The air wave surrunding an expanding sphere By G.. TAYLOR, F.R.S. (Received 5 December 1939) [Fr summary see p. 292.] NTRODUCTON When the surfce f a sphere vibrates in any assigned manner the spherical sund waves which are prpagated utwards can be represented by wellknwn frmulae prvided that the mtin is such that nly small changes in air density ccur. When the mtin f the spherical,surface is radial the velcity ptential f the sund wave is rp = r-1f(r-at), (1) where a is the velcity f sund and r is the radial c-rdinate. The velcity, U, and the excess, P - P, f pressure ver the atmspheric pressure P are U = r-2f(r-at)-r-1f'(r-at), P-P = -par-1f'(r-at). f R is the radius f the sphere which, by its expansin, is prducing waves, R is a functin f t and the surface cnditin is R = R-2f(R-at)-R-lf'(R-at). (4) Equatin (4) is an equatin fr finding the functinf. A simple case in which equatin (4) can be, slved is when R is cnstant s. that the sphere is expanding at a unifrm velcity. Taking t = 0 when R = 0 the radius at time t can be expressed in the frm R = aat, (5) where a is a nn-dimensinal cnstant. The limitatin that the changes in density are small implies that equatins (1)-(3) are true nly where a is small cmpared with 1. Writing w = R-at = (a-)at equatin (5) becmes a- (a-)2 -f'(w)- - f(w)+aa = O. aw aw The slutin f equatin (6) which is valid fr negative values f w is aas f(w) = -12W2+C( -W)<"-l)". -a (2) (3) (6) (7) Vl. 186. A. (24 September 1946) [273] S
274 G.. Taylr The cnstant f integratin c must be taken as zer in rder that few) may vanish when w = O. Hence aa3 (r-at)2 = 1-a2.r ' (8) u = (a2t2 _ 1) } 1-a2 r2 ' a 2 a 3 (at ) P-P = 2P1_a2 -r- 1. f at time t = 0, u = 0 and P - P = 0 everywhere, then at all subsequent times u = 0 and P = P in the regin utside the sphere r = at. t will be seen that bth u andp-p are cnstant when rjat is cnstant, thus pints where u and p - P have any assigned value are prpagated utwards at unifrm speeds which are prprtinal t distance frm the centre. Subject t the limitatins f the thery f sund therefre* the air wave prduced by a unifrmly expanding sphere expands at a unifrm rate and the velcity and pressure at crrespnding pints are cnstant at all stages f the expansin. This result might have been expected a priri but the slutin is here given in detail because it frms the starting pint f the wrk which fllws. (9) ANALYSS WHEN VELOCTY OF EXPANSON S NOT SMALL t seems likely that a unifrmly expanding sphere will be surrunded by a unifrmly expanding air wave, accrdingly a slutin f the cmplete equatins f mtin is sught in which u and p are functins f x = rt nly. Fr such mtins ( r a) (10) at+t r (u,p,p) =. The equatin f mtin is u au 1 ap -+u-=--- at r par' and in view f equatin (10) this may be written (u_x)du = _dp. dx pdx (ll) (12) * t is shwn later that the sund wave equatins themselves are nt valid in this case, even when (X is small, but this fact des nt invalidate the expressin (8) regarded as a slutin f thse equatins.
