ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Technical Report THE EFFECT OF FINITE-AMPLITUDE PROPAGATION IN THE ROCKET-GRENADE EXPERIMENT FOR UPPER-ATMOSPHERE TEMPERATURE AND WINDS Joseph Otterman Department of Aeronautical Engineering Approved: Lo Mo Jones ERI Project 2387 DEPARTMENT OF THE ARMY, PROJECT NO. 3-17-02-001 METEOROLOGICAL BRANCH, SIGNAL CORPS PROJECT NOo 1052A CONTRACT NOo DA-36-039 SC-64659 April 1958

The University of Michigan * Engineering Research Institute ABSTRACT This report presents an estimate of the effect of the finite-amplitude propagation on the travel times of pressure waves from the explosions to the ground and a method for taking this effect into account in the data reduction of the rocket-grenade experiment for upper-atmosphere temperature and winds. ii

The University of Michigan * Engineering Research Institute THE UNIVERSITY OF MICHIGAN PROJECT PERSONNEL Both Part-Time and Full-Time Allen, Harold Fo, PhoDo, Research Engineer Bartman, Frederick Lo, MoS., Research Engineer Billmeier, William G,, Assistant in Research Harrison, Lillian Mo, Secretary Henry, Harold F,, Electronic Technician Jew, Howard, MoAo, Research Assistant Jones, Leslie Mo, BoSo, Project Supervisor Kakli, Go Murtaza, B f., Assistant in Research Kakli, Mo Sulaiman, MoSo, Assistant in Research Liu, Vi-Cheng, PhoDo, Research Engineer Loh, Leslie To, MoSo, Research Associate Nelson, Wilbur, MoSoEo, Profo of Aero Engo Nichols, Myron Ho, PhoDo, Profo of Aero, Engo Otterman, Joseph, PhoDo,, Research Associate Schumacher, Robert Eo, BoSo, Assistant in Research Taylor, Robert No, Assistant in Research Titus, Paul Ao, B.So, Research Associate Wenzel, Elton Ao, Research Associate Whybra, Melvin Go, MoAo, Technician Wilkie, Wallace Jo, MoSoEo, Research Engineer Zeeb, Marvin Bo, Research Technician iii

The University of Michigan * Engineering Research Institute 1o INTRODUCTION The rocket-grenade experiment for upper-atmosphere temperature and winds is based on measuring the time of travel and the direction of arrival at the ground of waves set by charges exploding at high altitudeso It has been assumed in previous calculations that the waves travel with the velocity of sound, iOe,, the velocity of an infinitesimally small disturbanceo In reality, the wave from the explosion travels at a higher velocity, and the departure from the velocity of sound increases with increased amplitude of the wave. This effect of the actual propagation can thus be called the finite-amplitudepropagation effect. As part of the assessment of the systematic accuracy of the rocket-grenade experiment in determining the temperature and winds, an attempt was made to estimate the error due to the finite-amplitude propagation of the pressure waves from the explosion. The problem is rather complicated since it is necessary to estimate the difference in finite-amplitude propagation for two successive explosionso To the first approximation, the finite-amplitude-propagation effect on temperatures and winds computed for a layer between two explosions cancels out for two successive explosions if equal charges are usedo In other words, it is necessary to assess how the differences in anbient conditions between the bottom and the top of the layer between two explosions affect the shock-wave propagation. No experimental data could be found on this subject within the range of air densities involved in the experimento The examination of the problem shows that, other conditions (primarily the explosive charge and the distance from the explosion) being equal, the wave travels faster in a more rarefied mediumo This is so because for the same energy content, the wave in a rarefied medium has to exhibit a higher relative pressure (ratio of the wave pressure to the ambient pressure)o With increased relative pressure, increased velocity resultso Thus the travel time from a higher explosion is shortened more than the travel time from the lower explosiono If this effect is not corrected, the calculated temperatures will be too high. The calculations which have been carried out indicate that the effect of finite-amplitude propagation is rather small, but not negligibleo The effect increases rapidly with increasing altitude It is suggested that the following correction for the finite-amplitudepropagation effect be introduced into the data reduction: a certain interval of time tg should be added to the measured time of propagation~ That is to say, the measured time of travel from the explosion to the ground microphones -------------------------- 1 ---------------------------

