ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR FINAL REPORT (Period November 1, 1951 - October 51, 1952) PROPAGATION OF UNDERWATER SOUND IN A BILINEAR VELOCITY GRADIENT By R. M. HOWE Assistant Professor of Aeronautical Engineering Project 2002 U. S. NAVY DEPARTMENT, OFFICE OF NAVAL RESEARCH CONTRACT N6 onr 23223 March 1, 1953

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN PREFACE This report summarizes the results of some theoretical investigations and differential analyzer solutions for the problem of wave propagation in a medium with varying indices of refraction. In particular, the problem of wave propagation underwater is considered when the index of refraction changes as a function of depth and the effect of the bottom is neglected. The electronic differential analyzer solutions were limited to the determination of the eigenvalues and eigenfunctions associated with the depth-dependent wave potential. The above work was sponsored by the Office of Naval Research. Under the same contract, an electronic differential analyzer was designed and constructed to allow further study of the underwater-sound problem. The description of this equipment is given in a separate report.6 The author would like to acknowledge the assistance given by Dr. C. L. Howe in obtaining the differential analyzer solutions presented in this report. Dr. J. R. Sellars was mainly responsible for the approximate eigenvalue formulas developed in Chapter 2 from the asymptotic forms of the Hankel functions. The theoretical investigation was carried out by Dr. C. L. Dolph, who has written the portion of Chapter 1 which summarizes this effort. The author is also indebted to Dr. H. R, Alexander of the Acoustics Branch, Office of Naval Research, for his help in clarifying the derivation of the bilinear-gradient equations, and to Dr. H. W. Marsh, Jr., of the U. S. Navy Underwater Sound Laboratory, New London, without whose interest and support this research program would not have been possible. R. M. Howe i

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - TABLE OF CONTENTS Chapter Title Page 1 OUTLINE OF THE PROBLEM 1 1.1 Introduction 1 1.2 Equations to be Solved 1 1.3 Method of Solution 4 1.4 Review of the Status of the Mathematical Theory 6 2 APPROXIMATE NORMAL-MODE SOLUTIONS USING ASYMPTOTIC FORMS 10 2.1 Solution for the Linear Gradient 10 2.2 Solution for the Bilinear Gradient 11 3 PRINCIPLES OF OPERATION OF THE ELECTRONIC DIFFERENTIAL ANALYZER 16 3.1 Introduction to Operational Amplifiers 16 3.2 Solution of an Ordinary Differential Equation with Constant Coefficients 19 3.3 Solution of Differential Equations with Variable Coefficients 20 4 SOLUTION OF THE BILINEAR GRADIENT PROBLEM BY THE ELECTRONIC DIFFERENTIAL ANALYZER 22 4.1 Transformation of the Equation into Computer Units 22 4.2 Separation into Real and Imaginary Parts 22 4.3 Computer Circuit for Solving the Bilinear Gradient Problem 23 4.4 Measurement Techniques 25 5 COMPARISON OF COMPUTER RESULTS WITH THEORETICAL RESULTS FOR A LINEAR GRADIENT 27 5.1 Theoretical Solutions for a Linear Gradient 27 5.2 Method of Interpolation to the Exact Eigenvalues 27 5.3 Comparison of Differential Analyzer Solution and Havard Tables 29 iii -..

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN CONTENTS (Cont'd) Chapter Title Page 6 DIFFERENTIAL ANALYZER SOLUTION TO THE BILINEAR GRADIENT PROBLEM 31 6.1 Determination of the Eigenvalues 31 6.2 Rerun of Analyzer Solution from Surface on Down 32 BIBLIOGRAPHY 34 APPENDIX I - SAMPLE CALCULATION OF EIGENVALUES AND EIGENFUNCTIONS A 1 APPENDIX II - SUEIARY OF COUMUTER HESULTS A2 APPENDIX II- DIFFELRENTIAL ANTALYZER iRECORDINGS A3 DISTRIBUTION LIST -... —....i-.. —-.. —.. —..iv....

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN CHAPTER 1 OUTLINE OF THE PROBLEM 1.1 Introduction The propagation of waves in a semi-infinite medium having varying indices of refraction has been the subject of a very considerable number of researches. This report is concerned with a small part of that problem, namely, the determination of the normal modes making up the depth-dependent wave-potential function describing propagation of sound waves underwater. The problem is complicated by the fact that the velocity of sound varies with the depth of the water. In particular, we shall consider the special case where a positive velocity gradient exists from the surface to some finite depth, at which point the velocity gradient reverses and becomes a negative constant for all lower depths. For this reason the problem treated here is known as the bilinear gradient. The medium of nroragation is considered to be semi-infinite, i.e., the effect of the ocean bottom is not included. The electronic differential analyzer, along with tabulated solutions to Stokes' equation, is used to solve the depth-dependent equation. No attempt is made to normalize or interpret the eigenfunctions obtained; this task is left to other workers in the field. 1.2 Equations to be Solved Cylindrical coordinates will be utilized to describe the wave potential function \/(r,z,t) where r is the radial distance from the origin, z is depth below the surface, and t is time, The wave equation is V2t = l 92{ (1-1) r,z c2 at2 where c is the velocity of propagation, and is a function only of the depth z. Note that the wave potential Nr is assumed independent of the polar angle. Assuming that the time variation of 4f is sinusoidal with 1

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - frequency W, so thatf= lej_, we have 2 j + k2y= 0 (1-2) where k' = L4, (1-3) C and where 4 is a function only of r and z. By separating variables in the usual way the following types of solutions are obtained: H(1-4)(1) r (r,z = H(1 ) (r) U(z) (1-4) where U(z) sFtisfies the equation d U + (k2 _ y) = (1-5) dz Here Y is an eigenvalue to be determined by the boundary conditions on U(z)o These are U(O) = O (1-6) and lim U(z) -> outgoing wave (1-7) z - oo That is, the wave potential vanishes at the surface and corresponds to an outgoing wave at infinityo Next we assume a linear variation of velocity -ith depth. Thus c = cO (l+bz) (1-8) where co is the velocity at the surface and where b is a constant. Assuming that bz << 1 2 2 AJ2 k - - 2 (1-2bz) = k2 (1-2bz) (1-9) c cO

