ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Technical Report A SIMPLIFIED METHOD FOR COMPUTING UPPER-ATMOSPHERE TEMPERATURE AND WINDS IN THE ROCKET-GRENADE EXPERIMENT Joseph Otterman Department of Aeronautical Engineering Approved: L. M. Jones ERI Project 2387 DEPARTMENT OF THE ARMY PROJECT NO. 3-17-02-001 METEOROLOGICAL BRANCH, SIGNAL CORPS PROJECT NO. 1052A CONTRACT NO. DA-36-039-SC-64659 FT. MONMOUTH, NEW JERSEY June 1958

The University of Michigan * Engineering Research Institute TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iii ABSTRACT iv THE UNIVERSITY OF MICHIGAN PROJECT PERSONNEL v 1. INTRODUCTION 1 2. NOTATION 2 3. OUTLINE OF THE METHOD OF THE DATA REDUCTION 5,4o CALCULATION OF THE CHARACTERISTIC VELOCITIES FROM THE TIMES OF ARRIVAL AT THE MICROPHONES 7 5. ROTATION OF THE CHARACTERISTIC VELOCITIES 16 6. THE LAW OF REFRACTION EXPRESSED IN COMPONENTS OF VELOCITIES IN A CARTESIAN COORDINATE SYSTEM 19 7. EQUATIONS FOR TRACING THE SOUND RAY THROUGH A LAYER WITH KNOWN METEOROLOGICAL PARAMETERS 22 8. EQUATIONS FOR METEOROLOGICAL PARAMETERS IN A LAYER WITH A KNOWN SOUND-RAY PATH 24 9. THE DETERMINATION OF TEMPERATURE FROM THE VELOCITY OF SOUND 25 10. DIGITAL COMPUTER PROGRAM 27 11. CALCULATIONS AND RESULTS FOR SM1.01 AND SM1.02 ROCKET FLIGHTS 29 12. COMMENTS ON THE EXPECTED SOURCES OF ERRORS 29 13. CONCLUSIONS 39 14. ACKNOWLEDGMENT 40 ii

The University of Michigan ~ Engineering Research Institute LIST OF ILLUSTRATIONS Table Page I SM1.01 Data 30 II SM1.02 Data 31 III SM1.02 Balloon-Layers Data 32 IV Computer Output: Tracing of the Sound Ray 33 V SM1.01 Results 34 VI SM1.02 Results 35 Figure 1 Tracing of sound rays through successive layers. 8 2 The microphone array at Fort Churchill, Manitoba Province, Canada. 9 3 Rotation of the characteristic velocities. 17 4 Characteristic velocities in a Cartesian coordinate system. 21 5 The flow diagram for the computer program. 28 6 SM1.01 results. 36 7 SM1.02 results. 37 iii

The University of Michigan * Engineering Research Institute ABSTRACT This report describes a simplified method for computing temperatures and winds at high altitudes from data obtained as results of the rocket-grenade experiment. The method is exemplified by applying it to the data from SMl.01 and SM1.02 rocket flightso iv ___________________________

The University of Michigan * Engineering Research Institute THE UNIVERSITY OF MICHIGAN PROJECT PERSONNEL Both Part-Time and Full-Time Allen, Harold F., Ph.D., Research Engineer Bartman, Frederick L., M.S., Research Engineer Billmeier, William G., Assistant in Research Harrison, Lillian M., Secretary Henry, Harold F., Electronic Technician Jew, Howard, M.A., Research Assistant Jones, Leslie M., B.S., Project Supervisor Kakli, G. Murtaza, Bo.., Assistant in Research Kakli, M. Sulaiman, M.S., Assistant in Research Liu, Vi-Cheng, Ph.D., Research Engineer Loh, Leslie T., M.S., Research Associate Nelson, Wilbur C., M.S.E., Prof. of Aero. Eng. Nichols, Myron Ho, Ph.D., Prof. of Aero. Eng. Otterman, Joseph, Ph.D., Research Associate Schumacher, Robert E., B.S., Assistant in Research Taylor, Robert N., Assistant in Research Titus, Paul A., BoS., Research Associate Wenzel, Elton A., Research Associate Whybra, Melvin G., MoA., Technician Wilkie, Wallace J., M.SoEo, Research Engineer Zeeb, Marvin B., Research Technician. ——. —-------- v —--— v

The University of Michigan * Engineering Research Institute 1. INTRODUCTION The rocket-grenade experiment for high-altitude temperatures and winds consists of exploding HIE. charges (hereafter referred to as grenades) carried aloft by a rocket, and recording information on the propagation of the explosion waves. This information consists of the coordinates of the explosion (coordinates of the rocket at the time of the explosion obtained by DOVAP, and sometimes the coordinates of the explosions obtained by ballistic cameras), the time of the explosion (obtained by a disturbance in the DOVAP record, and sometimes by groundbased flash detectors), and the times of arrival at a number of points (array of microphones) on the ground. In the currently used method of the data reduction*,** the effects of temperature and winds are treated separately. More accurate results can be obtained by reiterationo The concept of virtual source is introduced in the calculations, which is theposition at which an explosion at height z would have to occur if there is no wind below z so that its sound wave would arrive at the microphone array from the measured direction, In this report a new approach is presented to the problem of the computation of winds and temperatures in the layers between the explosions from the recorded data on propagation of the explosion waves. The effects of temperature and winds are taken into account simultaneouslyo The method is simple both from the conceptional point of view and from the point of view of programming for a solution on a digital computer. As in. the currently used method, zero vertical winds are assumed A. G. Weisner, "The Determination of Temperatures and Winds Above Thirty Kilometers", in Rocket Exploration of the Upper Atmosphere, Pergamon Press, 1954, p.1355 W. G. Stroud, W. Nordberg, and J. R. Walsh, "Atmospheric Temperatures and Winds Between 30 and 80 Kmn," J. Geophys. Res,, 61, 45 (March, 1956 )* 1 __

The University of Michigan * Engineering Research Institute 2. NOTATION Ak Arrival time at the k-th microphone measured relative to the arrival time at the center microphone, and corrected for horizizontal and vertical departures of the microphone location from an ideal array of two horizontal lines. Ak Measured arrival time at the k-th microphone. A(x) Function relating the time of arrival at microphones nos. 6, 7, 8, and 9, as a function of distance x. A(y) Function relating the time of arrival at microphones nos. 5, 4, 2, and 1, as a function of distance y. AAh Correction in the time of arrival for horizontal displacement of a microphoneo AAz Correction in the time of arrival for a vertical displacement of a microphone. al Coefficients of the powers of x in function A(x). ak b Coefficients of the powers of y in function A(y). bk do Distance between the point (xo, you Zo) and the wave front passing through the origin of coordinates, measured parallel to the vector of propagation P. e East direction. et Short time interval. Gu Short displacement along u axis. ex Short displacement along x axis. cy Short displacement along y axis. ez Short displacement along z axis. i —---------.~2

The University of Michigan * Engineering Research Institute f Frequency. h Altitude above the sea level. i Index of the explosion. k~ Infinite characteristic velocity. km Minimum characteristic velocity. keY kr kn Characteristic velocity in certain horizontal directions. kx ky kx Approximate characteristic velocity. ky m Direction of infinite characteristic velocity. m Direction of minimum characteristic velocity. n North direction. P Propagation velocity vector. p Magnitude of propagation velocity. Pe Pn Px L Components of the propagation vector in a Cartesian coordiPy nate system. Pz T Temperature arrived at by Eq. (9.2). to Travel time of the sound wave through distance do. t Correction in travel time of the wave for finite amplitude propagation. ti Travel time of a sound wave from the explosion to the center microphone. l5