The University of Michigan * Engineering Research Institute should be increased (for the purpose of calculating the temperatures and winds) by an amount which is believed equal to the shortening of the travel time because of the faster-than-sound propagation. This shortening of the travel time has been computed for different altitudes for 4-lb and 2-lb explosive charges of the type used in the current experimentso It should be pointed out that such a correction is especially important in the layer between the explosion of a 2-lb grenade and an explosion of a 4-lb grenade NOTATION C velocity of sound E total energy of the explosion Ew energy of the shock wave Fo finite-amplitude-propagation effect (distance) at Ro Fg total finite-amplitude-propagation effect (distance) from the explosion to the ground Ho altitude of the explosion Hs atmospheric scale height L length of the positive phase of the wave Lo length of the positive phaserof the wave at distance Ro AL finite-amplitude-propagation effect (distance) from Ro to ground P ambient pressure on the path of propagation PO ambient pressure at the explosion, assumed unchanging to Ro R distance from the explosion Ro distance from the explosion at which JI =.075 tg total finite-amplitude-propagation effect (time) from the explosion to the ground t time to time from the moment of the explosion to the arrival at Ro T absolute temperature AT error in temperature determination V shock velocity x dimensionless distance from the explosion R/2HS xo dimensionless distance from the explosion at Ro, Ro/2Hs xg dimensionless distance from the explosion at the ground, Ho/2Hs y ratio of the specific heats II relative overpressure (ratio of excess pressure to ambient pressure) IIo- relative overpressure at Ro, equal to o7075 2, CALCULATION OF THE FINITE-AMPLITUDE-PROPAGATION EFFECT The calculations are based on the approach that has been outlined in the quarterly report of November 15, 1955, Report 2387-6-Po --- 2 -2

The University of Michigan * Engineering Research Institute The velocity of a shock wave V is related to the relative overpressure II by the following equation: V c (1+ r+ 1l ), (1) and for small overpressures: V C( + 42 II (2) where y is the ratio of the specific heats. These equations are based on the Rankine-Hugoniot relations and the equation of state of the perfect gas. In the calculations, symmetrical propagation at the explosion was assumed: the effect of the velocity of the grenade at the moment of the explosion (equal approximately to the velocity of the rocket) has been neglected This velocity will usually be of the same order of magnitude as the mean velocity of the molecules of the explosive gases. For grenades exploded on the up-leg of the rocket trajectory, this directed velocity of the sphere of the explosion products causes a pressure build-up in the upward direction, and a decrease in the amplitude of the wave propagating downwards. Thus, the directed velocity tends to lessen the effect of finite-amplitude propagation for a single explosion; but, on the other hand, it introduces a new element in the correction, ioe o the correction depends on the difference in the velocities of the grenades at the times of the successive explosions relative to the path from the explosion to the microphoneso Since the grenades are ejected on the up-leg, this velocity decreases with successive explosionso This tends to increase the finiteamplitude-propagation effect, since the difference in the finite-amplitude propagation between two successive grenades increases The calculation of the propagation for each explosion has been broken dowr into two phases: (1) propagation up to the radius Ro, up to which point the change in ambient density relative to the point of the explosion has been neglected; and (2) propagation from R0 to the ground, where the attenuation of energy of the wave has been neglected. The calculations are largely based on a dimensionless solution of spherical blast waves by Ho Lo Brode 2 A particular wave form, shown in Figo 1, has 1o For the derivation, see, for instance, J.WoM. DuMond, et alo, "A Determination of the Wave Forms and Laws of Propagation and Dissipation of Ballistic Shock Waves," J. Acoustical Soco of America, 18, 97-118 (July, 1946)o Note a misprint in Eqo (10) on po 104; instead of po, read po 2o Ho Lo Brode, "Numerical Solutions of Spherical Blast Waves," JO Appl. Phys, 26, 766 (June, 1955)o The numerical values describing the wave form of Figo 1 were obtained from Brode in private correspondence. - 5. —----