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - We are interested in the case where b = bl for z<zo, and b =b2 for z > z0. If we define a dimensionless depth variable ~ by = z (1-10) z0 equation (1-5) becomes d2U () 2 1-2blzof)- -2U) )= 0 (1-11) d20 and d2U —- 2+z0 [2 Z 2bz22bz) - =0 d2U +) Z2 2k (1-2b2z0 2b2 122blz0) -t ( d^ L l, 1 (1-12) Note that equations (1-11) and (1-12) are written so that there is no discontinuity in the velocity at 1. Equations (1-11) and (1-12) can be rewritten as 2 1 b 2b i^b d 2 0 ko 2b0z 2 - 2kb2 2 Zo. 2z0 k 2 0 f<1 (1-13) and d2 2z03 k2 r 1 2 d2U + kb2 1 b (f)=O0 d,2 1 (1-14)0 0 > 1 (1-14) Let us now define the following parameterso 1 (1-15) b2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 3 3 2 2 s3 -2z3 k 2 0 b 2 2 2 Y - Y (1-17) Y 0 2 2zok02b2 Equations (1-13) and (1-14) finally become dU f (f + y ) = O >o (1-18) where f(~) (1-a)+ 0 ~f 1 (1-19) -a f> 1 These equations are subject to the end conditions U(O) = 0 and Ut)) -* outgoing wave, Since norrjally b< O0, we see that s is a positive real parameter. For the reverse gradient case, b>0O and hence a is a positive real parameter greater than 1o The independent variable is real, but in general the wave potential U (~) and the eigenvalue Y will be complexo The boundary conditions can be met only for certain discrete values of Y. We denote these eigenvalues by m (m = 1,2,...), where m Y, is the smallest allowable value of Y- Y2 is the next smallest Y, etCo The wave potentials associated with each Y are called nrormal modes or eigenfunctions U ( )o The problem is to find Y and Um () for the lowest modes (we will consider the first three modes in this report)o 1.3 Method of Solution For J > 1 equation (1-18) becomes d52 m d2U 4 1 _________________ 3 ____'IT_____TT__

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN If we let p = s (f - + Y ), (1-21) equation (1-20) becomes d2U 2m + pU 0 (1-22) dp2 m dp which is known as StokeS equationo The solutions to this equation are modified Hankel functions of order 1/3o These functions (there are two types, hi (p) and h2 (p) ) are tabulatedo2 It turns out that the function h2 (p) satisfies the boundary condition at infinity, namely that U (p) correspond to an outgoing waveo Thus in terms of our original variable S we have as a solution U() h2 s ( P-a+Y)} Yml (1-23) where the solution is valid only for >1 and is subject to a prior knowledge of Y Thus the problem becomes one of patching onto the solution (1-23) a solution valid in the region 0 ft 1 and at the same time selecting the proper eigenvalue Y so that this second solution vanishes at. = 0O m The differential analyzer will, therefore, be utilized to solve the equation d2U m + s3 (1l-a) +Ym]Um O (1-24) d42 subject to the end conditions Um (0) = 0 (1-25) U(1) = h2 s(l-a +Ym (1-26) 1 dUm (1) "h2 s(l-a + Y ) (1-27) _______"_____s d — _____ " C m 1_5

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - The method of attack is to assume a trial Y, find U (1) and 1 dU~ ~m m - d —m (1) from theHarvard tables, and with these starting conditions at S 1, integrate toward j =o In general the resulting U (O) will not be zero. It is then necessary to assume new trial values of Y and rem determine U (O) in each case. In this way we can interpolate to the Y m which yields a solution for which U (O) = O, as requiredo 1.4 Review of the Status of the Mathematical Theory by Co Lo Dolph Although considerable time and effort has been expended on various aspects of the complex eigenvalue problem encountered in propagation theory, nothing like a satisfactory mathematical theory has yet been devised. Since there are a number of sources such as Sommerfeld (3), Kerr (4), Marsh (1), and Friedman (5) which develop this problem from the physical situation, this report will be limited to a few observations concerning ito In the course of examining the theory of anomalous propagation, an attempt was made to understand somewhat more clearly why the variational process introduced by MacFarlane (6) was capable of giving correct resultso An examination of the curves of Ament and Pekeris (7) who also used this same formal process led to the observation that the imaginary part of the eigenvalues, if different from zero, was of constant signo Subsequent investigation showed that the paper of Hartree (8) contained an argument, which, by a slight reinterpretation made it apparent that this must necessarily be the case. Although the usual Rayleigh quotient leads to a saddle point, the definiteness of the imaginary part of the eigenvalues has the nossibility o" leading to a one sided estimate for the imaginary part of the eigenvalues and a min-max principle as in the usual positivedefinite real case where a minimum is involvedo A corresponding estimate for the real part does not appear possibleo Moreover, it was shown that the usual theorems concerning the reduction of a real symmetric or hermitian quadratic form to the sum of squares by means of real orthogonal transformations were capable of generalization to this caseo Here the coefficients of the form are symmetric but complex-valued, and the complex sum of squares remains invariant provided that the matrix of the coefficients has a minimum function which possesses simple rootso It should be noted that ______________________6_ —----------------— 6

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN this condition on the roots can be shown to be automatically satisfied in the real symmetric or hermitian caseso It is interesting to note in the finite dimensional complex case that the proof of the above spectral theorem makes use of the fact that an orthogonal basis can be constructedo The vectors in it are orthogonal to each other in that the sum of their pairwise complex products vanishes so that a vector may be orthogonal to itselfo The fact that such an orthogonal basis can be chosen in an infinite dimensional srace has been established by JO McLaughlin at the University of Michigan. This result has apparently been obtained previously and independently by Kaplansky (9). The existence of this spectral theorem naturally led to consideration of possible infinite dimensional generalizations in a sequence space having an inner product of the above formo Such a generalization as well as a theory of operators in such a space appears necessary before the calculus of variations method can be considered rigorously establishedo A basic difficulty occurs at once in such a generalization in that the space of vectors with the property that the sum of the squares of their complex components is finite does not form a linear spaceo It is easy to see that the sum of two vectors of this space may not lie in this spaceo Although the subset of this space consisting of all vectors whose components are zero except for a finite number at the beginning does form an infinite dimensional linear sub-space9 it appears to be a difficult matter to complete this space in the proper wayo Work in this direction is still continuing but little progress is expected until the right completion has been foundo The difficulty can perhaps be sumrmarized by remarking that these considerations apparently lead to a conditionally convergent situation rather than the more usually treated one of absolute convergence0 As has been suggested by J. Dieuionne, one possible way out of the difficulty might be to employ the sraces of Kothe (10)o Here the basic approach would be to start with the denumerable set of eigenfunctions and build ur the largest complete space. That there are possibilities in this direction for the conditionally convergent situation has alreadyv been indicated by Kotheo Many researchers(fll) believe that the occurence of complex orthogonality in these problems can be best understood from the viewpoint of a hermitian inner product and the introduction of an adjoint differential problemo If this is properly done, the complex orthogonality can be shown 7