The University of Michigan * Engineering Research Institute t Measured travel time of the explosion wave from the explosion to the center microphone. At Time interval spent in a layer. u Arbitrary horizontal direction. V Velocity of sound vector. v Magnitude of sound velocity. v e Vn vx} Components of the sound velocity vector in a Cartesian coordinate system. vz vph Phase velocity. vg Group velocity. W Wind vector. We Wn Horizontal wind componentso Wx wy X Unit vector in x direction. x Horizontal axis. xo Displacement of one of microphones nos. 1, 2, 4, or 5 in x directiono Xk Distance from the center microphone along x axis of one of microphones nos. 6, 7, 8, or 9. Y Unit vector in y direction. y Horizontal axis. you Displacement of one of microphones nos. 6, 7, 8, or 9 in y direction. ---------------------—. 4 —----------

The University of Michigan * Engineering Research Institute Yk i, Distance from the center microphone along y axis of one of microphones nos. 1, 2, 4, or 5o Z Unit vector in z direction, z Vertical axis. zo Displacement of a microphone in z direction. a Angle between the velocity of sound V and x axis. Angle between the velocity of sound V and y axis. r Angle between the velocity of sound V and z axiso T Relaxation constant, <JD Angular frequency. G Angle between m direction and u direction. Angle between x direction and u direction. 3. OUTLINE OF THE METHOD OF THE DATA REDUCTION In the calculations, the physical quantities are expressed by their components in the Cartesian coordinate system and the computation of angles is avoided. The origin of the coordinate system is taken at the center point of the microphone array with two axes horizontal and the z axis up. As a first step, arrival times at the microphones are used to compute characteristic velocities (trace velocities) at the center point of the microphone array. The calculation is presented in Chapter 4. These characteristic velocities are computed for two directions, x and y, in a horizontal plane, along the arms of the array, which has the form of a crosso From these, the characteristic velocities in the north and east directions are computed, using formulas presented in Chapter 5. The sound ray which arrived at the center point of the microphone array from the lowest grenade explosion (explosion no. 1) is then traced through layers with known meteorological parameters, i.e., layers in which the temperature and winds were obtained by balloon measurements shortly before or 5

The University of Michigan * Engineering Research Institute after the rocket flighto* The winds and the temperature are assumed constant in a layer. The appropriate formulas for tracing of the sound ray in the layers with constant, known temperature and winds are presented in Chapter 7. The formulas are based on the law of refraction, which is stated in terms of a Cartesian coordinate system in Chapter 6. The tracing through layers with known meteorological parameters is continued up to the altitude of the balloon flight; but in the rare case that balloon data extend to the altitude of the lowest explosion, explosion no. 1, the tracing is stopped 3 or 4 km under the altitude of the lowest explosion. This altitude up to which the ray is traced through layers with known meteorological parameters can be conveniently referred to as the altitude of explosion no. O As a result of the tracing we know the east and north coordinates of the point at which the ray from explosion no. 1 to the center point of the array intersects the altitude of explosion no. 0, and we know likewise the time spent in propagation from the center point of the array up to the altitude of explosion no. 0. This time is subtracted from the determined time of sound propagation from explosion no 1 to the center point of the microphone array,** to yield the time of propagation (At)l from explosion no. 1 to the altitude of explosion no. Oo The distances (An)l and (Ae)l that the sound ray travels in the north and east direction within the layer are likewise computed. Knowing the geometrical path through the layer between explosion no. 1 and the altitude of explosion no 0, the meteorological parameters in the layer between explosion no 0 and explosion no. 1 are computed using equations derived in Chapter 80 Then the sound ray from explosion no 2 is traced, starting from the center microphone, using equations of Chapter 7, up to the altitude of explosion noo lo The coordinates north and east of this point and the time of travel from this point down to the center point of the microphone array become available. The difference (An)2, (Ae)2, and (At)2 for propagation of the ray from explosion noo 2 to the altitude of explosion no. 1 are computed. Knowing *It will be shown later in Chapter 12 that the end results, i.e., the computed temperatures and winds in the layers between the explosions are not at all sensitive to the accuracy of those balloon datao **The measured time of the propagation of the explosion wave, i.e., the difference between the time of the explosion and the time of arrival at the center microphone, is incremented by the estimated theoretically finite-amplitudepropagation effect to yield the determined time of sound propagation. The finite-amplitude-propagation effect has been discussed and the amount of increment as a function of altitude computed for 2-lb and 4-lb grenade explosions in The Effect of Finite-Amplitude Propagation in the Rocket Grenade Experiment for Upper-Atmosphere Temperature and Winds, by Joseph Otterman, Univo of Micho Engo Reso Insto Report 2387-34-T, Ann Arbor, April, 1958. -6

The University of Michigan * Engineering Research Institute the geometrical path through the layer between explosion no. 2 and the altitude of explosion no. 1, the meteorological parameters in the layer are computed using equations derived in Chapter 8. The process is continued for successive explosions yielding each time the meteorological parameters of successive layers. Reiteration of the tracing is therefore unnecessaryo The method is outlined again in Chapter 9 as a Flow Chart for a computer program~ The pertinent equations are included in the Flow Chart. The process can be visualized by considering Fig. 1, where the order of tracing through the layers is indicated by successive numberso The solid lines indicate the layers in which tracing is done by means of the equations of Chapter 7, and the dotted lines indicate the layers in which the meteorological parameters are calculated by means of equations in Chapter 8. The equations of both Chapter 7 and Chapter 8 are based on the assumption of constant temperature and constant winds in the layer. It will. be shown in the discussion in Chapter 12 that this assumption does not lead to appreciable error in computing the average parameters in the layer. The equations are based on the assumption of zero vertical wind velocity. This is a very crucial assumption, and it can possibly be the source of significant errors. This is also discussed in Chapters 12 and 135 It should be pointed out that the average meteorological parameters computed for the layer between explosion no. 0 and explosion no. 1 do not possess the accuracy of the rocket-grenade experiment. Actually, the accuracy of these parameters is worse by an order of magnitude than the accuracy of the data from balloon flightso 4, CALCULATION OF THE CHARACTERISTIC VELOCITIES FROM THE TIMES OF ARRIVAL AT THE MICROPHONES The microphone array, which has been used for the IGY series of the rocket-grenade experiment in Fort Churchill, Manitoba Province, Canada, consists of nine microphones arranged in the form of a horizontal cross (Fig. 2). The purpose of this chapter is to describe the computational procedure for obtaining the characteristic velocities in the directions along the arms of the cross from the arrival times of the wave at the microphones. To the first approximation, the characteristic velocities kx and ky at the center microphone, nOo 3, in the x and y directions are given by: X9 - X6.XT' X8, /\ ^ kx = ~9 x6 ~ ~ and (4.01) As -A A A - A --------------— 7. —---------------

The University of Michigan * Engineering Research Institute |-_ -Exp/ #3 Exp/ #2 I / EXPl#I ___,__/__/} / A/titude of Expl +0 H -1 x1:/ /DZ Bolloon dato 1r21 i Center of microphone arroy Fig.. -Tracing of sound rays through successive layers. --------------------—. 8

-- The University of Michigan * Engineering Research Institute +y ~^ X N 2 -'21.71' Fig. 2. The microphone array at Fort Churchill, Manitoba Province, Canada. This array has been used during the IGY series of the rocket-grenade experiment, -------------- 9. —

