use a point source and a moving shock waveo Therefore it is natural to carry out the extension to this case also. Such generalization will be made at a later date. The theory of the interaction of plane and cylindrical waves with a stationary shock could be carried out in a wind tunnel. A schlieren photo

105 graph should reveal the distortion of the shock as well as the surfaces of constant phase of the sound wave, The density field could also be measured with the aid of a single-fringe interferometer. Photographs of the interaction taken with the aid of such an instrument give quantitative measurements of the density which could then be compared with the results of Section V. In the limit X -=., the Cerenkov limit, the line source becomes a stationary object in the gas~ Such an object will give riSe to a diffraction effect similar to that studied in the paper of Ludloff8 mentioned in the introduction. One could easily set up such an experiment to check the theory in this limiting caseo

APPENDICES

APPENDIX A THE DIFFERENTIAL EQUATIONS AND SHOCK CONDITIONS In this appendix we shall discuss the equations of compressible hydrodynamics and the conditions which apply across shock waves in a compressible flow. The laws governing the motion of the fluid may be written in integral form as: Conservation of Mass: ( -- ) Conservation of Momentum: g( U +~AX %)) ) ~ -Ojt (Al) Conservation of Energy: ( l( [t ) X ^s (H)) v 0, where e = specific internal energy andH = specific enthalpy of the gas. All these equations are'O the form: (, (I X g )&\1 = ~ (A.2) where i= g J 5 = )U", y^or UY 5%A= UE- - (A.3) or 5 = We now have a general equation which governs the fluid flow. We would like to deduce from this equation the behavior of the fluid both in portions of the field where the properties behave continuously, and across surfaces of discontinuity (shock waves). Let us assume that the fluid is continuous except for a single surface 107

1o8 moving through the fluid, across which the properties may change discontinuously. We can describe this surface parametrically by means of the equations X = i (io,,; t) o The velocity of the surface is U i t[( o Y -t), and the tangent vectors to the surface are to t = i.r The normal vector is (t 1)/ sttl, IThe normal component of the velocity of the surface is U0, which is related to the Jacobian T = (T T t by means of the determinants I~ f~ I fa' G o ^ \ o T. O by= I 3 = j^ o; tcr, To; l" i?.x^,in^.) 8 (A.4) We find the time at which this surface passes a point ( X, X> X ) by solving the equations X = f Cot ~ for t. These equations may be solved provided 3 * 0O i eo, U' ^f O, It will be necessary to calculate Hi, where t = (') is the solution to the equations' = tO'0C, t) for to For this purpose considero zLc, \z L O 8 = ^<, OosL + c Ts + ft He ( 81- =,cO 4Xt E TX it; (A.^5) = / LX * + i o)t; + i T, s; + it "Cx; The coefficient determinant of these equations is to \t U whereas the determinant: S,; i, s8 l A/X; = I A, sTr s" _ 1 - ltt n1 (A.6) or xx;= - * (A.7)

109 Now since the surface is to represent a discontinuity in the fluid, we let: L = "'', t) h (t -x) +2. X X3st) - t; = tio ix'3t) h-t - ) + X,5L (At.^^t~t-x^ 5\V^)) (A>8) where h(x) is the unit step function defined by: h(x) = 0 x < 0; h(x) = 1, x > 0; then dh/dx = 6(x), the Dirac delta function, which is an even i isi il a frti i function. Hence for t < T, yi = y Siy J i~ _ and for t> r, S = SO. t < T represents points ahead of the surface, whereas t > T represents points behind the surface. Now L - t hl tt- + ^ - tm) - t) + to s-)^ (tand (A.9) _S~" _S Lot- ) + -. + (%S Ct -j't, __ = ht-t) +, dh('-t) + (S Thus our general hydrodynamic equation (Al10) becomes: + i^^+3a-i - ^ S1) ] - )D =0 at ax'"\ -.t a&$' (A.lO) + ~tt4, -, )loo-t ]sjt-~) v -O where VO is that portion of V behind the surface of discontinuity and V1 in front of the surface, If we take V to lie wholly behind or in front of the surface, the fluid is continuous, and hence we may conclude that: at ax4 u and " ~-* =, (A.ll) +tax