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN to result from the bi-orthogonal relation that exists between the given problem and its adjointo Friedman and his student, M. Kotik, have obtained some interesting unpublished results in this direction for a special class of problems; in fact they have been able to deduce a point-wise convergence theorem for some twice differentiable functions' Their results are also interesting in that they have investigated a case of isotropic eigenfunctions in some detail and have made a start toward an elementary divisor type of theory, R. So Phillips (ll), working in the usual Hilbert space framework, has given a discussion of second-order differential equations subject to complex-homogeneous linear boundary conditions, In this he has shown by use of Weyl's notion of the limit point and limit circle that in some cases the operator may be essential real while in others its spectrum may not be contained in any strip of one half of the complex eigenvalue plane, R. Phillips has a student continuing this work but he has reported that no progress has yet been made toward an expansion theorem. The basic tool in all of these investigations appears to be that of the resolvent, This can be constructed, formally at least, if all the eigenvalues can be confined to one-half of the complex planeo The use of the calculus of residues and Cauchyts integral formula are then available provided that the necessary estimates can be made. These have been successfully accomplished by Titchmarch (12) in the real case and work is continuing on the problem of finding appropriate estimates for this problemo From the above it is apparent that considerable doubt exists as to the proper framework for the complex eigenvalue problem even though it dates back to the work performed by Watson in 1910o Thus, no one knows for sure whether it would be better to view problems of this sort as nonhermitian, non-normal operators in the usual Hilbert space framework or whether it would be better to treat them as symmetric operators in a space with a symmetric complex valued inner product. It is also not known whether the difficulties encountered in the evaluation of the norm of the eigenfunctions of ref, 7 in the plane-earth approximation are basic or whether they are another manifestation of the anamalous behavior of the wave equation in two-dimensions. In any event there are many unanswered questions concerning the whole problem, In view of the quite wide spread interest and importance of problems of this type it is to be hoped that some answers will soon be forthcoming, The present writer has had the keen

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN interest and help of his colleagues, Dr. I. Marx and Dr. J. McLaughlin and present plans call for our continued collaboration on these problems. 9

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN CHAPTER 2 APPROXIMATE NORMAL-MODE SOLUTION USING ASYi:TOTIC FORMS 2o1 Solution for the Linear Gradient By use of the asymptotic forms of the modified Hankel-function solutions, we can get approximation formulas for the eigenvalues Y for the bilinear gradient problem. These approximate eigenvalues give an excellent point of departure for the differential analyzer solutions. Let us consider first the simplified case of a linear velocity gradient (a=O), The depth-dependent equation becomes from (1-18) and (1-19) d2( ) + s +Yj U(f) = O (2-1) d~2 with end conditions U(O) = 0 and U( o) -> outgoing wave, The solutions of equation (2-1) can be rewritten as 3 2 i }h~l4~j-Y j (2-2) U(Q) = h2 s( +Y) (2-2) where h2 is the modified Hankel function of the second kind and satisfies the outgoing-wave boundary condition at 5 =~ o To determine the eigenvalue Y it is necessary to impose the boundary condition U(O) = 0. The h2 j s(~ +Ym)l function given in equation (2-2) can be represented approximately by the first term of the asymptotic expansions for h 2 Since the imaginary part of the eigenvalue Y is known to be for h2o m positive,7 the argument s ( Y +Ym) always lies in quadrants 1 and 2, The asymptotic expression valid in this region is -1 -2 i3/ 12 5ji 2 1 ip+ +1 3 ip 1 h2(P) po( p e+ 1 3 p (2-3) or u( L)~ j 32 i+Y)}J }/2 e12 -3 i { Y + (2-4) ________________________________ 10 _____________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The boundary condition U(O) = 0 A.; met when -3/2 (sY) + 2 - 2mTT = 0, m50,;l,2,t2,.. (2-5) The equation for the eigenvalues Y is then m 2 2 r -(m - 3 i Yms L 2(m & e 3, m=192,,.., (2-6) where we have discarded m=O, and negative m values, since the imaginary part of Y is always positive. The accuracy of formula (2-6) can be seen in the following table9 which compares the eigenvalues Y gotten m from equation (2-6) with the exact values obtained by interpolation from the Harvard Tableso Comparison of Eigenvalues Obtained from Approximation Formula Equation (2-6) -ith Exact Values for Linear Gradient (a=Oj sY from Exact Value m Mode Equ (2-6) of sY 1 -1. 16+ 2.01 -10.170+i2 025 2 -2o 04+io` 53 -2 044+i3 540 3 -2 47 -2 6i6+iL8 -2.61+ 781 Evidenta.llyr the eigenvalies obtained from the first terms of the asymrptotic expressions are nuite accurate for the case of a linear velocity gradient. 2.2 Solution for the Bilinear Gradient Consider next the case of the bilinear velocity gradient, From equations (1-18) and (1-19) d2U + s3 [(la) +Y ] U=O 0 1 (2-7) d52 and dU + s3[ -a+Y UO 1'I (2-8) 11

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN with the end conditions U(O) = 0 and U(oo) --- outgoing waveo The solutions can be written as -2/3, -2/3 U(~)= Ahl {s(l-a) I (l-a)f +Y] Bht 2 (-a) 2 1-a) +Y 0 _S -1 (2-9) and U(Q) Ch2 {s( -a+Y) I (2-10a) Note that for f>1 orlv the h2 function is included, since it satisfies the outgoing-wave boundary condition at = C o The modified Hankel functions given in eouations (2-9) and (2-10) can be represented approximately by the first terms of their asymptotic expansionso Again since the imaginary part of Y is always positive, it is necessary to use the expansions for h1 and h2 valid in the first and second quadranto Taking only the first terms of those expansions, we have 3/2 1 2. TT hl(p) = p e (2-11) and equation (2-3) for h2(p)o Thus 3/2 3/2 3/2 3/ -1/4 3-1 -a [l-a) 1-a1 y U(S)' A' [(l-a) +YJ e o0 f1l (2-12) where the phase constants in the exponents have been included in the constants A' and B'o Differentiating equation (2-12) and neglecting the contribution of d ~ (l-a) +Y ] /, we have d2 2L 3/2 3/2 23/2 34 2 3U1/2 l/a -fil-a)g +Yi -1- +Y dU( ) is Alp(l-a)+Y e +B'e 0of- 1 (2-13) 12