The air wave surrunding an expanding sphere 275 The equatin f cntinuity is p p (OU 2U) t +u r +p r +r = 0, which in view f equatin (10) may be written u-xdp du 2u pdx+dx+x" (13) The gas equatin pp-y = cnstant, tgether with the expressin fr the velcity f sund, namely c2 = ddp = YP, give p p and dp 1 dc 2 j' pdx = ')'-1 dx' dp 1 dc2 pdx = (1'-1) c2 dx Substituting frm equatins (14) in equatins (12) and (13) dc2 du dx = -(y-l)(u-x)dx' u-x dc2 du 2u 0 (' - 1) c2 dx + dx + x = Fr cnvenience in calculatin equatin (16) may be replaced by (14) (15) (16) :: = - 2;{_(UXrrl. (17) Equatins (15) and (17) may be expressed in nn-dimensinal frm by substituting the variables g = ux, } 1J = C 2 X 2, (18) Z = lgex. The resulting equatins are d1j 21J1J+(y+l)g-yg2-1 (19) dg = T 31J - (1 - g)2 dz 1 1J- (l-g)2 dg= -g 31J- (l-g)2" (20) The slutin f equatin (19), which cntains tw variables nly, will cntain ne arbitrary cnstant. Withut attempting t express this slutin in 18-2
276 G.. Taylr mathematical frm it is pssible t cnstruct by numerical integratin a cmplete set f (;,1]) relatinships each crrespnding with a given value f the arbitrary cnstant and t set them ut graphically in a single set f curves n a diagram whse c-rdinates are; and 1]. This diagram is shwn in figure 1. The arbitrary cnstant a is defined s as t crrespnd with the cnstant a in equatins (5), (8) and (9), and the value f a crrespnding with each (;,1]) curve is shwn in figure 1. The single curve which cuts acrss all the graphical slutins f equatin (19) in figure 1 will be explained later. BOUNDARY CONDTON AT THE SURFACE OF THE SPHERE At the surface f the expanding sphere u = rjt = x s that ; = 1. (21) n the sund-wave slutin the cnstant a specifies the velcity f radial expansin f the sphere as a fractin f the velcity f sund. n the cmplete slutin 1] represents c2jx2 at any pint s that at the surface f the ( lcal velcity f SOUnd)2. sphere 1] represents l't f.. Crrespndence between ve c! y 0 expansn the cmplete slutin arid the sund-wave slutin is therefre attained when the arbitrary cnstant a is defined by the relatin where 1]2 is the value f 1] at ; = 1. The curves in figure 1 were cnstructed fr a series f values f a starting at the pint (; = 1, 1] = a-2 ) and calculating the change in 1] step by step fr small decrements 8; in ; thrugh the range; = 1 t; = O. The change 8z in z in each interval 8; was als calculated using equatin (20). When 1] has been fund as a functin f; the slutin f equatin (20) is f the frm Z-Z2 = (functin f ;), where Z2 is the cnstant f integratin. f Z2 is chsen s that Z = Z2 when ; = 1 then (23) (22) OUTER BOUNDARY CONDTONS At the uter bundary f the expanding air it must be pssible t cnnect the still air cnditins with thse which btain in the disturbed regin. This can be dne in tw pssible ways: either (a) it might be fund that u = 0 at radius r = at, s that the (;,1]) curve passes thrugh the pint (; = 0,1]= 1); 01' (b) it might be fund that at sme radius the pressure temperature and
The air wave surrunding an expanding sphere 277 velcity are attained which crrespnd with the pressure, temperature and velcity immediately behind a shck wave mving int still air with velcity rt. n case (b) the expanding regin wuld be bunded by an expanding O '-+-----+-+--+--t----j'---t---r-o OL-O.'--2O 3--O4-O5O6--O7--aO9'-,.a f., FGURE 1* spherical shck wave and the air utside this sphere wuld be at rest. The sund wave slutin (equatins (8) and (9» satisfies cnditin (a), fr at f' = at bth 'U = 0 and P - P = O. * The area f figure 9 is indicated by a brken line n the right f the area..