The University of Michigan * Engineering Research Institute Y5 - Y - Y4- 2,o) =.....~..... (4.02) J A.5 - AI A4' AA where Ak denotes the recorded time of arrival at microphone no. k, The above equations would determine exactly the characteristic velocities if the waves were planar and if the microphones were located on two horizontal lines. The actual locations of the microphones depart from an ideal array of two horizontal straight lines. The arrival times at each microphone have first to be corrected to allow for these departures, in both the horizontal and the vertical directions The horizontal correction AAh is rather easy to compute. Let xo be the horizontal displacement of one of the microphones nos. 6, 7, 8, or 9, and YO be the horizontal displacement of one of the microphones nos. 1, 2, 4, or 5. We have then, from definition of the characteristic velocities, the following equations for the correction of arrival times: AA = h - (4.03) kx kx for one of microphones nos. 6, 7, 8, or 9, and Yo Yo AAh = ~ - (4.04) ky ky for one of microphones nos. 1, 2, 4, or 5. These corrections, in the case of the Churchill array, involve only fractions of a millisecond and the use of approximate characteristic velocities [Eqs. (4,01) and (4.02)] is completely adequate, The derivations of equations, to be presented now, for correcting the arrival times to allow for the vertical displacements of the microphones are much lengthier. The basic equations expressing the propagation vector P and the velocity of sound in the medium V in terms of their components are: P = PX X + py Y + pz Z (4.11) V = vx ~ X + vy. Y + vz * Z (4.12) P = V+W (4.13) Px = x +wx (4.14) ----— 10 —,-...10

The University of Michigan * Engineering Research Institute Py = vy + Wy (415) Y = vy+w3T (4.15) PZ = vz (4.16) -- 7 = 2.2 2 2 2 I E -- - 2.2 2 (4.18) p IJ Px + P +p (4.18)PZ We develop first a formula for the distance do' between a point xo, Yo, zo and the plane tangent to the wave front at the origin of the coordinates, where the distance is measured along, i.e., parallel to, the vector of propagation P. The equation for points in a plane passing through the origin of the coordinates and tangent to the wave front will be (x cos a + y cos j + z cos 7) constant = 0, (4.19) where cos a, cos P, and cos y are the directional cosines of the normal to the wave front. The validity of Eqo (4.19) can be seen from the following: components of a point in the plane are x, y, and z. The scalar product of the vector xX + yY + zZ representing this point and of unit normal vector cos OX + cos PY + cos YZ must be zero. The equation of the line passing through x0, yo, zo and parallel to P is x x - Yo = - Zo = A, (4.20) Px Py Pz where k is a parametero The point of intersection of this line with the plane tangent to the wave front shall be designated by xl, yl, and zl (parameter.k)o We have, therefore, x - XO Yi - Yo Zl - ZO _xxo = yl-yo = za- ^ = z-, (4.21) Px Py Pz and since the point lies in the plane defined by Eqo (4.19), cos ox4 + cos Py4 + cos yzl = cos a(Oipx+Xo) + cos P(XiPy+yo) + cos 7y(ipz+Zo) = o (4.22) 11

The University of Michigan * Engineering Research Institute?l o (cos apx + cos 3py + cos yPz) = -(cos Co + cos yo + cos 7zo) (4.23) ACO = c os Oyo + cos yo + cos 7z (4.24) cos oCpx + cos Bpy + cos ypz We have for the distance do~ do = [(xj-xo)2 + (y-lyo)2 + (z1-zo)2]1/ (25) From Eq. (4.21): x - XO = Px - (4,26) Yl -Yo = Py Xl (4.27) z - Zo = PZ \1 (4028) and therefore (if the negative sign root is chosen), 2 2 22 1 /S Cos Oxo + Cos YO + Cos yZo do = P- (x+py+z)' = p _,, cos Ctpx + cos ppy + cos 7Pz (4.29) and if both numerator and denominator are multiplied by v, we have I= p x v xo + vy Yo Y +vz o (4 30) Vx o Px + Vy o py + Vz o Pz The difference in arrival times to between the origin, of coordinates and the point xo, Yoa zo will be (the two points are assumed to be close together, so that the direction of arrival does not vary): do _ cos G o + cos PyO + cos YZo VXXO + Vyoo + Vz o P cos Px+ cos Ppy + cos YPz Vxp Vypy + VzOpz (4531) This rather cumbersome formula is used in a simplified form, to correct for departures from the ideal array in the vertical directiono This correction is denoted by AAzo vzz o ZO Vz o ZO 432) A Z Vxopx + Vyopy + Vzz x( x) + y(y+y) + 12

The University of Michigan * Engineering Research Institute The corrections AAZ in the case of the Churchill array amount to as much as 10 msec; it is thus desirable to compute the correction with an accuracy of about 1%. It has been found that the terms vx2wx and vy"wy contribute insignificantly to the denominatoro Thus, unless the ground winds are strong and the direction of arrival departs considerably from the vertical, Eqo (4032) can be replaced by a simpler relation: AA z Vo 0 (4 33) v2 The distances zo are known from the survey of the microphone array, and v2 is determined by the average temperature at the arrayo The vertical component vz of velocity still remains to be determined separately for each arrival. This vertical component is determined from: v = (v2 - Vy _ v2)1/2 (4.54) iThe horizontal components are determined from the following two equations. 2 v2 vx = kx - x- (kx/kWyWy kx vy = -- (4~36) ky - wy - (ky/kx)wx ky The above equations are based on Eqso (7~4) and (7o5), which will be derived later. It should be pointed out that components vx and vy do not need to be determined to an accuracy of 1%, and still vz can be known with 1% accuracy from (4034)o The corrected time of arrivals, which are given by Ak = Ak - AAh - Az ( 7) could be used to determine exactly the characteristic velocities kx and ky at the center microphone in accordance with Eqso (4o01) and (4o02) if the waves were planar. However, since in reality the waves are not planar, a more sophisticated.approach is necessaryo Regard the arrival time Ak to be a function of x, or of y. Then the characteristic velocities kx and ky at the center microphone will be 1 - aA(x)| (458) kx x =O 13 -

The University of Michigan * Engineering Research Institute and 1 = aA(y) 49) (4.39) ky 6y y=0 Each of the functions A(x) and A(y) is known at five points. And obviously many approaches can be used for determination of a reasonably fitting function. Three approaches which were tried by the writer are outlined below. In the following, arrivals Ak are measured relative to the arrival at the center microphone, no. 53 A function of the type A(x) = aix + a2x2 with least-squares fit has been rejected, since the value of the function at the microphones differed by more than 1 msec from the actual values Ako A function of the type A(x) = aix + a2x + a3x3 with the least-square fit and a function of the type A(x) = aix + a2x2 + a3x3 + a4x4 gave virtually identical results in two caseso This last type of function, which leads to computationally rather simple determination of the characteristic velocities, has been used. Only coefficients ai and bi in the functions ~I,n,~A(x) = aix + a2x2 + a3x3 + a4x4 (4.40) and A(y) = biy + b2y2 + b3y3 + b4y4 (4o41) need to be determined for our purposes. To fit the corrected arrival times, the functions of Eqs. (4o40) and (4o41) are rewritten as followsA(x) (x-x7)(x-x8)(x-x9)(x-O) (X6-x7)(X6-x8) (xx)(X6 ) (X6-0) + A7.(X-x6)(x-x8) (x-x9)(x-o) (Xx-X6)(X-X8)(X-X9)(X-0) + A7 (X7-x6)(X7-Xs)(X7-x9)(x7O) (x-x6)(x-x7)(x-x9)(x-O) + A (X8-x6)(X8-x7)(X8-x9)(X8-0) + A (X-X)(X-X7)(XXs)(X-O) 6 (4 42) (X9-X6)( X9X7)(' 9sX8)(X9-0) and. 14