The University of Michigan * Engineering Research Institute to_ 3 ( E / tO= 0.075 PO Lo= 0.44(E ) \ PO/ PO Distance from the explosion RO= 3.63 P-)OFig. 1. The wave form at the distance Ro from the explosion. been used as the starting point of the calculations. This wave form with an overpressure of about.075 occurs at the distance R, Ro = 3.63 (E/Po) 1/ (3) and at the time to to = (3/C)(E/Po)1/3, (4) where E is the energy of detonation, and PO is the ambient pressure. The finite-amplitude-propagation effect (the distance from the center of the explosion minus the distance that a sound wave would cover in the same interval of time) at this stage, i.e., up to Ro, amounts to Fo = Ro - toC = (363 - 3) (E/Po)/3 = 63(E/Po)/ (5) The length of the region of positive overpressure Lo is Lo = 0o44(E/Po)1/3 o (6) Brode's solution takes into account the viscosity of the air. Thus, up to the distance Ro, the attenuation of energy is not neglected. The energy of the explosion of a 4-lb grenade was taken as 1.472 x 106 kgm and of a 2-lb grenade as 0736 x 106 kgm, corresponding to the assumed specific energy of 1900 cal/g for the explosive. The value of (E/Po)1/3 is thus 5.22 m, 4

The University of Michigan * Engineering Research Institute for the sea-level pressure Po = 10332 kg/rm2. Different values of (E//P)1/3 with Po changing by a ratio of 2 are compiled in column 2 of Table I for 4-lb grenades and of Table II for 2-lb grenades o In columns 3, 4, and 5 the lengths Fo, Lo, and Ro are tabulated; the calculations are based on Eqs, (5), (6), and (3), respectively, It will be noted here again that up to the radius Ro, the change in ambient pressure with the distance from the explosion is neglectedo Since Ro is at most of the order of 1 km, the effect of changing pressure is very small. In column 6 the altitude Ho corresponding to the pressure in column 1 is tabulated, using the ARDC 1956 Model Atmosphere.3 For propagation from Ro on, the total energy of the wave is assumed constant and equal to w R2I2 (R) L(R)P(R) Ew L=, (7) where R is the distance from the explosion, L(R) the length of the positive overpressure region, II (R) the relative overpressure (the ratio of excess pressure to ambient pressure), and P(R) the ambient pressureo4 The finite-amplitude-propagation effect (lengthening in L), is computed by means of Eqo (2) for the shock velocity as a function of the relative overpressure. The relative overpressure is o075 at Ro and decreases in accordance with Eqo (7) with increasing radius. For such small overpressures, the use of Eqo (2) as compared with more exact Eqo (1) involves only a very slight inaccuracy. Considering the fact that the zero-overpressur-e point moves almost exactly with the velocity of sound, and using Eqo (2), we express the rate of increase in the length L of the positive phase of the wave as the function of the relative overpressureo dL(R) dL(R) dt ~ dL(R) 1 dR dt ~ dR dt ~ C(R) (8) dL(R) V(R) - C(R) - C(R) I (R) (9) dt 4 aL.(R) y+l dL(R) - + ~Z[I (R) o (10) dR 4.7 5. R. Ao Minzner and Wo S. Ripley, "The ARDC Model Atmosphere, 1956/ AFCRC TN-56-204, ASTIA Document 110233, Geophysics Research Directorate, Air Force Cambridge Research Center, December, 1956. 4. Ho Ao Bethe, et alo, "Shock Hydrodynamics and Blast Waves," AECD-2860, October, 1944o Equation (7) of this report can be obtained from Eqo (367) by Bethe by substituting the value of C. ----------- 5 -------------------— 5