278 G.. Taylr Fr values f ex. larger than thse t which sund wave analysis can be applied it appears that it is nt pssible t satisfy cnditin (a). f the slutin fr ex. = 0 7 fr instance is fllwed (see figure 1) fr values f decreasing frm = 1 it is fund that z reaches a maximum while is still psitive and fr smaller values f, z decreases. Thus the same value f z wuld crrespnd with tw different values f which is physically impssible. t remains t find ut whether the alternative cnditin (b) can be satisfied. Fr this purpse it is necessary t express the apprpriate shck wave cnditins in terms f the variables and 'fl. FGURE 2. Diagram f symbls. A shck wave is an extremely thin regin within which the pressure, density, temperature and velcity change frm ne set f values t anther. The ratis f density and temperature n the tw sides f a shck wave depend nly n the rati f the crrespnding pressures. f P is the pressure immediately behind a shck wave and P is the atmspheric pressure in frnt f it, y = PPO may be regarded as the independent variable in terms f which all ther changes ccurring at the shck wave may be expressed. Figure 2 shws the psitins in the field t which the varius symbls apply. The shck wave frmulae were first given by Rankine (1870) and later independently by Hugnit (1889). The rati f densities n the tw sides f the shck wave is P y-l+(y+l)y (24) P = Y + 1 + (y - 1) y'
The air wave 8urrunding an expanding 8phere 279 where y, the rati f specific heats, is the same as the" y" which appears in equatin (19). n the present calculatins y is taken as 1 405. Cntinuity requires that PPO must be equal t the rati f the velcities f the air n the tw sides relative t the shck wave itself. Hence if U 1 is the velcity f the shck wave and u 1 that f the air behind it r, using equatin (24), U 1 -U1 P --u;- = P' U 1 2(y-l) U 1 = y-l+(y+l)y The cnditin that the shck wave may expand unifrmly with the rest f the system is = r 1t, where r 1 is the radius f the shck wave. Hence U 1U 1 = U 1 tr = 61> r 2(y-l) 61 = y-l+(y+l)y S far as this cnditin is cncerned an apprpriate value f y may be chsen and a crrespnding pssible shck wave fund at any pint in the field. Equatin (26), hwever, is nt the nly necessary cnditin. The velcity f sund in the air behind the shck wave must als satisfy the cnditin and 11 = ct2ri = cijui, c2 a2 {y+ 1 + (y-l)y} l-yplpl- y y-l+(y+l)y' where a is the velcity f sund in the undisturbed atmsphere. expressed in terms f a making use f the mmentum equatin POUl = P-PO = (y-l)p, substituting a2y fr PP and U 1U 1 frm (25) a 2 2y Ui=y-l+(y+l)y' hence frm equatins (27), (28) and (30) 2yy{y+ 1 + (1'-1) y} 11= {y-l+(y+l)y}2 (25) (26) (27) (28) U 1 maybe Values f 6 and r; have been calculated frm equatins (26) and (31) fr a sequence f values f y. These are pltted in a curve in figure 1. The intersectins f this curve with thse which describe the flw in the expanding (29) (30) (31)
280 G. T. Taylr air determine the values f and 'fj, and hence the value f y, crrespnding with pssible shck waves. These have been taken frm figure 1 and are given in cls.. 2 and 3 f table 1 fr values f OG ranging frm 0 5 t 2 1. The crrespnding values f r 11 R btained by numerical integratin f equatin (20) are als given in cl. 4 f table 1.. Crrespnding values f yare given in cl. 5 f table 1. TABLE 1 a: 2 3 4 5 () 1 8 gl 7fl r1r Y p y' P2PO 0 0 2 4 93 1 000 0 203 0 928 1-075 0 4 0 0021 0 998 2 44 1 003 0 410 0 775 1-295 0 5 0 033 0 974 1-950 1'050 0 523 0 750 10400 0 6 0'103 0 916 1 763 1-169 0 638 0'749 1-569 0 7 0 198 0 833 1-503 1 365 0 761 0 755 1 808 0 8 0 291 0 749 1 392 1-629 0 891 0'774 2 105 1 0 0 453 0'597 1 256 2 400 1-180 0 8U 2 959 1 2 0 575 0 474 1-182 3'59 1-520 0 847 4 250 1 4 0 662 0 382 1-135 5 60 1 953 0 887 6 32 1-6 0 727 0 313 