The University of Michigan * Engineering Research Institute A(y) = A, (y-y2) (y-y) (y-y) (y-o) A(y) (Y-Y2)(YI-Y4)(Yl-Y5)(Yi-O) A( (y-y)(Y2-y4)(Y2-y5)(y2-o) (y —y) (y-y2) (y-y5)(y-O) (y4-Y1)(y4-y2)(y4-y5)(y-O) (Y-Yl)(y-Y2)(Y-Y4)(y-) (4 43) (y5-yY)(y5-y2)(y5-y4)(y5-) (44) The characteristic velocities for each explosion wave are given byo 1.L = al = - X x 9 + A7 X6 xS x9 kx (X 6-X7 ) (X6-X8) (X6-Xg9 )X (X7-6 ) (X7-x8)(x7 —x )x7 X+ A x7 +A X 7 X7 X X8 (xs-x6)(x8-x7)(x8-x9)x" +A9 " (X8-X6)(X8-X7)(X8-X9)X8 (x9-X6) (X (-X8)( x9-x / (4o44) sand 1 J _ bi = -A Y2 Y4 Y5 + A2 Y Y4 yY5 ky \ (Y1-y2)(yl-y4)(yl-y5)Y1 (y2-yl)(y2-y4)(y2-Y5)Y2 Ya_, Y Y2 Y1 Y Y4 N 4 (Y4-Y1)(y4-Y2)(y4-Y5)Y4 +A5 (y5-yl)(y5y2)(y5-y4)5 (4.45) Thus, the evaluation of one characteristic velocity consists of multiplying four corrected arrival times Ak by coefficients of the type X7 X8 Xg (X6-X7) (X6-X8) (X6-X9)X6 which are determined by the geometry of the array, and adding the results * *The resulting coefficients are such that the arrival times at the interior microphones are weighted by a factor of about ten more.heavily than the exterior microphones. It has since been demonstrated to the writer by Captain William R. Bandeen that this method of computing the characteristic velocities is 15

The University of Michigan * Engineering Research Institute The presentation in this chapter has been aimed at describing the computational procedures for determining the characteristic velocities for the IGY series of experiments in Fort Churchill However, the method is readily applicable in any array in the form of a cross (or a hexagon), provided the departures of the microphone locations from straight horizontal lines are not largeo The approach outlined here brings out the fact that, for a truly horizontal array, it is not necessary to record the temperature and winds at the array during the arrivalso And this holds true for winds determination in the case of an array with departures from the horizontal, unless the winds are very strong and the waves arrive at a relatively low elevation angle 5o ROTATION OF THE CHARACTERISTIC VELOCITIES The quantities kx and ky that are obtained from the arrival times at the microphones are the characteristic velocities in the horizontal directions along the lines of the microphones, as explained previouslyo For purposes of ray tracing, it becomes necessary to determine the characteristic velocities along different directions (in the case of the Churchill array, in the north and east directions), Therefore, the way of computing the characteristic velocity in an arbitrary direction in a horizontal plane, when kx and ky are given, will be presented nowo Consider Figo 30 At t = O the wave front arrives at the origin (0;0). After et the wave front arrives at points (ex;O) \and (O;ey) on the x and y axis, respectively.o Assuming that the wave front is smooth, in the limit for Et - 0, the small portion of the wave can be assumed planar; and therefore at et the wave will intersect the horizontal plane along a straight line passing through (cx;O) and (O;~y) The characteristic velocities. along the x axis and the y axis are, respectively, k = ex/Et and ky = ey/et. Along an arbitrary direction u the characteristic velocity ku is given by eu/st, where cu is the distance from the origin to the line of intersection at et measured along the direction Uo The characteristic velocity in an arbitrary horizontal direction u is thus given by the distance along the direction u from the origin to the line joining the end points of kx and ky. It should be pointed out that x and y need not be perpendicular; but for greater accuracy of measurements, the angle should be close to 90~o The maximum characteristic velocity is always infinite and in the direction parallel to the line joining kx and kyo The minimum characteristic velocity, inferior to the approach through which the arrival times at different microphones influence the determination of the characteristic velocity more equallyo The differences can be significanto 16

The University of Michigan * Engineering Research Institute N kmNlj 1-k x (Ex,O) /km Fig. 3- Rotation of the characteristic velocities. 17

The University of Michigan * Engineering Research Institute knm, is in the direction perpendicular to the line joining end points kx and kyo The geometrical relationships inherent in Fig. 5 are grasped more easily if it is realized that the end points of inverses of the characteristic velocities map on a circle. This is because ku = k/cos 0, (5.1) where Q is the angle between the direction m of the minimum characteristic velocity and the direction. Uo Therefore the end points of 1/ku 1/ku = cos 0/km (5.2) are located on a circle with a diameter of l/km. It follows easily that, when x and y are perpendicular, the minimum characteristic velocity km is given by 1/km = (l/kx) + (1/ky)2 (553) or kx ky km =* (5.4) kx'.2+'k 2 To find the magnitude of the characteristic velocity ku in a direction u which is 0 degrees counterclockwise from x, when kx and ky are given (x and y perpendicular), we note that the equation for the line joining the end points of kx and ky is: y = _ x + ky (5o5) ~ kx and the equation for the line in the direction u iso y = tan 0 x. (5.6) The intersection of the two lines occurs at a point with the coordinates x l and y: given by: X1 =y x (5 7) x = - ky/kx + tan 0 ky + kx tan and tan 0 ky kx (5 8) ky + kx tan 0 18

The University of Michigan * Engineering Research Institute The characteristic velocity ku is given by ku = 1/ x2 + yl2 ky kyx + —- ky kx ky + k tan 1+ t = ky cos + kx sin This is the formula that is applied in the case of the Churchill array to yield the north and east characteristic velocities through a rotation by -21o71~ (see Fig. 2). 6. THE LAW OF REFRACTION EXPRESSED IN COMPONENTS OF VELOCITIES IN A CARTESIAN COORDINATE SYSTEM The refraction of the sound waves is caused by the changes in the velocity of propagation, ioe., temperature, and changes in the velocity of the medium, i.eo winds. The law will be stated in components of velocities in a Cartesian coordinate system. The different media of propagation consist of uniform horizontal layers. The vertical component of the wind is assumed zero. The law of refraction, sometimes called Snell's law, is based on the premise that the characteristic velocity of a ray (a small portion of the wave front) in any direction parallel to the plane separating two media is the same in both media. The law can be derived from the Huygens principle from which it:follows that a plane wave, after undergoing refraction in the plane separating, two media, remains a plane wave. The equations for the characteristic velocities in terms of the components of the velocity of sound and wind velocities can be arrived at from the basic Eq. (4o31) in the following manner: l/kx = 6to/aXo = Vx _ vx vx px + vy py + vz Pz vx P + vy py + vz2 (6.1) and vy vy Vy Vy 1/mky =ty=Vx Px + vy Py + z vXz Px + Py + Vz2 (6.2) However, the above equations for the characteristic velocities will be derived 19