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN - In the rerion > 1, the arnroximate solution is is2 3/2( 3/2 2 3/2 )3/2 -].,/4 - 2is (f-a+Y) is (~-a+Y)3 U($) CT (S-a+Y) Le +ie' 1 If we take dU(_ ) and neglect. d (f-a+Y): 4 as before, it follows that d- d 39 /2. 3 y)3/2 2. 3/2 3/k 2(_a+Y)3 1 S /2 (d~~~2ils (-a+Y)[-e = C Vis (f-a+Y) e +ie J >1 (2-15) We must now match at J =1 the solutions for U(s) and U' (S) for 04-l with the solutions for U(%) and Uv(~) forfl. Since our task here is to find the eigenvalues Y for which U(O) vanishes we are not concerned with evaluating the constant AMwhich sets the magnitude of the solution in the region 0-f^l1, but only with evaluating the constant BI in equation (2-12). The constant B? determines the ratio of U(1l)/U(l) from equations (2-12) and (2-13). This ratio must equal the U'(l)/U(l) given by equations (2-14) and (2-15), which is.3/2 3/2 Lics (1-a+Y) 3/2r 3 U'l) is 3 Lie -1 If (2- 6) U l --------— ^ —--— ^-/ —--- 21)2 32 3is (1-a+Y) [ie +1 J On the other hand, from eauations (2-12) and 2-13) 3/2 3/2 3/2 3/2 3,/2r ai a (la+Y) is (1-a+Y) U(I) = i(s Be3 a e -1(217) U I (1.) 4. as 4 -(l-a+Y) eis (1-a+Y) Be + Bqe e +1 After comparison of equations (2-16) and (2-17) it is obvious that -i as3/2 (l-a+Y) 3/2 3 1-a Bt = ie (2-18) 13

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN which, substituted in equation (2-12), gives 3/2 3/2 3/2 3/2 3/2 3/2 U-1Y -2 4 -li las (1-a+Y) 2i -i s 1 (0) Yi3 1-a i3 1la.... U(O) A,~ e +le e 3/2 3/2 3/2 3/2 3/2 3/2 -1 -2. s Y as -a+ + a.s Y 1i ) -4a -a a + ~i =AY4 e Na -e 3 1-a 3 1-a 2 (2-19) From the boundary condition U(O)=O it follows that 3/2 3/2 3/2 3/2 -4i as (l-a+Y) + S i- Tli -4i as (-a+Y) + i sl- _ Ti + 2mTi=O m=,+l,+2,..., 3 1-a 3 1-a 2 and 3/2 3/2 Y - a(l-a+Y) + 3TT (a) (1 -m) = 0 (2-20) 2 S 372 4 s For the linear velocity gradient(a=0), equation (2-20) reduces to 2/3 2WT Y 1 T 3(m-) 1 3.(2-6) s 2which agrees with our previous result for a = 0 from which we conclude that m=1,2,3,ooo In equation, (2-20), then, m=l yields the eigenvalue Y1 corresponding to the first mode, m=2 yields Y2 for the second mode, etc. Equation (2-20) is only an apnroximation, and its usefulness should not be overemphasized, since it can be solved for Y only by trial mnd error, in any case~ However, it leads to an important simplified formula for Y when Y ~l-ao In this case we can write, m m ___________________________________ 14

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 3/2 3/2 1/2 (Y +a) Ym + m (1-a) (2-21) m M 2 m Substituting this equation into (2-20), we obtain 3/2 1/2 3T 1 TT Y - aY l_ e (2-22) m 2 sm 2s 3/2 Taking the 2/3 power of both sides of equation (2-22), we find that 3/2 1/2 2/3 2/3 2 (2-23) 3 3fC^2 (hz- 1 r3TI ^(2-23) (a -Y aYm 3 -i In J 2/3 2,~l [3tm 1I 1; ~ ~'3~:T or Y= f V( )J e 3 +a, Y?>a, Y>].. (2-24) Thus for large eigenvalues (as is the case when s << 1 or when the mode number r is large), equation (2-24) gives a very simple approximate formula for the eigenvalueo To illustrate this, the following table compares the eigenvalues Ym obtained from equation (2-24) with the exact values (subject, of course, to computer errors) obtained with the differential analyzer Comparison of Eigenvalues Obtained from Approximation Formula (Eau, 2-24) with Computer Valueso Differential Differential Mode Equo(2-24) Analyzer Mode Equ.(2-24) Analyzer s=0.5 1 -1o22+i4o02 -1o23+i4o05 s=0.5 1 -0.32+i4o02 -0.32+i4.O a=ll 2 -2098+i70O6 -2,98+i7008 a=2o0 2 -2O08+i7.06 -2.06+i7.0O 3 -4.42+i9o56 4o041+i9o56 3 -3.52+i9.56 -3.50+i9.56 s=l.O 1 -0006+i2001 0o03+i1198 s=loO 1 +0O84+i2.01.01+il.91 a l.1 2 -0o 9+i3o 53 0 o 6+i3o45 a=2 0 2 -0.04+i3 53 0.13+i3.40 3 -1 o66+4o78 -lo57+i4 o 68 3 -0.76+i4.78 -0.65+i4.5% Evidentally for s ~ Oo5, equation (2-24) is an excellent approximation to Ym, 15

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - CH APTER 3 PRINCIPLES OF OPERATION OF THE ELECTRONIC DIFFEREN1TIAL ANALJYZER 31o Introduction to Operational Am.plifiers The basic component of the electronic differential analyzer is the operational amnlifier, which is shown schematically in Figure 3-1. It consists of a dc voltage amplifier of high gain, an input impedance Z., and a feedback impedance Zfo Z f e, L Z e D.Co APLIFIER eo I Figure 3-1> Operational Amplifiero If we neglect the current into the do amplifier itself (i.eo neglect the current to the grid of the input tube), it follows that i1 = i2o Let us also neglect the voltage input e' to the do amplifier in comparison with the output voltage e2 or the input voltage eI to the operational amplifier, We then have 1 or e e2 Zf' ZZ from which Z ^e2 -Z~ e1 s (3-1) I —---------- ~~~16' -----------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN which is the fundamental equation governing the behavior of the operational amplifier~ In general Zf/Zi is made the order of magnitude of unity, We shall now consider the scheme by which the operational amplifier can be used to perform three different functions: (a) Multiplication by a constanto If we wish to multiply a certain voltage el by a constant factor k, we need only make Zf/Zi k. From equation (3-1), then, the output voltage e, of the operational amplifier will be given by e2 = k eL (3-2) Thus the required multiplication by a constant has been achieved, except for a reversal of sign, For example, if we wish k to be 10, we may let Z. = 1 megohm resistance, Zf 10 megohms resistance. If we also desire the sign of e2 to be the same as e1, we must feed e2 through an additional operational amplifier with Z = Zf = 1 megohm. This second operational amplifier merely acts as a sign changer by multiplying any voltage by -1. (b) Additiono In order to add a number of voltages, say ea, eb, and ec, the arrangement shown in Figure > 2 is usedo Hence i + ib + i = i, and a b c 2 if we La +- Z Z 4 ea I Lb em Z"-b* g' ~ DCo o AMPLIFIER eb e l 1' —-i Figure 3-2o Operational Amplifier Used for Summation. 17