1-103 9 06 2 560 0 917 9 89 1 8 0 779 0 256 1-083 17 95 3'598 0 92 19 7 2 1 0 832 0 197 1 060 c c 0 93 c REDUCTON TO MORE FAMLAR FORMS The physical cause f the mtin f the expanding air being the mtin f the sphere, the results are mre cmprehensible when expressed in terms f the rati f3 = velcity f expanding spherical surface = U2 (32) velcity f sund in undisturbed atmsphere a ' rather than in terms f OG. Since U 2 tju1t is Rlr1 fj = U2 U1 = RJ(Y-1 + (y+ 1) Y), (33) U 1 a r 1 2y values f fj are given in cl. 6 f table 1. The rati is related t c 1 c 2 by the equatin, pressure behind shck wave P1 y = pressure at surface f sphere = pz' (34) y'(l-y)y = cm. (35) Values f y' are given in cl. 7 f table 1. By definitin f 'fj c2 'fj R2 R2 i==-2--' C1 'fj1 r 1 rlog1
s that The air wave surrunding an expanding sphere 281 P2 _ JL _ ()1'(1'-1) P - y' - y ricx211 Values f P2PO namely the pressure at the surface f the sphere expressed in atmspheres, are given in cl. 8 f table 1. The way in which P1PO and P2PO vary with fj is shwn in figure 3. 21 (36) 19 7 5 13 1'2 p. 1 A. pyp 9 7 5 3 1-0 1'7'P.--; 1'yp. h -- 0-5 1-0 1-5 2-0 2-5 V (3 1 1 ; 3-0 3-5 FGURE 3. Pressure P2 at sphere and P behind shck wave as multiples f atmspheric pressure P' LMTNG VALUES FOR VERY HGH RATES OF EXPANSON The limiting values f;1 and 11 when y are ; = y! 1 = 0.8316,} 2y(y-1) '1J = (y+1)2. (37) Starting frm these values equatins (19) and (20) were integrated numerically fr increasing values f;. At ; = 1 the value f 1 s fund was 0 226
282 G.. Taylr crrespnding with ex = (0'226)-2 = 2'103, and the value fr1rwas 1 0602. These are given in the last line f table 1. t seems therefre that as the velcity f expansin becmes infinitely great the pressure at the surface becmes infinite, but the density remains finite. The thickness f the layer f expanding air is never less than 6 0 % f the radius f the sphere. VARATON OF VELOCTY WTH RADUS The numerical slutins f equatins (19) and (20) give 1] and rr in terms f g. The mst cnvenient variables fr describing velcity distributin are ua and rat which are cnnected with g, 1] and rr by the relatins ua = pgrr,} (38) rat = prr. Sme calculated values f ua and rat are given in table 2. These velcity distributins are shwn in figure 4 frvalues f p ranging frm 0 20 t 1 95. X = 0'2, fj = 0 203 rat ua pp 0 203 0 203 1 0752 0 214 0 182 1 0745 0 228 0 159 1 0727 0 253 0 127 1 0671 0 300 0 090 1-0571 0 374 0 356 1-0431 0 425 0 042 1 0362 0 594 0 018 1 0196 0 766 0 008 1 0087 1 000 0 000 1 0000 X = 0'6, fj = 0 638 rat uq, pp 0 638 0 638 1-569 0 660 0 597 1 566 0 723 0 489 1 539 0 787 0 405 1 494 0 850 0 332 1 437 0 936 0 249 1 349 0 978 0 209 1'300 1 020 0 167 1 245 1 042 0 145 1-186 1 067 0 114 1-169 TABLE 2 X = 0'4, fj = 0 410 rat ua pp 0 410 0 410 1 295 0 430 0 369 1 293 0 451 0 334 1 286 0 471 0 303 1 280 0 512 0 211 1 263 0 614 0 162 1 213 0 697 0 113 1-173 0 799 0 069 1-122 0 901 0 035 1'068 0 984 1 015 1-000 1-003 X = 0'7, fj = 0 761 rat ua pp 0 761 0 761 1 808 0 782 0 717 0 826 0 640 0 890 0 541 0 934 0 484 1 000 0 404 1 043 0 353 1 087 0 302 H09 0 275 1-130 0 240 1 145 0 225 1 786 1 736 1'692 1 612 1'550 1 480 1 442 1 399 1 365 X = 0'5, fj = 0 523 rat ua pp 0 523 0 523 1'400 0 544 0 481 1 397 0'564 0 444 1 391 0 586 0 411 1 386 0 627 0 353 1 363 0 669 0 304 1 338 0'711 0 262 1 310 0'774 0 209 1 265 0 836 0 162 1 219 0 900 0 120 1-171 0 940 0 093 1-137 0 983 0 065 HOO 1-017 0 031 1-050 X = 0 8, fj = 0 891 rat ua pp 0 891 0 891 2-105 0 935 0 805 2 096 0'980 0 729 2 067 1 025 0 662 2 022 1 068 0 598 1 965 1 114 0 537 1'898 1-158 0'478 1-820 1 203 0 417 1 133 1 225 0 384 1'677 1 242 0 357-1'630