The University of Michigan * Engineering Research Institute here again from geometrical considerations depicted in Figo 4. Let A, B, and C be the intersections of the wave front with the x, y, and z axes, respectively, at t = Oo At 1 sec the wave intersects the x and y axes at points A* and B*, where A*B*IIABo The distance DO represents V the vector of sound velocityo The following relations hold~ V AO = cos (6030) BO = v (6031) cos S cO = (6,32) cos Y OA* = wx + y X = + Wy cos 64) -BO cos a BO cos a, ) OB* = wy + W = y + Wx cos (6.41) The characteristic velocities are defined by: x = AA* = AO + OA* V Cos $ = +v, + wx + wy Cos a cos O Vy __ +Wx Vx + Wy Vy = - + wx + Wy - = v + wv+yvy Vx vx Vx z2 + (Vx + wx) v + (vy + wy)vy Vx 2 Vz +Px Vx + pY vy - - v, +Px vx+py vy (6.5) VX and similarly ky = BB* = BO + OB* V cos a V2 vx - + Wy + wx -- + W + cos p, cos P Vy Y Vy v2 + wx v + wy Vy = vz2 + Px Vx + y v y. (6.6) vy vy 20

The University of Michigan * Engineering Research Institute:z. Fig. 4. Characteristic velocities in a Cartesian coordinate system. 21

The University of Michigan * Engineering Research Institute The same results have been obtained in Eqs, (6.l) and (6.2)o The above two equations constitute the basic equations of refraction we will use. 7. EQUATIONS FOR TRACING THE SOUND RAY THROUGH A LAYER WITH KNOWN METEOROLOGICAL PARAMETERS In this chapter the equations are presented for tracing t~he sound ray through a layer with known meteorological parameters. The horizontal axes are taken to be in the north and east directions. The knowledge of meteorologi;al...p:rameters implies the knowledge of. the wind componeint wn and we and the knowledge df the velocity of sound v. (The formula by which v is computed as a function of temperature is given in Chapter 9.) The characteristic velocities.of the ray, kn and ke,are known. The basic equations for finding the components of v, ioe., vn, ve, and vz are Eqs. (4.17), (6.5), and (6.6). The equations are presented here again for convenience, in the north and east horizontal coordinates. ~2 2 2 2 V2 = vn + ve + v (4.17) 2 2 vz + PnT + Pe _a v + vn Wn + Ve We65),vn vn v, 11Vn ke = + Pn +p Pe = v +n wn + Ve e (6V6) -e z Ve ve It follows that vn ke (7.0) Ve kn Equations (6,5) and (6.6) can be rewritten vn (kn - wn)-vee = v2 (701) ve (ke - e)-Vnwn = v2 o (7,2) Introducing the value of ve from Eq. (7.0) into Eq. (7.l), we obtain kn 2 Vn (kn - Wn) - Vn - e we = v, (7) or v2 vn = kn' n - (kn/ke)we' (74) 22

The University of Michigan * Engineering Research Institute and in a similar manner, introducing the value of vn from Eq. (7.0) into Eq. (7.2), we obtain vS Ve = ke - e - (k/kn)(n Equations (7.4) and (7.5) are used for the determination of the components vn and ve. Subsequently vz is determined by means of Eq. (4.17)o The following can be noted here, Two perpendicular directions, i and m, can be chosen so that kQ is infinite and km is minimum. This is shown graphically in Fig. 3. If a set of equations equivalent to (7.4), (7.5), and (.4.17) is written for directions m, I, and z, we have: vm = k2m. (7.60) km - wm v~ = 0 (7.61) 2 = 2 - V2 (7.62) These equations are simpler. However, this approach, although theoretically attractive, is rather cumbersome, since the rotation has to be effected through a different angle (and the angle has to be computed first) for each grenade explosion. Therefore the north and east coordinates were used. Once the velocity components are determined, the path of the ray is traced by the following equations. The time At in the layer is At =. (7,7) vz The north-south direction traveled, Ae, is Ae = At ~ (ve + we) A At * Pe (7.8) and the east-west direction traveled, An, is An = At. (vn + wn) = At,n ~ (7.9) In the above, pn and pe denote the components of the vector of propagation p in the north and east directions, 23

The University of Michigan * Engineering Research Institute 8. EQUATIONS FOR METEOROLOGICAL PARAMETERS IN A LAYER WITH A KNOWN SOUND-RAY PATH Once the ray originating from the i-th explosion has been traced from the microphone array up to the altitude of the lower, i-l, explosion, the path of the ray between the upper and lower explosion becomes known. In the following, ni and ei denote, respectively, the north and east coordinates of the explosion relative to the center microphone and ti denotes the determined time of sound propagation from the explosion to the center microphone. The quantities At, An, and Ae, when reference is made to layers with unknown meteorological parameters, are indexed as follows: (At)i (An)i, and (Ae)i, where the index refers to the upper explosion of the layer. i-1 (At)i = ti - Z At (8.1) i-1 (An)i = ni - Z An (8.2) i-l (Ae)i = ei - Z Ae (8.5) The summation in each case extends for layers from the ground (microphone array) up to the altitude of the lower, i-l, explosion. The thickness of the layer (Az)i is known. We have for the layer between the explosions =2 (z) (8.4) (At)i P vn = n = (n)i (85) (At)i Pe = ve +we = ()i (8.6) (At)i The quantities pn and pe become known, but the quantities vn, wn, and ve are determined first on the basis of Eqs. (6.5) and (6.6), which are restated here. Vz2 + vn Pn + Ve ) (65) kn and 24

The University of Michigan * Engineering Research Institute 2 Ve = z + Vn Pn + Ve Pe. (6.6) ke It follows that vn (kn- Pn) - Ve Pe = vz (8.1) Ve (ke - Pe) - n Pn = 2 (8.2) Substituting the value of ve from Eq. (7.0) into Eq. (8,l), we obtain n) Vn kn2 (8 n (kn - Pn) - Vn Pe = z (83) and therefore VZ2 n =VZ (8.4) Vn =kn - Pn - Pe (kn/ke) In a similar manner, introducing the value of vn from Eq. (7o0) into Eq. (8.2), we obtain VVZ e ke - (k/) 5) Since the velocity components vn, ve, and vz are known, the velocity v can be determined, Eqo (4o17), and hence the temperature, Eqo (9.2). The wind components can be now determined from the following equations: Wn = Pn n (8o6) and we = P e e (8o7) 9o THE DETERMINATION OF TEMPERATURE FROM THE VELOCITY OF SOUND The temperature of the layer between two successive explosions is determined by means of the formula* Bo Gutenberg, "Sound Propagation in the Atmosphere," Compendium of Meteorology, edited by Thomas Fo Malone, American Meteorological Society, Boston, 1l9,Ppop 366-375o 25

The University of Michigan * Engineering Research Institute v = 20.06 / (9o1) or v2 T= 402' (9o2) where temperature is measured in ~K and velocity in meters per secondo Considerable departures from this formula occur at both low and high temperatureso.Since temperatures as low as 165~K are involved in the data reduction, consideration was given to the formula suggested by Quigley-* v2 = 387o62 T + 180430 T-1 - 20364000 T-2 + 806 + 0o03007 T2 o (9o3) If use is made of Eqo (9o3), the determined temperatures would be higher by about 1.5~o Formula (903) is possibly more accurate in the range of temperatures involvedo It appears that the velocity of sound changes by a negligible amount only because of the change of density0 -This is gathered first of all from the following calculation0 Use is made of a formula developed by Ro Bo Lindsay for the phase velocity vph of sound at low pressures.** The formula, in a much simplified approximate form, reads as follows. Vph = v (1 + 3/8 ai T2), (9o3) where w is the angular frequency and T is the relaxation constant The group velocity, vgroup. can be obtained from the phase velocity by the formula 1 — = --. (9o4) group and therefore 1 d[r Cl 1/8 22) 1_ = _Lv (1 +3/8_ d2( / T2) 1 - (3/8 ) (1 9/8 2T 2) Vgroup drn v dto v (9o5) *T, -H Quigley, "An Experimental Determination of the Velocity of Sound in Dry C02z-Free Air and Methane at Temperatures Below the Ice Point," Physo Rev., 67, 298-303 (1945). Ro.Bo Lindsay, "Transmission of Sound Through Air at Low Pressure," Amo Jo Physo, 16, 371 (October, 1948) o The formula referred to is Eqo (16), po 372o 26