110 If the surface is included in V, then we have by the well-known property of 6(x) ) tl~ K) s& = $(o), at t = r t ^ -1) - ( -iS'- = 0o (A.12) at( -\ ) = -t T ). or Thus we conclude that in the continuous portion of the fluid, the differential equations (A.1l): b = must be satisfied, whereas the relations (Aol2) (So yi U) = (S1 - y3 U) n must connect the variables on the two sides of a surface of discontinuity. We wish to return from the general form of the differential equations and shock conditions given above to the equations in terms of the original variables. We would like also to put the energy equation into a simpler form by use of the laws of thermodynamics. In terms of the original variables the differential equations are: 3- +:= o ~'W.'~W,;~ t -J3' a ms ^ J^ a_ +^ to.Cp 8 O; (A.15) If we now write the combined first and second laws of thermodynamics for systems in equilibrium in the form:'J >g = Q~ - & en Q (A.l4) where 2 is the specific entropy, the third equation of (A.13) may be simplified to':

111 & = o Oat - The second equation may be written: $./aU' ni^'SU^ 4. 1-$ - o Hence in the continuous portion of the fluid we have: ( ^-t )t+ O) To these we must add the equation of state of the fluid: The shock conditions, too, may be written in terms of the original variables. The first becomes: letting f — J; then &o 0') = tfJ) Y) The second becomes: DO W, v o') + +T = -.' + 1

112 Multiplying scalarly by n we find: and multiplying vectorially by n: - PMOI, Xn = U, x; The third may be written: M + (J ) O i L.'i 1 Thus across the shock we have: 90+ &olMf~)l= |+$(B? ^) (A.16) ^Xn; =.,X; Now we have the standard differential equations (Aol5) and the shock conditions (A.16) of compressible hydrodynamics. The differential equations apply to the flow in regions containing no shock waves, and the two sides of a shock. Notice that in the above treatment the shock waves are not necessarily planes. We shall put the shock conditions into a form which will be useful to us for later calculations. In doing so we shall also discuss some of the important properties of the shock transitiono Let us consider a two-dimensional situation in which fn denotes the normal component of velocity, and. denotes the tangential component. If we assume that the gas is an ideal gas with the equation of state a( / A as^ C /Cv t then the enthalpy 4 -4 I, where I =/ CVc Y-1 jT(/ /

113 Thus the shock conditions are: ) + PoUn = 5 ) +'Vt j (A.17)'Uot USt t We have here four equations in eight unknowns We may solve these equations for four of the unknown terms of any other fouro Let us decide to find the quantities Hi ) al n,nd d i+t in terms of the quantities D o ) a jon and oLt. To do this most easily, let us solve the first two of Eqs. (A.17) for a and T in terms of LJ. Then.we may solve the last equation for l5n, using these values for i and j o Thus: al = _~ Urn ) (, =? + tSo (,,-I,,) and i- = n + ru +~~o~ -- -~~~-'~~~-(t lit el U 2. - 9 -.-v^ cLoUS t na, Now we let - = C2, Co" being the speed of sound in the gas on side "Oo" Then we find: L + ++ n = r-r L> + A la-n For convenience let i = Y+1 then we have: 7-1 A n(/A^ ) o^ on/. o X (Al.18) Now we define the normal Mach number 2B)oby means of 1 "r, and note that 71 is dimensionless. In terms of'o,, our Eq. (A,18) be

114 comes: /AU -co +o tt )7]SlnCo[ - vanl + which may be solved to give: U -n = ~.on ) or /In = n il A ] We rule out the first possibility since it does not represent a discontinuity in velocity, and we confine our attention to the second. The second may be written in the form: lt In =, ] +(A.l9) We are now in a position to calculate r and e Since Vonl-UJn ='5 o -rl'] we have Corn - Min = onor) hA. Thus T o C9O tI + no VI on n becomes: T = A~ kt+4) - ifo I (A.20) We calculate A from (Ao17) and (Aol9): rB - ft 2YO 0 (A-21)

115 We have now found -E and in terms of rY,. Let us eliminate >W0 from these equations and find S_; in terms of o To do this we notice that - = ( +)T -, hence =, so we have: (A,22) This is the well-known Rankine-Hugoniot equationo The sound speed on side "1" satisfies the equation: + (AA-0 (A.23) C = y 22 = 2 ( (L ^ -li;+(^ ) (A25) Hence: and thus we have: t^Z a n-~,,. (Ao24) Solving for -10 we have: "M 2 = m 1)-\ a (Ao25) The symmetry of these two relations is due to the symmetry of the original equationso Notice also that: =X c 5 \ implies z Now let us consider the entropy difference on the two sides of the shock wave, A, no =c, Jv * a

Thus: CA,, -— }^<AA (Ao26) We give a summary of the equations which are important to us in the textO These are: m 5 = e I \I - A. (A.27) A= h These equations may also be written: din='to F t'm ) j ), = % - &(Tfo); (Ao28) tF(I t where Notice that: Let us now investigate a particular solution of the differential equations and the corresponding shock conditionso As was pointed out in the introduction, the problem with which we wish to deal concerns the