The University of Michigan * Engineering Research Institute 0N\ K> co l- Ct- CO 0O.-t Ar- c\ O. O- t 0 -O C - - \D CO "\ CO r- X e o c o e * X o. A A O',',.- "',\,' ", C', X 0c \. 0d\ C "- C O. C! N _n ".-\ cO I CO, O O b'" L~ — r OC 0 -: r, CC n C C CO \O CU OC rON^ O-O - O O O OKr\; -'lr\ CM j N ".CNz r~-.' I C Mc CO c c O O c CO n nCO n - r-\ cO C- Ln CO rC, O O C', Ln CU -\ - Z \ j- t-n\ n O Ln CO L0 K co Cn C00 c co 1K - co LF O O O O0 rO O r- 0O O O O O\ O-\ C O O O O lO O- O O O O O CO \O 0 r LOC r QJ 1C\0 j LC0 0NK -0N0 0 C\Q LrN-f i-CU 0 0 00 V - 0 rt * o _ * e eO*O\ - e * er- 10 r-A 0\0\0\O C 000000 \0D 0D \, CO CC) \ 000 00000A A n n ~: j \ r' r4 \ — ) cO \ I. \F 0 -c::- \..t Oo \ <\ \9 O\o ~o o; o~ GI o Q -coco oe- on'\ co o\ 0 o rL s H\D "O-d- CUKOKO On K0 C\J-O KO O K \ 0 00 9 0 \ 1 C\ rI N N 0 L \o r-A r- N CO 00 n. \ t uz s\ S'n U —.\ 1t C,-.Oi,).l\ oN \0 \ co \ \ - o rH r —O r-\ A CM Kj L\ LO OK t-Ln ON ON ON 0 C rpq r-i r- o-4 r-4 E- C m CQ oN-O OO c JOnOL\ O C O H +^ P^4- l A\,D l CO Ci n -\t LQn\ O CO ri COD -. 0 0 co cr-4 r-4 r-4 J l CU' U2OO J O C CO O O 0 00 OO 00 n o 0000.q O ), \O CO 0 - — n O CO O O O O C0 r r-4 r-4 Nd r -OJ L 0 D Co O O Cd O L OO o cO 0 1 1 H N Ck n > \o Z 0 CU' O n C0 O O X ~- A O C CC Ct rrcc> -— vL0000 0" N) O O O O L CUrL4 n CO ch o ro H O C ood cc o-o-.Lroccooooo K^ l F| _d Co t D C- O LI',d r — d d d \-O CO — d-t r r -1 r —4 r — 1 N O 0 oocco.. o. N.Oo _C CD AOAACUOCUMNr LLA t O CO