The University of Michigan * Engineering Research Institute Vgroup _ v (1 + 9/8 2T) o (9o6) Taking T = 10-10 second for air at normal atmospheric pressure and room temperature, and assuming that T changes inversely with pressure (temperature change and a possible change in y are neglected), we observe that for 95-km altitude T= 10-10/106 = 10-4 second l For frequencies of under 50 cps, the group velocity will depart by less than 9/8 (2o f T)2 = 9/8 (21t 50 10-4)2 0 13 = 0ol from the velocity of sound under normal atmospheric pressure and under the same I temperature..This conclusion, that rarefication by a factor of 106 has very little ef-: feet on sound velocity for f < 50 cps, is to a certain extent borne out by experimental evidence from measurements of propagation of very-high-frequency sound under the conditions of normal pressure. Such a propagation resembles, in the ratio of the wavelength to the mean free path of molecules in the medium the conditions of high-altitude propagation with which we are concerned. Experiments indicate that the velocity is practically independent of frequencyo 10o DIGITAL COMPUTER PROGRAM The computation of winds and temperatures has been carried out on the LGP-30 computero. This is a desk-size general-purpose digital computer with 4096-words magnetic-drum storage. Floating-point subroutine has been usedo The program consists of 170 instructionso The computer-solution time for a rocket flight has been from 2 to 3-1/2 hours, depending on the number of grenades and the number of balloon layerso The flow diagram of the program, including the appropriate equations, is presented in Figo 5o The flow diagram is presented in detail, since it is relatively easy to rewrite such a program for any general-purpose computero It should be pointed out that the program will be somewhat simpler for computers with index registerso A backward-propagation convention has been followed, ioeo the velocities and distances are taken positive in propagating from the microphone array to the explosiono By this convention the meteorological notation of winds results, ioeo0 if the north and east are positive directions, the winds components will be positive if blowing from east and north, respectivelyo 27

The University of Michigan Engineering Research Institute, d,;~ ~ GJ I:1 I i 00 i ~'i~ II ~ b OI'IO -c 4 1' I.,di - dc <iT ~~ \^ A43? d 7i Clq a? 04 0 d i'i' 0 dto 4 H d 0 Pi ^M ~ ~ H. / ad HI- 4 ol9 ~ -1 1 1 IV019 "di LO *ph\~ H <^~Oi Hu/Hup~ ^ap Odi ~ d* \ p/ -'.-* B*f ddi o id atoo di Cdl,.$ 4,. \| —. ~:1 P|d O. CTi'~ o 3 - 5.-~~~~~~~~~~~~~~~~~~- PEI T a^1 No!) ~ ~ did H 1i ON )H4A O+i d* ir ~ ti-^-'w d Pi rS o Cl N, i+.di 0, d I 1 -- -- I I -- - -- I I~~~~~~~~~~~~.1.S -.A r. pOdi tfr[ ^1o 4 I fr rd ~ ~ ~ 9 0 -1 -~~~I +:Hdi di &pl ^ a~p<' ^ is ^ ^Pdl — ^ C+!i Co~~~~. + h~,C p |.a 0 di di04~~~~~~~~~~~~~~~- -4 H 43 0' f lti. Cl'.|t~~~Z-C, 0 oOi 0 4315 43.0 s? -1.,a diC 4 P di3 )4 di.l^ Coi_ I i8 -- y - di ^.9 Cl ~ di CmiC di 4311 It di t4- PIi I " ^i ^ ~ - ^ 8& | — ri — 4ai 1 di (I ~ f ~~~~~~~~~'i, " 1,,,: 5.^ 0 I00 15 C. ^ ^ o> d&^'?^ ~ ^~~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~3 41~ - U-\ 43 430 En A, "~~~~~~~~ ^Ii~ rt^^M ~ ~ std Hp d lI 4;> LO 4)~~~~i I iH43 1 m - C) 43R + rd C -P CH 0 di - d di S S^ l^ ___ ___ *. di 43. O 4- 15 43 di - pi Pi-1~ -1 43 C 0~~~~~~~~~ Oi 14 1.0 1 ~3'a..4,C i i0 di -'Cl Cl + 3S 1i -.) 443 d i ||t|'ddi. I | - n di $|'H1 a 0iii3 4- i +1T ~ 5H -P — 1 -— P E-i _0 EC -H'H l Id 0 1 - T-& 43 H 31 4i3 ON O d i OOO~-i43di Cog.ri||. 4,-N $..,). H 48'g53 i- P o" ~" d i 0 (oil: ~,-' 4 P pi - 0 L'; TT,^^a~. ~~ O,L 0, 4 3 bO 4) 41 1 43i rprif ^6Co di bO di Cl H *H~~~~~ ~ ~~~~~~~~~idi N43 oN ^ M ^+, ^^^^(a~~ ~, S.gg ~',^.g^-Pri^^<l'P15+43 CCl..014 Odir. 43Ff + 011 d h R I ~ I l 4-1 + Pa~ I1 -— l. 4I':....~,,,_P 4-3 C) 0) -H~~~~~~~di$4 43 1 ~,.-~~~~~~~~. 10 H Cditi4 ~ Hi Or,,a..........~~~~~~~~~~4 4- -8-3

The University of Michigan * Engineering Research Institute 11. CALCULATIONS AND RESULTS FOR SM1O1 AND SMlo02 ROCKET FLIGHTS Aerobee SM1oOl, carrying eighteen 4-lb grenades, was fired at 0548 CST on November 12, 1956. Grenades nos, 8 and 18 failed to explode. Grenade no 15 provided very weak arrival signals, making accurate determination of arrival times very difficulto The weak arrivals could be interpreted as caused by a subnormal detonation of the explosive. In such a case, the finite-amplitude propagation could introduce considerable errorso For these two reasons it has been decided to omit the data on grenade no. 15 and to consider a single layer between explosions noso 14 and 16. Aerobee SMl.02, carrying nineteen grenades (of which the first twelve were 2-lb and seven were 4-lb), was fired at 2216 CST on July 21, 1957. Grenade no. 14 failed to explode, and the arrivals from grenade no. 19 were masked by the rocket re-entry waveo.The input data to the computer, consisting of north and east coordinates ni and ei of the explosion relative to the center microphone, layer thickness (Az)i between the explosions, time of sound propagation ti,* and characteristic velocities kn and ke are presented in Table I for SM1oOl and in Table II for SMlo02. The balloon data for SMl.Ol, where 21 balloon layers have been used, are omitted, since they are thought to be almost completely immaterial to the solution (see Chapter 12)o The data on five balloon layers for SMlo02 are presented in Table III to facilitate understanding of the computer tracing of the sound rays, which is presented for the first three explosions of SMlo02 in Table IVo The computed temperatures and winds are presented in Tables V and VI and in Figso 6 and 7 for SMloOl and SMlo02, respectively. 12. COMMENTS ON THE EXPECTED SOURCES OF ERRORS The discussion in this chapter on systematic errors is kept short relative to the importance of the question since there already is an ample analysis** of the accuracy of the experiment. A few remarks are presented here on points which the writer has considered in more detail or points which are especially.significant in the suggested method of data reduction, *The time of sound propagation is computed from ti = tm + tgo See the footnote on page 60 -*.*Wo Nordberg, A Method of Analysis for the Rocket-Grenade Experiment, Techo Memo NR. M-1856, U.S. Army Signal Engineering Laboratories, Forth Monmouth, No Jo, February, 1957 (now the UoSo Army Research and Development Laboratory)o 29