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN neglect e' as small compared with e2, we have ea eb e e2 Za Zb Zc Zf or e2 - ea +z b + (3-3) 2 Za a Zb c Z Thus the output voltage e2 is the sum of the three input voltages, each multiplied respectively by a constant - -(n = a,b, or c). The operational amplifier can, of course, be used En general to sum any number of input voltages, (c) Integrationo If we make the input impedance Z. a resistor and the feedback impedance Zf a capacitor, then the operational amplifier serves as an integrator. Referring to Figure 3-3, we see that if we neglect e' and let ii = i2 as before, we have (i dt e e2- JC - and il - from which e2 -RC e dt (3-4) The output voltage e2 is then the integral with respect to time of the 1 input voltage el (multiplied by a constant factor - RC- ). / /- D.C.AMPLIFIER e. e' Figure 3-30 Operational Amplifier as an Integrater --------------------— 18 ____________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 3.2 Solution of an Ordinary Differential Equation with Constant Coefficients In order to demonstrate how operational amplifiers performing the above three functions can be combined to solve ordinary linear differential equations, we will now set up the amplifier circuits required to solve the following differential equation: d2x dx a + a dt x = O (3-5) dt subject to the initial conditions x(O) = V and d (0) = V2 (3-6) dt 2 The constants a2, al, and a are assumed positive. Since the electronic differential analyzer integrates with respect to time, the independent variable t in equation (3-5) above will be timee The dependent variable x is represented by voltage. The computer circuit for solving equation (3-5) subject to initial conditions (3-6) is shomw schematically in Figure 3-4o II II III r-v I B, *..A I-~-' All Resistor Units are Megohms All Capacitors are lMfdo Ground Connections are Omitted for Clarity Figure 3-40 Computer Circuit for Solving a2x + alx + a x = O. 19

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN If we assume that the output of amplifier A is a2x then this voltage gets multiplied by - - and integrated once in passing through amplifier~ A3, the output of whic? is therefore -Xo This voltage is multiplied by -1 and integrated once to give x as the output of A4o In order to obtain +x instead of -x it is necessary to pass -x through sign-reversing amplifi Al x and x are then multiplied by -al and -a respectively and summred in amplifier A2, the output of which is now -alx -a X But we originally assumed the output of A2 to be a2xo Hence, a2x = - alx -a x, vhich is jus: the equation which we wish to solveo The initial conditions (3-6) are imposed as voltages impressed across the integrating condensers of A3 and A, in Figure (3-4)o When the 3 4 two switches holding the initial voltages across the condensers are simultaneously opened, the solution of the problem as a function of time begins, i.e., the voltage output of Al represents x(t)o 3.3 Solution of Differential Equations with Variable Coefficients Suppose the coefficients a2, a1, and a in equation (3~5), instead of being constant, are functions of the independent variable to Then it is apparent that the resistors marked a2, 1/al, and 1/a in Figure 3-4 must vary as a function of time6 If we can accomplish this, then the differential analyzer can solve the more general problem of ordinary differential equations with coefficients which are functions of the independent variable [for example, the bilinear gradient equation (1-18)]o It is often more convenient to vary a resistance with time in discrete steps instead of continuouslyo 1415As an example, a resistance varying linearly with time can be approximated by the staircase function shown in Figure 3-5o The staircase function is arranged so that at the end of each time interval At the area under the stepped curve is equal to that under the linear curveo The accuracy attainable by using such a crude approximation is surprisingly good, even when the time increments A t are made relatively largeoL It is this step-method of approximating variable coefficients which is used in obtaining the differential 20

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN analyzer solutions to the bilinear gradient problem. For a complete description of the circuitry involved, the reader is referred to other reoorts.,15 6 - -LINEAR FUNCTION - STEP APPROXIMATION TIME t Figure 3-5. Sten-Method of Arproximating a Linear Function, 21

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN CHAPTER 4 SOLUTION OF THE BILINEAR GRADIENT PROBLEM BY THE ELECTRONIC DIFFERENTIAL ANALYZER 4.1 Transformation of the Equation into Computer Units In the bilinear gradient differential equation (1-18) the independent variable ~ starts with the value 1 and runs to the value 0. Actually, the computer independent variable, which is time t, will start at t = 0 and run to t = 1, where one computer time unit is now the length (or duration) of the solution. Thus we make the following change of independent variable 1 - t (4-1) and 2 2 d -d d2 d2 = - (4-2) d dt' d 2 dt2 Equation (1-1) remains du(t) + s3 f(t) + Y U(t) = 0 (4-2a) dt2 but now f(t) = (l-a) (1-t), 0 - t 1lo (4-3) 4.2 Separation into Real and Imaginary Parts We have already pointed out that in general the wave potential U is complex, as is the eigenvalue Yo To solve the problem with the differential analyzer it is necessary to break the complex functions into real and imaginary parts. Thus we let 22

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - U I U + iU. (4-4) r 1 Y = Y + iY (4-5) r 1 After substituting (4-4) and (4-5) into equation (4-2a) and equating real and imaginary parts to zero, we find that d2 U dt..r + f(t)u + Y U Y.U. = (6) 3 2 r r r 1 i and d2U. I- + f(t)U. + Y U. + Y.U 0 (7) 3 2 1 ri ir s dt Initial conditions for the above equations become from (1-26), (1-27) and (4-1) Ur(0) = R [h2 {s(l-a+Y)}] Ui(O) = I [ h2 Is(l-a+Y)] (4-8) - U r(0) = R Lh'2 s(l-a+Y)l] s i(O) =I [h'2 s(l-a+Y)3 4.3 Computer Circuit for Solving the Bilinear-Gradient Problem The electronic differential analyzer circuit used to solve equations (4-6) and (4-7) for the case when a = 0 (linear gradient) is shown in Figure 4-lo Note in this case that Y < 0 and Y. >0O The r l circuit required for a21 (bilinear gradient), Y r0, Y > 0 is shown in Figlre 4-2. If Y < O, it is only necessary to restore the R/sY input to the positions shown in Figure 4-1l 23