The University of Michigan * Engineering Research Institute l- r< l- \9 K^0Q\ K ^ \ O J rN Kc r EH 0 V C' r,\c - L-t \ CO r- \ 0 - C - - CO CO CO. OJ <I o... 0 o * * * o *. 0 C) HC \O c>O r i CO O OOr-i MCoO\ C CJ Q O~ r-, L( co o c' Lr co 0.. oco a-,O i o C\L\nc r — r — 01 O). O * * N O * o. o Hr 0V U2 n n r r COI- r N; OCOO \ rH rH H- cO o\ \ o\ 0 Or-O C N GKO\ a N A 0- \ CO CO CO CO CO CO N a J CN J WI \ rc \ W W^ K N NJ CJ Ca N C\j C C\JJ bhO 0? rHlCO CC r- J CO J \ r-1 CO D( (a) * r- hK ^4 \, 6 K r r4 K t-rI 1c \l N -i ir -~- — r- l I LJ i - r l r- C r Wr O- -lC Zl - n F Jt r O CO Q CO CO N CO r-l 0 K\ N N \OD H' \D W\ Wn W a \, 1 H —4 [x<+ H4D OJ O 0 d. — n Ck0 CO- \iK C) KL\ -H —I \' 0'x r- - i'0OJ0 ( ( D r-r r 4 r- r- r CQ LA C\ K CO r \ \,Og CO aJ g Lil i oJ ol1 CO 0 0 o o - c O I I rO HHH0J01C\ CO rN \ID n r:- 1 r4 o 0!. b * * 0 o Q- EL r" r, -0- KO rP r0 C',' C\0 C\ \O D K0.-d-0\ t o 1 o V cit br r- l r- C' CO N \ KI Ln \r -O- CU0 \ 0 0 0 H, C1 ~ 0 0 00 O I. L I EH H CU H CLC oE H KZ-n \J c -t —c — 0' \ X (H H n c O L O& 1 H t r-4 r —4 G0JH 1 \j LC\' 2 r- C\ J -P H, H C -U KO > K- - t — l\ - \O' C - -OO 0C -' -o, O\.\- Llf C O, a Q j n X -1. 0 H b o — I CO i\ -..-t L0 \ r-, r-. in. n - CO -t r — L11 k * ~;S da - c d r r-I - -1c 0N 0 0. O1 lC a tq a:~~~~~~~~~~~0 N,' 0" 0..I r —4 I 0 r x L N CC O\ r- N 0 O - \ -I r- r —I nC r-i-lr —ir —1GI04OJ'^_.~..f'I-hO- C~ O r-l' r — r- r - -4* *C\L-. tJ > X \C\JCMHCMN 1H j H C O \LO, \ O O0- O N-I' C1 0 Od N\0 C \ O O4\ CC )O O ~~~~1 1H 1d* hnD r^c r-4 r-4 — + Ln -l 0i 0 r0l r lr r-^ - ~UX l~ -in —HO)lO t-t r-I -t —!-O- ---- 7- -H HHHHHHHHH *3 ~d ed~~~

The University of Michigan * Engineering Research Institute r CO Ct,-O 0 9 -t r-H \ O b0 D- O C. — 0 9 cO cO CO Hr- X o o o e o *. e r- r-I N CUN N CU K 0 X c,D \o D (O',,CO\ CO Ln n.-" t-: - - CO - CC -. CO I Cj OCO L n t rr- 0'c\H 0 lr\0' —- j 0 CO,0 01\I Lp ir ^ ^ ^ ^ i^ rcvl 0 c01 01 ra rC L\ cO cO CO -t \.o - cO L n r-4 r — -0 \ CO CO C\JO D O C O O.:.:. -..n,l O -C- COO O'- 0 t- C fr — >-CO - CO o O O O O OH ri rI. I Cl. \ -:O - O' r-! GO 0 0000000HH0 0 0 \0 i 0 0-0 NH O000O00000000000000 r- - cO OO - O C O LO,,O I.. -I OH 0,._.-O O co o\ oO o nI>- O CoO L( CO gC _- O r- cQ Kl _ -- ru c r'3 0n O..- - \ O —t -0i O, \ C'\ CO' h C \ PX * M o 0 * * e0 o. * * * * * * @ C \I r-I \ -' C I _ D \ID O -' - 0 -- _ \ - \IDO KIx K l\c h-OJCO 0"\ CN L I- lx\ LN r'c C' — CO O O r-H t\ \..O....... c WI CO CH 0 r-\HL0 C\j -N CNO- OM HH_00 \ H C)J C\J LC \\ _:J LN n H n l\ \ t) > C0CO CC) 0\ oI CD r \-O \O Dn L C\ O OD10 n O C\I r -- NO O *... o... e * *.. H U) H-l ~~ ~n -, 1o- r, I-. o CO \) CU \ \ | ( - r-I r- rl CO l C\U N -- L\ O\ r -t o[x] HHH0101VL(N^ 0~~~~~~~~~~- I r — PSQ ~ ~ ~- C C n -c r m0 _- 0 o 0 cO c',1 O'x 0 \ID cO 00 N —. 0...- Ur"X\ 0 CO 0 0 \ID 0 0 (1) c * e o * o d O* r* o - (U (-l(-4r-10J~U~~~jrQ -t L^ ["~r- r-l r —i OC J Cl O C0 0 COO 0n \C 0O C o — 0 LN nCO O O O V O O -P e * * e *r O L *- oO K * 0 0 eO -o r —.i r.-.i r.. \A -i OO O i x. i 000O 0 L nOO US O *O O O cCOdr L F OO r~ O O O O. 0 l d Cd % 4 ON O 4N ^. 000000 LF D O O O O O LOC L t-COi-OO H O O O OL P O O Lr\.O.r 0 -^ 0 \OJ CO r4 V:L O OrCOO -- 0 CO \ C >-d- C1 \Hr-l L' C'j r-4 r-H 0C -r' LN\ I-CO OC O r0-I CH 01 - LN \'O t 8 -