The University of Michigan * Engineering Research Institute eO i. t: — -. r.-o O 0, CO r-o CO CO o — or 0 0 (1 I O OO C O -O-OlD I Ol t- - I t - rO 0) Ct O CO \U - O C fJ CO — in -f- O l\-CO r- \D X^ ~ W h\ Zr cO h^ - \O Kl to 4- \o C 0\ aO\ \ CLJ CJ I j CJOj CJ OJ t —cO rlC\J L'L~ ~- r — r-I.I I I I I I I I r-! r- r- r-I I.I I. -.- C — n N O\,\ _- - 7i L~ - ~'1n 0 C, 0 tr \, 00 t- b- O O c 0 — t L rl- - O 0) a a a a a a 0 a a a a a a a 0.ri O O r-! OO O LCL" r-! \ \ LD r- 4 Q \-!O \ co ON o C\ Kr\ 0o C L\c L \OoD DC C CO C~ 0 - OD — -- CO o'II O'cO c O \ CQ O - co4 E-.- 0,- \C- H H' H -H H W' D C.) C'O \ L'-\O I r.LC\O t O N H l'-' 0OJ C\j C! t-' 1 ~ \ o',o, \D LC\ h"\ ro',A \,oC \C\ r- \,Q n 0 H -,. - c'- -' ln, - O rl O N C JC CO\l Q ^ ^ O r 9 Or nr X O cU tO X o Hi | cO t ~ v O e0 O OOC O O C o O Oo O,rI l r- I D coD 0 - co C) Ot 1 r'- I t- -- r-l 2tH 0) H H- Oc\O - 0\ O H H -4 HcO -- N; 0 -) C\H,\ N \ 0 t O Hl\ C- n 0C M n \ID O \ O — 0 n -- r- O'\ t-'- 0 LC C,0 Q C;o e Io- o 0 o t- C- O O 6r ) D 4 4-< C0 -'-L —0 r-0 - r-O n LCO r- LC 0 r-O C r- Lr- r- I 0)- CH\' OJ HC N\ H G HLN n n L \ L. -- LI I I I I I! I I I I I' I I I -H) 1 > n O O L CO n rC \O OL (CO C JL ~HCQ Vr- C_:-I \ Oc- CO O -OO O CJ- - rI a- t- h O\ co LO n CO N rq -- CJ n O o,-I ( r I - 0r'~ NU N O LCt rn a) r- n - WIOC\r- r-4 r —4 O r-4 - CO \. oH CU V X lc\j r k\ = — co - n\ i 0 r n C\j ot: C i c t N t- t — ^C\ \ C r-K K \o -O::Jc- o\ C\ C3Od (UO r -p -:I- G'\ C\ t — -N ir -\ n \ lD Ol\r a!C\j hC KN ^K^\ Kl -:- _i- c-_- Ln Ln in r\LC\ \ o rt r l r -l ~ r -l ~ r -I

The University of Michigan * Engineering Research Institute rH ON \O:- R.'O \ cz t - > ON ON i.- 0 -o.. 0C o o a c e o 9 o 9 o o o o Oa o q o ~) 2 r-4 -: ^ _ L> n O CO - \ CO O CO N' C CO O l OOin r- O-c a\ t- \ C O o -t CO O 0 Ln I r\L r\-i —-- ft rc\ KIN KCN"C\ C\Mj C M Oj C CM I I I I t I 1 I I I I I I I I 1 I ~t n-z'l. \1. Lt ~ L -n c- -. \ I.0 1 - Hr C\M CM O co Ln CO CO Ln O C O\ r<N\ KC C\J Kl\ - a,\ r O \ r-l r-:d 0 \O JOJ K^ t tD- P- n N > r-4t - -lir\ r,-I,.-I 4 r C- r- C'U' CU! C IK{" WI' _.' \': b- -i -L P- O Xt nr CMl\ L-Q C 0 Ck N (I _-: O ON 0 -t in O\ O\N Ln Ln -- COt - — t O \O n n Ir *rl O K n \ O i o oc- t>- O L \ oa\ -9 O 4H ) -'-O 0 L L- O0 t-t-0 O L \C O O\-: 00 -00 ~ CO Q1 I rC C a CU CMC\j K c \ -C-t- L\ O\ C\J \o ON 0 rH CM "- Ln \D t-CO ON H r-I l \ - Hr- H HHr- r-lH rH r-l r-l H C CU CO J CO C cD0 ) f- n Lr\ D L O r- \D L O\ - O O L C O 4n (DD — 0 C ON\O CJK' " I.n - 1 — 1 — t \O," Ckj LO Q U, H, — r-, —,-! C CI " Le::n O - ON O Lf\ O'\O,-I: - O\ " i C' - O. n N. -. \E, O 0 A -4 \ Ci O GO C\ ON CO O O CKl O rH H. o o o o o o o o o o H 0 9 9 9 9 9 99 0 9 9 9 9 9 9 9 G9 Q o1 1 Cflj C\i j CN \i Cl\j o j\i r L) ch OL co ip C j O r-o N K C-t Lrn\ \ L- C O 0 r41 n 1\ HH r- H - r-l - Hr- r- r -— I C N N C CM O N0 CM N0 KNiQ —- LC.- - E'' *rXl,?- o o o o e, o e' 0 a o o o' o f O C) C: c O - O - 12-0 C) NO..- 0_ L-\ cO C1 CON N - O ) C\ ) r-1 Ln t - n L 4 1 i-t to re r1 c- O O m (DO Co Cto o- r-l t L o oC> C on o-e- o r-! t O o-''H O.o <* a) Lo cr\_ -:,\-t a n \ o n c\ Q) + O, CU n,\ _: r-l co 0n _:t -i- td - rf t- c(N n (I cn H) O C\l 0 >-t 0 C- L CO rt l — CC) zr O\ -\ LC n I I i 1 i Io \ I I I I\ I I I \ I I 4-) - c. O c\ -C- CO O Ck' N CL\ O - \o L\ Cj co - ra. Ln C\ CO L -H C- 4 O - r C MtC - \. ON\ C -t (1 O o c \n \ con CO QO tC\ O \ D n C\jt r- rC Nl) CO ) CO\ Lcn 0rln r JO O rL\ k \ \0 C l k o o' o o v e o~ 4 o~- te- op ~-,o,.o o' o,, o \rl co ) 0 _ 01 t- F N q 1, C, ON N -HO ln - O'C\ 0ln f C \O \ \f. C\j 1 () C ON - 4 \ -:I- COO \ t- \D Ln O Ln O \!D, \, O O t rt O. t — O t.~'~ t O.. OCM CM r ^t_:-t - P n n Ln \ 0 > \OD t> — COCO r, H CNNt:lN tV -CO O o H CM Oc\ln L\o t-CO