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN.R~"Y C'(-t) C I A, A- -- (-f(t)U 1l - R/2 Y c(L- t) C C C R/s a=o, Y<o, Y.>o >/5'_ Figure 4-1.... R/2 /52 Yt, _,,, u,>o'/~......... a>1, y. > o, Y,: >o FR/s2Y I Figure 4-2. 2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The (l-t) factor in the f(t) function of equation (4-3) is incorporated into the feedback resistor of amplifiers Al and A5 as a staircase type of function (see Section 303). The interval 0 t 1l is broken into 20 equal time intervals, The first resistance step is 0~975 megohms, etco, finally down to 0~025 megohms on the 20th stepo Thus the constant CI in Figures 4-1 and 4-2 is actually unity0 The initial-condition voltage circuit is omitted in Figures 4-1 and 4-2 o Actually, the initial conditions are applied in a somewhat different manner than that shown in Figure 3-4o6 The unit of computer time is RC seconds, so that one unit of t corresponds to RC seconds in real timeo For the work done on the bilinear gradient, a feedback capacity C of 5 microfarads was employed, along with an R value of 2 megohms, Thus one unit of t corresponds to 10 seconds, and the length of a computer solution is 10 secondso The interval of time between sters on the staircase resistance simulation of f(t) is 10' 20 or 005 secondso 4.4 Measurement Techniques Initial conditions were set in as voltages across the integrating capacitors by reading the output voltages U r Ui -s r9 - with a -r 1r t r U I type K-2 Leeds and Northrup potentiometer, This allowed the voltages to be set with a precision of 0o01l Since the K-2 potentiometer can only measure voltages up to 1l6 volts whereas we might wish to set in initial voltages as high as 100 volts, a potential divider arrangement was connected across each output whenever that particular voltage was reado Then only about 1/50 of the actual output voltage was read by the potentiometero The general technique for obtaining the normal modes was discussed at the end of Section 1,3. It involves measuring Ur (5 ) and Ui( ) when ~ 0, or for the computer variable t, measuring Ur(t) and U,(t) when t = l1 The computer voltages representing Ur(l) and Ui(l) are measured by actually stopping the solution at the end of one unit of computer time (10 seconds of real time) and by reading the U and U. voltages held on r 1 integrators A3 and A7 respectively, These readings can again be made with the K-2 potentiometer to a high precisiono The integrators are made to "hold" their output voltages by means of a relay which disconnects the 25

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN input resistors to the respective dc amplifierso This "hold" relay is energized 10 seconds after the solution is begun by control circuits originating in the synchronous equipment running the resistor steps (see Ref 16 for complete circuit descrinti.on). If we have chosen the eigenvalue Y + iY, properly9 then U (1) and U,(l) should both be zero (corresnonding to zero wave potential at the surface). Actually, a finite value for U (1) and Ui(l) will in general exist, and the interpolation method described in the next section must be employed to find the correct eigenvalueo The resistors in the computer circuit were calibrated to the order of Oo01% accuracyo Capacitors were calibrated by connecting three amplifiers (two integrators and one summer) to solve the equation (RC)2X + x = 0 and by measuring very accurately the resulting period of sinesoidal oscillation15 A 100 kilocycle frequency standard stepped down to 12,2,5,5, or 10 cycles per second was utilized as a time reference and as a means 16 for driving the resistor-stepping equipment Most of the solutions were obtained with the amplifiers balanced manually; in the unit being delivered to the Navy the amplifiers are drift stabilized9 and frequent rebalancing should be unnecessary0 26

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN CHAPTER 5 COMPARIOISON OF COkPUTER RESULTS'.'IThTH THEORETICAL RESULTS FOR A LINEAR GRADIENT 5o1 Theoretical Solutions for the Linear Gradient In order to check the solutions of the electronic differential analyzer, we considered first the problem of a linear gradient (no discontinuity)o This is equivalent to letting the parameter a equal zero in equation (1-19), so that now f(t ) = 5 over the entire range in depth variable The Hankel function solution (1=23) which in general is valid only for >l is now valid for the thole range 0 < ~<ofor a = Oo We have seen in Section 21l that in this case the eigenvalues Y are given m anproximately by 2/3 2 T /3 y [ (m- e (2-6) Thus for a given value of the parraeter s wne can calculate theoretically the approximate eigenvalues Y and compare these eigenvalues with those obtained by the computero Furthermore, for the a=o case we can directly cross-check the U and U1 values obtained from the computer with the entries in the Harvard tables 5.2 Method of Interpolationto the Exact Eigenvalues The computer was set up to solve the a - 0 case in the range 0.- lo The circuit of Figure 4-1 was used, and initial voltages were set in according to equations (1-26) and (1-27)o In order to compute these initial voltages it was necessary to go to the Harvard tables of the h2(z) function, where z = x + iy. The increments in x and y for these tables are 0.1, and since it is inconvenient to interpolate the functions, the smallest increments in assumed eigenvalues Y which we used were 0.1/s, Thus for s = 1, Y1 (for the first mode) should be - 1.16 + i2o02 according to eouation (2-6), Instead of trying to get the computer solution for Y; -lo16 + i2,02 (this necessitates the determination from the tables of 27

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Complex U(o) Plane- I 1r nerpola-ioo s= O, F;'rs lMode Y =-I.168 + L 2.023............. — o.3 _- ------ 0 tt/ I -o. 3 >2 -0 -:o*2/ o. r / o|l.oS0.ii~ ~~~ Figure 5-1 Interpolation Diagram.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN h2 [- 0.16 + i 2.02] ard h2 =[0.16 + 12.02] ), we obtained computer solutions for Y = - 1.1 + i 2.0, - 1.2 + i 2.0, - 1.1 + i 2.1, and - 1.2 + i 2.1. This entailed setting in initial conditions involving h (z) and h' (z) where z = - 0.1 + i 2.0, - 0.2 + i 2.0, - 0.1 + i 2.1, and - 0.2 + i 2.1 respectively, which involve no interpolation in the tables. By bracketing the true eigenvalue Y1 in this manner we ought to be able to interpolate to Y. This has been done in Figure 5-1, where U(O) for various assumed eigenvalues Y has been plotted in the complex U plane. In each case the U(O) value was taken from U and U. computer solutions using the r 1 technique described in Section 4o4. The eigenvalue obtained from the differential analyzer results shown in Figure 5-1 is Y1 = - 1168 + i 2.023 compared with a theoretical value of - 1.16 + i 2.01 from equation (2-6). Since equation (2-6) is only approximately true, it seemed desirable to check the computer Y1 against a more exact theoretical value for Y1 With this thought in mind a plot similar to Figure (5-1) was made using the values of U(O) obtained from the Harvard tables. By interpolation from this graph a value Y1 = 1.170 + i 2.025 was obtained, which shows better agreement with the value obtained from the electronic differential analyzer, 5.3 Comparison of Differential Analyzer Solution and Harvard Tables.!Te have seen in the Previous section how, by starting at f = 1 and integrating to 0 = O, we can obtain U (0) and U.(0) from the differential r 1 analyzer for a given Y + iY.. By chosing a number of different values r 1 for Y + iY., we can make the plot shown in Figure 5-1 and interpolate to the correct eigenvalue for which U (0) - Ui(O) = 0. A check of the analyzer solution for a = o, s = 1 with the theoretical eigenvalue showed agreement to the order of 0.1%. A more direct check of the computer accuracy is obtained by comparing computer values for U(O) and U'(0) for a given Y + iY. withthe equivalent entries in the Harvard tables. This has been r 1 done in the following table: Comparison of U(O) and U'(O) as Obtained with the Differential Analyzer with Values from the Harvard Tables, s = 1, a = o, First Mode 29