The University of Michigan * Engineering Research Institute H > cO.S C. r 0 0C c sa E y u H ~ r) o c o o oc o, e o o o o ~,o 0 0 0 0 O 0Hr- - C\J C\K N jLr\ C CO O COO Co C0 U I- I _ -r-H O N C c Hr- Or- O r O O J ON 0 r - 1 H Ct N Q LrE 90 FC r1 n n r-4 L — CO \n J C) 00 0 C )0 r0 r- r- r- CO H 0\ O O r- C\J \ CU r- C\0 CO C CO CO CO CO 0 N' —. O kO NO O HOdo J H O ^c O xJ ( o c \cO Jc O N HOf C C o K K c\ \ K H \ N j Cj C C CN C | CX b | r r-i-K r4-I r-r1 CI lo C I cO bo Oo ooC IO UCD)~~~ 0 1 Ci OO C0 CC o) CC) O",\ W r 0,1\ 1, n1 N _:J I c o -: no o o\ o WI \ ( _t L\ CC-C O) 0 W\ \ION\ KNO\ r- L\ l 0 > — C\ HrHQ r\r-q t-li-4OJOJK K\-t4 - C\U\O C\j CO -J0O ~! O O r' I O b O0 C cO \ (D 0 0' Co\ C) KO b O r3 i-t Gl\ 0 r-;4 _-zf C~j - KI r-l4 O C\JLn r-4 r-l C \ K^ K11i \ a) 0(Dr-r r0 o C\J JC\ - L _ O n o - o\ or- o o 0 - rd - r-r- r- CI C\J bO 0 U m o (-l O _ O cO O LO o CQ 0 co C,\ nH JO r.-. r-i i,. T r'Cl H -0C?I bid) rr 1-ir K\- co 1 H r-J ll -I *Co r4 C 1-^1-1^> rl~~~~~~~~~-.r-. I C -.-I 0 00HHHC\JO J ClOlO' 0 >- -., 00 b +.C~~~~~~~~~r-I)~~~~~~~ ~C\r k 0 CO _:-'COCr-lcr-COC\O0 cO \n\LrN-Hc0 r-l \ H 0'0 0 0 0 0 00 XC CO b~ 0 0 -1 >J d, C 0 d- LD - H CC) 4\ C\J \O O\ — t- \O: -n - rd- rd- r- CO -rd Cr rd bO H-~ \O \o -J c\ r- G\ \- CCO C) CQ 0 CC) \D H I r- r —4 C) 0 0 0 0 0rH --------- 9-o ----------— O —O -C — H - HHHHHHHH *~~~~~~~~4