The University of Michigan * Engineering Research Institute TABLE III SMlo02 BALLOON-LAYERS DATA Balloon Layer Velocity of North-Wind South-Wind | Layer Thickness Sound, Squared Component, Component, L_........... m (m/sec)2 wn, m/sec we, m/sec I 700 115000 - 12.0 - 2o0 II 6000 108000 6.0 3.0 III 7000 100000 9o0 2.0 IV 7000 90000 2.0 - 8o0 ----- 32

The University of Michigan a Engineering Research Institute ooooteN tc\ cm o. ml o K \ 1N KN N K~ \ oo oN ~ o 0000 0 00000 0 000000 0 0 o o ^ ^g goo o o o t 0 44j\ N- 0al\ t- O -- h4N4 oM tT\ t- -t t-l lAOO O~R O t t r\ 4) V K4\ N 4 4-N I'D 4cm 4)40(44(44 CU 0000(44 CU 000(440444 (44 4. 0000 0 00000 0 000000 0 oo0i\ - o o ojr~t-oo\ () r C j - oN io' o ael 44444 oD~n X a aC \0444^~ \ l cu -4-' 4L RU 0000 0 00000 0 000000 0 g \ NN 14 C? U-\ S S g S S 8C 4 g I1 0 C~aC~41-40 K4: cl UQ~'~ 010.... 40 H 04044(4U 4-4 64 0 0 0 0 0 00000 0 000000 0 cijco?7 ^o o o o o o mo lxl _ K oca >r\ t^ <o ko — P d _o l nt- co p r-i r- a)tvi -t Kg. "'1 0 0, ~ 444 \Q Ot r —4-t0- 44 4-4 4.N 44 ^44 4 \ pi 4-P 00 00 tf - *: i 00 000 01 8 K 0 000000 0 0 t. \0 rq rc'\ l Cu C - O — r- F )..... 4... 4\ 0.........0 1 0J H~~522 ~ ~ M ~ ~ 4,,,,a, d ~ P ~ CM 0\K ONf 4 A Kc\ _K t K 44 UN -\5 1 \ 000 0 0 0 000 0 0 0 00000 0 0 o i'^ o\ ^ y U U <, c \J \ a\ Sk 2 N A c N < P, R N 4- CU K\ HHH Nm 04 H I d" 44444444 (00 CCM cm044 (44 CM 4044CCUtU0 -044 (44 t~ Nt lP\>)44a, r' 8 4 H (UI K2 cvj r-joo ir\ oo i-i r-lot4i!0 (,~3 ado oCo.o \Cs o. o b7 14 W\ cm 4) CT\ UM ---------------------- 55 ---------------------- 33~~~~~~~~~~~~~~~~~~~~~

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The University of Michigan * Engineering Research Institute - In the method described it is assumed that the temperature and horizontal winds are constant in a layero In this connection the following can be said: if (a) an atmosphere finely divided into relatively thin uniform layers with parameters. changing slowly from layer to layer is assumed, or if (b) a uniform atmosphere is assumed of the same thickness and of average parameters that represent the same travel time and the same horizontal travel distance for a sound ray with characteristic velocities kn and 1e, then a ray with characteristic velocities km = -' and ke k ~e will spend practically the same interval of time to traverse atmosphere (a) and to traverse atmosphere (b); and the horizontal distances traveled will be practically the sameo This statement has been investigated by means of computer programso For instance, in one case (SMLoO1), the tracing up to the zero grenade explosion was carried out through 21 thin balloon layerso Subsequently, the zero explosion has been taken at the center microphone level; io.o, a uniform layer extending from the ground up to the first explosion has been assumed. The temperatures and winds obtained on the basis of those two assumptions differed by less than a fraction of a degree and less than 0~5 m/seco Similar results were obtained in SMLo02, where only five balloon layers were used in the first place o These computations indicate that no loss of accuracy occurs if balloon data are inaccurate or are unavailable. Layers of standard atmosphere can be conveniently usedo Moreover, it is the writer's contention that the above statement will still hold true if (a) is an atmosphere with linearly varying parameterso Thug data reduction based on linearly varying temperature and winds in a layer will not improve the accuracy of the resultso The question of vertical winds, discussed by Nordberg, seems to be of special interest in view of the fact that Groves* in England proposes to compute vertical winds by means of the rocket-grenade experimento In his method, data from four widely spaced microphones are used to determine the three-velocity components and the speed of soundo The microphones must not lie in a straight line, and the times of travel to different microphones must not be all equal, ioeo, the microphones must not be placed in a circle centered below the explosion. Groves plans to use more than four microphones to increase the accuracy of the experiment by deriving a least-square solution A point which has not been discussed by Nordberg is the assumption that the conditions remain constant through the experiment o The difference between consecutive arrivals is of the order of 15-20 seco It seems possible that conditions of propagation, especially winds, might change somewhat in the low atmosphereo No real estimate of the possible errors is currently thought feas ible *Go Vo Groves, "Theory of the Rocket-Grenade Method of Determining Upper-Atmosphere Properties by Sound Propagation, " Jo Atmoso Terreso PhySo, 8, 189-203 (1956) o

The University of Michigan * Engineering Research Institute The question of finite-amplitude propagation (FAP for short) has been discussed by the writer in a technical reporto* It will be pointed out here that the suggested correction, which can amount to about 5~ for high-altitude layers, is thought to be known with an -accuracy of possibly 30%o Thus an error of about ~ lo5~ (again, only for high-altitude layers) is probable even after the correction has been carried outo The analysis of the effect of random measurement errors on determination of the temperatures and winds based on the simplified method has not been concluded and may be presented in another technical reporto One point will be made here relative to the importance of measuring the ground temperatureo In the currently used method,** the angle of elevation at the ground is computed, using the times of arrival and the velocity of sound at the ground. Both the arrival-times measurements and the temperature measurements contribute to the error in the elevation angleo (The errors in the determination of the array dimensions are negligibleo) The error in the elevation angle contributes to the errors in temperature and winds Using the concept of characteristic velocities, the ground temperature needs to be determined only to correct for departures of the array from the horizontal (see Chapter 4)o These corrections are of the order of 10 msec at most; thus, the error in the determination of the velocity of sound at the ground, which is of the order of 0o3%, does not contribute to the error in temperature and windso 53o CONCLUSIONS The suggested method seems to offer the advantage of being simpler, both conceptually and computationally, than the current method of data reduction: the refraction due to changes in temperature and winds is taken into account simultaneously. However, it does not offer the advantage of determining the vertical winds; and in this respect it is lacking when compared to the method suggested by Groves. It is the writer's opinion that Groves' method, even.though it has disadvantages in that the computations are much more elaborate and the possible variations in the horizontal direction penalize the accuracy of the results, is an extremely interesting development. The first results of the British experiment, in which Groves' method of data reduction will be used, will be available in the near future; and the method can be assessed more conveniently theno *See footnote on page 60 Nordberg, opo cito, po 280 59

The University of Michigan * Engineering Research Institute 14o ACKNOWLEDGMENT The rocket-grenade experiment has been conducted jointly by the Uo So Army Research and Development Laboratory, Fort Monmouth, New Jersey, and The University of Michigan under Contract No, DA-36-039 SC-646590 The temperatures and winds from SMoOl1 and SMlo02 rocket flights were obtained as the result of the joint effort, and should by no means be regarded as obtained solely by the author of this reporto The University of Michigan personnel participating in the experiment are listed on page vo The Uo So Army Research and Development Laboratory group has been headed by Mr. William G. Stroud and included Dro Wilhelm Nordberg and Captain William R, Bandeen. All of them participated in all the stages of the experimento 40