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Y U(O) u'(o) Differential Analyzer -l.l+i2.0 -0.098+iO.110 -1.819+il.075 Harvard Tables -l.l+i2.0 -0.100+iO.119 -1.829+i1.070 Another method of checking the accuracy of the differential analyzer involves solving the linear gradient problem exactly. With s = 1, a = o, for example, interpolation from the Harvard tables indicates that Y1= -1.170 + i2.025. If we wish to run the solution from f = 1 to.= 0 with this eigenvalue, we must find h2 (-0.0170 + i2.025) and h2' (-0.0170 + i2.025) from the tables To do this, the following interpolation formulas are used: h2(z0+t) = h2(zO) + h2(z )t (5-1) and h2 (zO+t) - h2(zO)t + h2(zO) (5-2) where z0 is the table entry and z0 + t is the desired entry. Equations (5-1) and (5-2) are accurate to the order of 0.l/1 or better. The initial conditions obtained from these equations, along with the correct eigenvalues, are then set into the computer. The computer solution for U (0) and Ui(0) should now vanish, The table below shows the results obtained with the differential analyzer to corroborate this, Summary of Differential Analyzer Results, Linear Gradient(a=o,s=l), for Correct Eigenvalues. Solution Started at, = lo y (Interpolation A Analyzer Solution Mode,from Harv, Tables U (0) Ui(O) 1 -1.170+i2.025 -0,0055 -0o0005 2 -2.044+i3.540 -0,0027 +0.0100 Evidentally U(~ ) practically vanishes at 0 = 0 as required, when the correct eigenvalue is utilized. Having thoroughly checked the differential analyzer solutions against the theoretical solutions for the linear velocity gradient(a ~ o), we next proceeded to solve the bilinear gradient problem with the analyzer. __________________________ 30 __________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN CHAPTER 6 DIFFERENTIAL TAALYZER SOLUTIONS TO THE BILINEAR GRADIENT PROBLEM 6.1 Determination of the Eigenvalues In Section 5 we described the method for obtaining differential analyzer solutions when a = 0 (linear gradient). The method of solution for a 7 0 (i.e., a bilinear gradient) is exactly the same. The approximation formula (2-24) for the eigenvalues is utilized as a starting point, Note that this formula can be written as Y = Ymo + a, (6-1) m mo where Y is the eigenvalue for the mth mode when a = 0. For a given mo a and s the eigenvalue from (6-1) is computed, and then bracketed with integral eigenvalues so that the corresponding initial conditions given in equations (1-26) and (1-27) can be looked up directly in the Harvard tables. Note that the computer variable t = 1-, so that - 1 d = Hence s dt s d Hence the initial condition applied to amplifiers A? and A6 in Figures (4-1) and 4-2 is merely given by h2' s(l-a+Y )], as shown in equation (1-27)o Analy7er solutions for each of the trial eigenvalues are run off, and U(O) is recorded by "holding" the solution 10 seconds after it has begun and by reading U and U. with the K-2 Potentiometer, A plot similar to that r I in Figure 5-1 is made, and the correct eigenvalue for which U(O) vanishes is obtained by interpolation. This eigenvalue, along with the appropriate initial conditions interpolated from the Harvard tables with equations (5-1) and (5-2), is set into the differential analyzer and a solution run off. This solution should then represent an exact solution to the problem, that is, U (O) and Ui(O) should vanish. Again this is checked with the potentiometer while the solution is being "held" at 5= 0 (t=l)o At the same time, 1 U' (O) and 1 Ui.(O) are carefully recorded with the K-2 9 ~s r s i potentiometer. By knowing the derivitives of the wave potential at the surface, we can later turn the problem around backwards and integrate from the surface on down. 31

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The reader is referred to Appendix I for a complete sample calculation for a given a and s. Eigenvalues were obtained for a = 1.1 and a = 2 and s values of 0.5, 1, and 2. The results are tabulated in Appendix IIo For s <0o5, the eigenvalues are given quite accurately by equation (6-1), and the normalmode solutions are practically identical with those for s = 0.5, except for the necessary scale change in independent variable. For s = 2 the solutions are somewhat critical, particularly for the higher modes. There is some question as to how practical it would be to attempt solutions with the differential analyzer for s values much above s = 2. The asymptotic solutions ought to get better for these high values of s, however. 6.2 Rerun of Analyzer Solution from the Surface on Down Originally it seemed desirable to have plots of the U(s) and U'(. ) functions over the range 0- 5 42o The method for finding the eigenvalues involves differential analyzer solutions from & = 1 to S = 0. However, by recording the derivitive of wave potential at the surface (i.e., 1 dU(0) we are able to turn the problem around backwards on the analyzer and rerun it from 5 = 0 to &= 2, since now we know the eigenvalue and all initial conditions at the surface CU(O) =.0 Note that the computer time variable t is now actually J, and not 1 =. The only change needed in the computer circuit involves readjustment of the f(t) function which appears as a variable feedback resistor in amplifiers A1 and A5 of Figures (4-1) and (4-2)o The f( ) given in equation (1-19) is shown in Figure (6-1)o For a = 2 the function f(S) is always negative, and hence - f(3) can be represented by the staircase approximation described in Section 3.3. For a = 1.1, the function f( ) changes sign, but by setting up 0.1 + f(s) with the stair case approximation, we are able to simulate a function which is always positiveo The net result is an output [- 0.1 - f( )J U from amplifier Al, to which OolU must be added to obtain ( )U. A similar method is utilized for obtaining f( )Ui when a = 1.1. Again a computer time constant of 10 seconds was employed, so that now the length of computer solution is 20 seconds, corresponding to 32

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN j going from 0 to 2. A total of 20 steps were utilized to simulate f(' ) as before, but in this case the resistor values were changed once per second instead of twice per second. The U ( ), Ui(E ), - 1 U'( ) and - 1 U.'(5 ) curves obtained r s r S'I from the electronic differential analyzer for 0 of 2 are shown in Appendix III. The first three modes for a = 1.1 and 2, and s = 0.5,1, and 2 are included. The computer output voltages were recorded with a Sandborn, Model 60, Two-Channel Recorder. Accuracy of the recordings should be the order of 2% of full scale. Io I. O 0. _ Figure 6-1. Bilinear Velocity Gradient. 33