The University of Michigan * Engineering Research Institute Introducing the value of 11(R) from Eq. (7), we have dL(R) 7+ ( 1/2 1 dR hy ) R[P(R)L(R)]1/2 0 It will now be assumed that the propagation takes place vertically downward and that the pressure on the path explosion-to-ground changes exponentially with altitude, the scale height Hs being constant and equal to the scale height at the altitude of the explosion; therefore P(R) Pe s /H 0 (12) Introducing this value of pressure into Eq. (11), we obtain dL(R) 7+1 1/2 e-R/2Hs dR - hy BPI/ R P 2[L(BR)1/2 (5 1/2d -y+l 7Ew1/2 e-R/2Hs [L(R)] dL(R) dR 47 VPO) B e-R/2Hs e-x K d K dx (14) R/2H5 2H5 x where the dimensionless variable x is defined as x= (15). 2H, and the value of the constant K is K = ( )/ - ZB- ROL1/2L (16) The second equality in Eq, (16) follows from Eq. (7)o Integrating Eq. (14) between Ro and the ground, we obtain: Xg (Lg3/2 - L03/2) = K - dx 5 xo x 3 9 K{-Ei(-xo) - [-Ei(-Xg)]3 -y XB0Lo1/2Eo-0Ei(X)1 [Ei(Xg)]}, (17) 47 1 10

The University of Michigan * Engineering Research Institute where Lg is the length of the positive phase at the ground, and where xo and Xg are given by Xo (18) 2HS xg - H (19) The exponential integrals -Ei(-x) have been tabulated, Columns 7-16 of Tables I and II represent steps in computing Lg by means of Eqo (17)0 In column 17, the finite-amplitude-propagation effect from Ro to the ground A L = Lg (20) is tabulated. In column 18 we have the finite-amplitude-propagation effect Fg in meters, from the explosion point to the ground: Fg = Fo + AL (21) In column 19 the velocity of sound C at the altitude of the explosion is tabulated from the ARDC Model Atmosphere, 1956. In column 20, the time tg, the shortening of travel time, is computed by the formula F tg = og (22) C The results, Fg and tg, are plotted, respectively, in Figs. 2 and 5 both for 4-lb and 2-lb grenades as a function of altitude Ho of the explosiono The effect of neglecting the problem of the finite-amplitude propagation on the temperatures derived from the rocket-grenade experiment can be computed readily under certain simplifying assumptions, Two examples, one involving 4-lb grenades and one involving a 2-lb grenade at the lower explosion and a 4-lb grenade at the upper explosion, will be presented in detail. Assume that 4-lb-grenade explosions occur at 82 km and 86 km. In this altitude interval the speed of sound is approximately 281 m/seco The difference in travel times from the explosions to the ground, for close to vertical propagation, will be of the order of (86,000 - 82,000)/281 = 14.2 seco The difference in the finite-amplitude propagation for the two explosions will be 5. Ao N. Lowan, Techo Director, Tables of Sine, Cosine, and Exponential Integrals, Vol. I, 1940, prepared by the Federal Works Agency, Work Projects Administration for the City of New York, 11

The University of Michigan * Engineering Research Institute 400 -- 360 320 ------ _ 280 4-lb grenade 240 200 l 160 ll 120 80 / --- 40 ""' -2 -I b grenade 0 20 40 60 80 100 Ho k m Fig. 2. The finite-amplitude-propagation effect (distance) as a function of the altitude of the explosion. 12

The University of Michigan * Engineering Research Institute 1500 1400 1300. 1200.. 1100 1000 900 800 700 600 4-lb grenade 300,_,_, 200.___.._____ 2-lb grenade 100C ----------— 13 I ------ -------------— 153 —-----------