117 interaction of a sound wave with a shock wave. If we wish to deal with a shock wave alone, we should first investigate the simplest solution of the differential equations, that in which all the dependent variables are held constant. To do this, we set = p, S_-= D,.n=U, and J= 0O. P, D, and S must satisfy the equation of state p = A',e5/C (A.29) The shock conditions then imply that the variables on the two sides of the shock wave are related by: U = U. F(Mo); Pt = Po G&T o) v (A 30) The entropy equation may be written as: ese ns s dte + F t s " These equations thus determine the state "1'" if the state "0" is known, and conversely. In this particular solution the flow is essentially one-dimensional, since we have set = "OJ — o =

APPENDIX B CHARACTERISTICS OF THE LINEAR FLOW EQUATIONS The characteristics of the differential equations (2.4) are defined as curves across which pu, \, and J may be discontinuous. In Appendix A the method of describing such curves is, given in detail. We apply this method to Eqs. (2.4) and find that the following equations result: ((Tc-T - On*) [ C, n, W n C jyvl =o; -c l n, =o (B.1) -Cnr P3 ~+( -UUr I O0 (i.cn-Un. =- o.0 In these equations the [ ] symbol is used to represent the amplitude of the discontinuity. QJ represents the velocity of the characteristic curve in question, and n represents the normal vector to that curve. Equations (B.l) have solutions for [-p], [n], [%T], and [A], provided that: (0i — un ) -cn. -Cny o 0Ui= 0 -cn, o (.U-U) 0 o o o ~ ( uu. ) 118

119 or n-0 -un -C)J =0. (B.2) There are two cases to consider: (Bo3) and C(:n-unZ =^ o. 0(Bo4) In the former case, (B.3), the solutions to Eqso (Bol) may be written: tl] = A lul ='nA; (B.5) V1 - n A,; l) f = o where A is an arbitrary constant, and the choice of sign depends on the choice of sign in taking the square root in (B-3)o In the latter case, (B.4), the solutions to Eqso (B.1) may be written: L^] = o; (B.6) v = n*C, [o = C where B and C are arbitrary constantso If we describe the characteristic curves by means of a function

120 x,-,t)=, then n -= n A X,and Equations (Bo3) may be satisfied by choosing (p [X )) t) C -t - (y-Ut- ), for which Eqo (Bo3) becomes I\' fr o The choice of the sign in the second and third of Eqso (B-3) is the same as the choice of sign in \V \ -- 1 + Equations (B.4) may be satisfied by choosing ) ( X )"t) = X MX-^,. This type of expression satisfies (Bo4) identicallyo

APPENDIX C SIMPLIFICATION OF THE LINEAR SHOCK CONDITIONS To simplify Eqs. (2.14) of Section II, we must first calculate NTo in terms of U, I qS J and o o Since. o = P, we have: J^^ ^, I P~'U,,, -^ ~%S~~I~ll~t ~CL9"-^^ * (C.1) Differentiating (C. ), we find: M o 2lko t Oo -+ \ which becomes, by virtue of the fifth of Eqs. (2.4): rmo = M0iu -o^^ 0. o. (C.2) We now use (Co2) to eliminate TY from (2.14). We find: +, 1 = f4. L ffI Ga^ ^ tou,- ^r - o PPO- fo) - +Az4 o MGG) 0 A 2O ~ GdY=0)- (C5) F'2 - (..t, (8) G(' ) /C.o C-* u'~~ = ~O + a (g.u0- ^ ^^-0^ -^w+m += +A( I M1 0F).F (d.-) W ( L + t0- FtMo-Ms F'(M.>) ^; 121

122 FW.) \ F(,4 )(O = F 0 1^- - (,- F(M) and (Co5) - Q- — (^ ^ ^!)0^ rl(M- ) r).t Now i = -F( and Go (o o. Substituting in the first of Eqs. (C 3), we find for -i ~6fi2 * > kD Ao - e' > % ~ ) F ^(Mo) l' Fm or (C. 4) in the second of Eqso (Co5), we find for U1t: vt =Mo~ ^^F ^^ ^- 4 (CM.Ii ) Noting that i-F'(t -' \ azK (f- __ MD-~))u.+lM.~.+r e~t~~ __,? o 0o Using the value sove for F in the last hird of Eqs (C), we find for g 2lxF 4_ 1) (t4 2_ _)(9 ) (C.4) Fl(^o) _ _t(kc-l) GIk to) _2U+_) u o Noting that 0 ( 0\) and GC U~) softs)Mt - we find that >+ G CM3) F (^v)| >t F t0o) Xo L Using this in the last of Eqs (C 3), we find:

123 + ((2J+^s+ - )(mo Wks^-.- -I)(%t- L))tz ) ( -1)i _~ ( re't +2A t+ ^-)(t -1)) 0 -,.e.) (..MCo t Mb +1 M- + to 0~~~~~~~~~~~'~

APPENDIX D TRANSFORMATION OF THE "FRESNEL" COEFFICIENTS FOR THE REFLECTED WAVES Equations (3037) for the "Fresnel" coefficients of the reflected waves may be rewritten in terms of o( and A^ alone by using the reflection laws (3o22) and (3026)o We note first that by virtue of the reflection laws:..... - (D.1) and that:'" " ~__ _ _ _, (D12) O(~- O~, - _ \+~ ~ — 0 i' - 81\ Imltiee _ __ _ - _ _' (D+ ) Using the expressions (Dol) to (D'o4) Eqso (3537) for the "Fresnel" coefficients, we find: a = k n=, t,) 6/; 4 =, ( tj + nt. fl )L;v (D * 124 A, = ~, /~,T +2,,',,)t,, i Ml~Mo~.(C ~S7i;~1 C: - z ~, ~,,~,.) ~ ~/~ a h

125 where LtL -,*v b,). Using the values for 6,. b0~ b3, and bq from (35533), we find: ZZY^ b.n s n b, = _ >-z5 ^-'^^.o ( M.-MZ)-, MlkA O+ M) M- +O+M. - (-fbIx,Fix r, i 0(VMwd M4 1 + (D.6) L 2 M\lo9M>\ e - b +. 2 tX,+io \ t ok)' The amplitudes O, A, B, and C become: = t a tA (o2 A 0 M.)(6 lM o d) __ I( 1-O, ) WI,\+mnO)'l^^^^ (Il+MKDt} 4 _ (9+ ( 2t o)) ( *. MU roe \ t o -2 ) -4 M, l+,do)4] tD P? + (C \ t! M ol+ (D.9) r -_ 4_^^l,-t+M. K v ( o)(m -i(l- + M, _

126 We shall transform the functions O, A,P, and C into forms more useful for the calculations of Section V. In particular we use the relations (4533) to find the appropriate formulae in the G-plane of Section IV, Part A, Fig. 4. We use the relations: \+ J# ot o I l+l'Mdoo n- ) \+moo I (D.10) I +, d0= I^+i > + +^ o - ) to find: 2 2A 2 Z 2-0+.^ oC- 4 (D.11) Q - 2.{ ( T? I t Bd o d e c e2. dtr1 Z o/\^c -t 0 C KM 0 0L o~O ~?- 2 M k M2- o(

APPENDIX E TRANSFORMATION OF THE "FRESNEL" COEFFICIENTS FOR THE REFRACTED WAVES In Section III, Part C, (3.64), we have shown that the "Fresnel" coefficients for refraction may be written: (E.l) C = B =- ^'u-.Iflab -; C = ( n'"^h (^ be - b, 4 < A 4 (^>^ - b4 (^ t ) It is convenient for the calculations of Section V, Part B, to transform cL, D, and C to the aberrated angles of Section IV by means of (4.64), and to transform A to a completely different angle variable. Before discussing'the transformations of A, let us first transform the other three quantities to the G-plane. The transformation (4.64) of Section IV, Part B, when applied to (3.43) and (35.52) give: i _ ^\ g 2 V.Tk t. (E.2) 127

128 2 {W-_A )- VA2, > (E.5) @-.^n? ) t(Eo) O>zou P l(o)l = P mo -15)( +ZMot o^; We also find from (3 61) (E.5) Ll3] \ /f (o6i;' L J A{#r M F(M) ( \ + Moo) bv; ^M F(olg =2 A2 V, o - A\) b3LFA =4I-. 831IX W (MoFk < =' r -j t;