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN BIBLIOGRAPHY 1. Marsh, H. W. Jr., Theory of Anomalous Propagation of Acoustic Waves in the Ocean, USL Report No. 111. 2. Harvard Tables, Vol. II (1945), Modified Hankel Function of Order 1. 3. Sommerfeld, Partial Differential Equations, Appendix II 4. Kerr, D.E., Propagation of Short Radio Waves, MIT Radiation Laboratory Series, Vol. 13. 5. Friedman, B., "Propagation in a Non-Homogeneous Atmosphere," Communications on Pure and Applied Mathematics, Volo IV, No0 213, August, 1951, 6. MacFarlane, G., "Variational Method for Determining Eigenvalues of the Wave Equation of Anomalous Propagation," Proco Cambridge Philo Soc., 43 (1947), p. 213. 7. Pekeris, C. LO, and Ament, S., "Characteristic Values of the First Normal Mode in the Problem of Propagation of Microwaves through an Atmosphere with a Linear Exponential Modified Index of Refraction," Phil. Mag, Sec. 7, 38 (1947), p. 801o 8. Hartice, D. R., et alo, "Meteorological Factors in Radio-Wave Propagation," Supplement, Physo Soco and Royal Meteorological Soc,, London, 1946. 9. Kaplansky, I., "Forms in Infinite-Dimensional Space," Anais Acad. Brasil, Ci. 22 (1950), pp. 1-17; Math, Reviews, April, 1951, p. 258o 10. Kothe, Go, and Toeplitz, O,, "Lineare Raume mit Unendlichvielen Koordinaten und Ringe und Unendlicher Matricen," Journal fur Mathematik, Vol. 171, po 193. 11. Phillips, Ro S., Linear Ordinary Differential Operators of the Second Order, New York University, Research Report No, EM-42, April, 1952. 12. Titchmarch, E. Co, Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford Press, 1946. 13. Kapur, PO L., and Peierls, Ro, "The Dispersion Formula for Nuclear Reactions," Proc. Royal Soco, Series A, Vol. CLXVI (1938), pp. 277-295. 14. Hagelbarger, Howe, and Howe, Investigation of the Utility of an Electronic Analog Computer in Engineerng Problems, University of Michigan, Engineering Research Institute, External Memorandum UMM-28, April, 1949. 34

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN BIBLIOGRAPHY (Cont'd) 15. Howe, Howe, and Rauch, Application of the Electronic Differential Analyzer to the Oscillation of Beams, Including Shear and Rotary Inertia, University of Michigan, Engineering Research Institute, External Memorandum UMM-67, January, 1951. 16. Howe, R. M., Theory and Operating Instructions for the Air Comp Mod 4 Electronic Differential Analyzer, University of Michigan, Engineering Research Institute, ONR Contract N6 onr 23223 Report AIR-4...... 35 --------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX I SAMPLE CALCULATION OF EIGENVALUES AND EIGENFUNCTIONS Third Mode s = 1.0, a = 2.0 From equation (6-1), Y3 -- 0.76 + i4.78 Initial Conditions from From Analyzer Trial Eigenvalues Harvard Tables Solution Y WU(i) U'(i) U(O) 3 ~ - 0.6 + i4.5 1.196 + i3.650 6.509 + i4.765 - 0.110 + i0.163 - 0.7 + i4.5 0.632 + i3.180 4.825 + i4.607 + 0.120 + i0.152 - 0.7 + i4.6 0.091 + i3.645 4.415 + i6.226 + 0.116 - iO.087 - 0.6 + i4.6 0.624 + i4.294 6.294 + i6.722 - 0.120 - iO.074 From interpolation plot similar to Figure 5-1, Y3 = - 0.650 + il4.566. For an exact solution, U(l) = h2(-1.650+i4.566), s U'(1) = h2'(-1.650+i4.566). In equations (5-1) and (5-2), let z = - 1.7 + i4.6, t = 0.050 - i0.034. Then U(l) = h2(z +t) = (0.091+i3.645)+(4.415+i6.226)(0.050-iO.034) = 0.524+i3.806 U'(1) = h'(z +t) = (4.415+i6.226)+(o.091+i3.645)(1.7-i4.6)(0.050-iO.034) s 2 o = 5.459+i5.941 Using these values for intitial conditions when Y3 = - 0.650 + i4.566, we find from the analyzer solution U(O) = - 0.012 + i0.007 (ideally, U(O) should vanish) and U'(O) = - 1.515 + i2.450 s To solve the problem in reverse, we use U(O) = 0 and - - U'(0) = 1.515 - i2.450 as initial conditions for the analyzer. We then proceed to integrate from t = 0 to t = 2 (i.e., from = 0 to 0 = 2). Al

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX II SUMMARY OF COMPUTER RESULTS ~YY.^ U v(0) - - U (0) s Mode a r s rs ui 0.2 1 0.0 - 5.850 10.125 - 1.835 1.060 1 1.1 - 4.750 10.125 - 1.835 1.061 1 2.0 - 3.850 10.125 - 1.836 1.062 2 0.0 - 10.22 17.70 2.121 1.214 2 1.1 - 9.12 17 70 2.121 1.214 2 2.0 - 8.22 17.70 2.121 - 1.214 3 0.0 -13.805 23,905 - 2.279 1.300 3 1.1 -12.705 23.905 - 2.279 1.300 3 2.0 -11.805 23.905 - 2.279 1.300 0.5 1 0.0 - 2.340 4.050- 1.835 1.060 1 1.1 -1.225 4.049 - i868 1o088 1 2.0 - 0.315 4.048 - 1.923 1.123 2 0.0 - 4.088 7.080 2121 - 1214 2 1.1 - 2.978 7.078 2.165 1,257 2 2.0 - 2.062 7.077 2.197 - 1.295 3 0.0 - 5.422 9.562 - 2.279 1.300 3 1.1 -4.408 9.558 - 2.322 1o341 3 20 - 3.500 9.556 - 2.375 1.367 1.0 1 0,0 - 1.170 2.025 - 1.835 1.060 1 1.1 0.032 1.980 - 2132 1.581 1 2.0 1.011 1.910 - 2.155 2 044 2 00 - 2.044 3.540 2.121 - 1.214 2 1.1 - 0.860 3.450 2.123 - 1.884 2 2,0 0.130 3.40 1.933 - 2.594 3 0.0 - 2.761 4.781 - 2279 1.300 3 1.1 - 1.573 4.677 - 2.056 2.126 3 2.0 - 0.650 4.566 - 1.515 2.450 2.0 1 00 - 0585 1,012 - 1.835 1.060 1 11 0.44 0.46 - 0.293 1.513 1 2.0 1.09 0.225 - 0.401 1.668 2 0.0 - 1.022 10770 2.121 - 1.214 2 1.1 - 0.161 1.468 0o580 - 00683 2 2.0 0.679 1.400 0.679 - 0o657 3 0.0 - 1.380 20391 - 2.279 1.300 3 1.1 - 0.583 2.171 - 0.553 0.332 3 2.0 0.263 2.173 - 0.656 0.298 --- A2

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