The University of Michigan * Engineering Research Institute (from Table I, column 20) tg upper - tg lower = 938 - 761 = 177 msec. Thus, the error in the determination of the velocity of sound in the layer between the explosions will be of the order of (.177/14.2) = 1.25o% The error AT in the determination of the temperature T will be of the order of +2o5%T, or about +4.8~Co In columns 21 of Table I and Table II the error AT has been tabulated for 4-lb and 2-lb grenades, respectively. This error has been calculated as in the above example, and applies to standard atmosphere and to explosion intervals corresponding to altitudes tabulated in columns 6 of the tables. Assume that an explosion of a 2-lb grenade occurs at 64.5 km and an explosion of a 4-lb grenade occurs at 69.3 km. The difference in the travel times will be of the order of (69,300 - 64,500)/304 = 15.8 sec, assuming the average velocity of sound to be 304 m/sec. The difference in the finite-amplitude propagation-will be (from Tables I and II, column 20) tg upper - tg lower = 389 - 248 = 141 msec. The error in the velocity of sound will be of the order of (.141/15o8) = o9%; and the error AT in the temperature will be about +1.8%T, or +4.2~C. Certain comments will be now made regarding the calculations, reflecting on the expected accuracy of the resultso The calculations by Brode apply to the point-source solution in an ideal gas. This solution resulted in a higher overpressure at the scaled distance 3.65(E/Po)1/3 than the other calculations by Brode, carried out for real air and for TNT explosions. The solution resulting in the highest overpressure was chosen because it corresponded more closely with the experimental data reported by Schardin6 and by Granstromo7 For the same scaled distance, Schardin reports a relative overpressure of about 0.10 and Granstrom, of 0.08, as compared with an overpressure of 0.075 for the point source in an ideal gas solution by Brode. The numerical values by Schardin and by Granstrdm would, if used in calculating the finite-amplitude propagation effect, produce results higher by about 335 and 7%, respectively, than the current calculations. The pressure and temperature profile, as stated, has been assumed according to the ARDC Model Atmosphere, 1956. It can be shown by evaluating the expression [d(E/Po)l/3]/dHo - AHo in terms of average atmospheric parameters in the layer, that the effective correction term for the distance (Fg upper - Fg lower) between explosions of equal charges is approximately inversely proportional to the average absolute temperature, and inversely proportional to the 1/3 power of the average ambient pressure in the layer. The effective correction term for the time (tg upper - tg lower) is approximately inversely proportional to the 3/2 power of the absolute temperature, and inversely proportional to the 1/3 power of the ambient pressure. The temperature is not expected to depart by more than 10% from the Model in the range of altitudes 6. Hubert Schardin, "Measurement of Shock Waves," Comm. Pure Appl. Math., 7, 223-243 (1954). 7. S. A. Granstrom, "Loading Characteristics of Air Blasts from Detonating Charges," Trans. Roy. Inst. of Tech., Stockholm, Sweden, No. 100, 1956. I. 956l. -------- -— 14

The University of Michigan * Engineering Research Institute considered, but the actual ambient pressure can be different by a factor of 3 or more. These possible very large departures in actual pressure from the assumed pressure may cause considerable changes in the finite-amplitude propagation, even though they are reflected through the 1/3 power only. It should be pointed out that recalculation of the results for a given atmospheric profile could be done in approximately one working day. This use can be made without changing the first 6 columns of Tables I and II. The computation of AL depends on Hs. The exponent 1/2 Hs is assumed to be constant on the propagation path from Ro to the ground, where Hs is taken equal to the scale height at the altitude Ho; AL is only about 1/4 of Fg; and Hs varies rather slowly. Most of AL is due to finite-amplitude propagation at the distance from Ro to 5Ro, that is, not far from the explosion. Because of these three points, it is thought that a constant Hs is a good approximation. Vertical propagation has been assumed. For propagation at an elevation angle a with the vertical, instead of Eq. (IJY the formula for the variation of pressure with the distance from the explosion will be P = peR cos /Hs (25) For small angles of elevation, and considering again that AL is only a quarter of the total effect Fg cos a = 1 is a reasonable approximation. The divergence or convergence of the wave, other than the spherical spread of the wave, has been assumed negligible for close-to-vertical propagation. 5. CONCLUSION The effect of finite-amplitude-propagation has been calculated for travel. times from explosions at various altitudes of 4-lb and 2-lb grenades. The results are presented in the forms of tables and graphs. The effect, if not taken into account, can cause errors in the determination in upper-atmosphere temperatures of the order of 5~C, The appropriate correction consists of adding to the measured travel times the calculated shortening of the travel times from the explosions to the ground due to faster-than-sound propagation. C —-------- 15>~ — 15