129 Hence: I k2.; = vr3 - (I ) e () ( 4 + PI(M0-l)+(\-M?)5? ) (I I+oMA) where r,= Ms,- + >,xo1; IX= 2AA; P3 = WA ~4oy& (, -') (o( M0o) 2 t'Z Mb( \+ od)+}^Mwg> (E 8) r4 = Mol (-V-> o+MoU2MOt+ (Y3I -; and B= t& 6Mol rs + rb A (o-(- )4- - 12 M (Eo9) MtA F 0oA kIA-_) + t —L_-,:) \+o -i)05t%\4 _%tt where F- = -4,tM0>1 pJ4fi^0+ (-oW-')to(^o —t)tWo(i+] _BS-^ A M?- o^ t 0+1+ +(A-l\-\(l,d))(t+) 20l — d = = - M^t o o tW-^) |^^^- ^^^"2 ^-^)^(^h )] _ 2AlM (t-2W-L-)\b)4 ((Th-() o MlO)) ( 4o + I)

130 and C = 2 A, ( g -> 4 t al-b * ( ^m?-\) ^"" ( l E (E. 11) Ad Me FltMo [r, + rPI(- + M),' where r- = t\- t-(- >-^H\)(d+o + 21Mo M+Mod + X>U}\Wwc +C _ t- dzedM o + Mhi \ wfed s)+ \ +? M ol;i (E,12) re = MDtypo2Mo >-l)(d e Mc (\+od)) -2(+ - L) ho i+^ o) t W I-_) ^I (I+ od)].O ) Thus we have the quantities Ad, B, C expressed as functions of G. To express A in the appropriate form, we first transform G0 to ~ by means of the aberration relations (4o64), and then transform G to GI by means of the relations: ^'-'tSF ) i= W^^0 -, _ _= ~71-^ (_E__ o l ) where the + sign is chosen for path P~ and the - sign is chosen for path x of Fig, 18. In terms of d and 6, the expression for A becomes: whA - - ho. + 1\t 4 8 + 1- i ) (E.14) /t ( M N M cm ~<) AA U~+Moo) where

1351 +[ (ve+ \- OWNM-) -^NU^Q(^]LvU9^- ^ }. I = 2 Who t} t\- me 5 t Flo + toC\)) +~~C O ~\~' C~ P) v\,rlo\ O'Luk'3k~ ~i i (.1?

REFERENCES 1. R. N. Hollyer and 0. Laporte, Am. J. Phys. 21, 610 (1953). 2. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGrawHill Book Company, Inc., New York, 1953), Part I, Ch. 7. 3. A. Sommerfeld, Lectures on Theoretical Physics, Vol 4, Optics, trans. 0. Laporte and P. Moldauer (Academic Press, New York, 1954), p. 319. 4. L. Landau and Eo Lifshitz, The Classical Theory of Fields, trans. M. Hamermesh (Addison-Wesley Press, New York, 1951), po 158. 5. C H. Fletcher, A. H. Taub, and W. Bleakney, Revs. Modern Phys. 23, 271 (1951)6.. Bargmann, Applied Mathematics Panel Report No. 108 2R (Applied Mathematics Group-Institute for Advanced Study No. 2), 1945. 7. M. J. Lighthill, Proc. Roy. Soc (London) A, 198, 454 (1949)o 8, H. F. Ludloff "On Aerodynamics of Blasts," Advances in Applied Mechanics, ed. R. von Mises and T. von Karman, Vol. 3 (Academic Press, New York, 1953). 9. G. F. Carrier, Quart. Appl. Matho 6, 367 (1949). 10. Ho S. Ribner, Convection of a Pattern of Vorticity Through a Shock Wave, NACA TN 2864 (1953). 11. F. K. Moore, Unsteady Oblique Interaction of a Shock Wave with a Plane Disturbance, NACA TN 2879 (1953). 12.. G. Tamm, J. Sci. U.S.S.Ro 1, 409 (1939)o 13. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inco, New York, 1941), p. 573. 14. A. Sommerfeld, Ann. Physik 28, 665 (1909). 15. H. Weyl, Ann. Physik 60, 481 (1919). 16. H. Ott, Ann. Physik (Lpz., Folge 5) 43, 393 (1943)o 132

133 17o Bo Lo Van der Waerden, Applo Scio Res. B, 2, 33 (1951)o 18o Ro Courant and K, 0. Friedrichs, Supersonic Flow and Shock Waves (Interscience Press, New York, 1948), po 12o Also see Appendix A of this study. 19. Courant and Friedrichs, opo cito, p, 121o Also see Appendix A of this studyo 20o Courant and Friedrichs, opo cit, p, 297o 21o A.Sommerfeld, Lectures on Theoretical Physics, Volo 1, Partial Differential Equations in Physics, trans. E. G. Strauss (Academic Press, New York, 1949), p 182o 22o Ibid., po 195o 235 Eo Janke and Fo Emde, Table of Functions (Dover Publications, Inco, New York, 1949), p. 148. 24. Morse and Feshbach, opo cito, p. 891.