The air wave surrunding an expanding sphere 283,,= 1 0, fj = 1 180 rat ua pp 1-180 1-180 2 959 1 227 1 088 2 939 1 274 1 004 2 889 1 321 0 927 2 822 1 368 0 853 2 731 1 415 0 779 2 621 1 463 0 704 2 485 1 482 0 670 2 413 " = 1'6, fj = 2 560 rat ua pp 2 560 2 560 9 89 2 603 2 474 9 87 2 649 2 385 9 82 2 696 2 291 9 72 2 750 2-198 9 50 2 824 2 050 9 07 TABLE 2 (cntinued),,= 1'2, fj = 1'520 rat ua pp 1-520 1 520 4 250 1 570 1'421 4 231 1-620 1'330 4 169 1 671 1 233 4 067 1 722 1-157 3 927 1 772 1 071 3 747 1 800 1-029 3 636,,= 1'8, fj = 3 60 rat ua pp 3 60 3'60 19 7 3 66 3 47 19 7 3 73 3 35 19 5 3 79 3 22 19 0 3 86 3 09 18 3 3 90 3 03 17 9,,== 1'4, fj = 1 953 rat ua pp 1 953 1 953 6 317 2 010 1 843 6 286 2 065 1 742 6 191 2 120 1 643 6 033 2 175 1 544 5 811 2 215 1 470 5 607 2' 0,. &,. 6 2 %,,. 0 8 6!" 2, "" ",,1' " ", ",,.,.0 7'0:& 'S: =0,+ M2' f> " 1."20 ' D 0 2 0'+ 0 6 0 8 1 0 1 2 "4- G, 8 2'0 2 2 a.t FGUR1il 4. Distributin f velcity. "
284 G.. Taylr The sudden jump in velcity which ccurs at the shck wave is represented in each case by a vertical line and the subsequent increase frm the shck wave t the sphere by a slping curve behind it. The pints crrespnding with the surface f the expanding sphere lie n the line u = rjt because this is the cnditin which must be satisfied at the sphere. Fr high rates f expansin the velcity distributin is practically linear. When the thickness f the layer f expanding air is small cmpared with the radius f the sphere : des nt differ appreciably frm its value at the spherical surface which is, accrding t equatin (17), equal t -2. The mean slpe f the velcity distributin curve fr P = 1 95 is infactfund frm figure 4 t be tan -1 ( - 1 8). The mean slpes fr p = 2 56 and 3 598 which are utside the range f figure 4 are still clser t the apprximate value tan-1 ( - 2). The velcity distributin is shwn n a larger scale in figure 5 fr the case when a = 0 7, P = 0 761. The calculated pints are marked in figure 5. The calculatin has been carried beynd the pint where the shck wave ccurs and the crrespnding part f the velcity distributin curve is marked in figure 5 with a brken line. t will be seen that in the virtual part f the curve the velcity fr a given radius is n lnger single-valued. PRESSURE DSTRBUTON The pressure p at any pint is given by p = JL (rjr2a2)y(y-1l P y' R2 (39) Values f pjp fr a selectin f values f rjat are given in table 2. The pressure distributins fr a = 0 2,0 4,0 5,0 6,0 7,0 8,1 0 and 1 2 are shwn in figure 6. t will be nticed that thse fr a = 1 0 and 1 2 appear t be nearly parablic. t can be shwn in fact that the distributin is parablic near the sphere s that when the thickness f the layer f expanding air is small cmpared with the radius f the sphere the distributin is nearly parablic thrugh the layer. f 8=(r-R)jr sthat x=u 2 (1+8), (40) and 8 is suppsed small the apprximate linear distributin f velcity is u = U 2 (1-28),
Q. The air wave surrunding an expanding sphere 285 75 '7 65 6 5 i 1 1 " 5,,..4- % O' 55 '3 '"" ex:.. 0 70 '" z (3 = 0 76 2 O 15 SOCK WAVE.,. r 4 1 " _.t- O.-.- 0015 0 8 0 85 0 9 0-95 1 0 1 05 1-10 S 'ZO rja.t FGURE 5. Distributin f velcity fr " = O 7 equatin (15) therefre takes the frm dc2 ds = - 6(')'-1) U8, s that C-C2 = 3(')'-1) U82 r (41)
286 G. 1. Taylr f 8 is small y C-C2 ----- y-l c (r-r)2 s that 1-- p = 3ya28 2 = 3ya2 --, P2 R (42) which represents a parablic pressure distributin. As an example f the applicatin f the apprximate expressin (42) the values f y' and a crrespnding with infinite rate f expansin may be 4-0 3-5 --. ' 3-0 fal f 2-5 z- 1-5 1-0 --.. -0 '" - "" ----. - ce- 0-2 -- t:::::; ""- 1 0-2 0-4 0-6 0-8 1-0 '-2 1-+ 1-6 18 jlat FGURE 6. Distributin f pressure. calculated. n this caseg1 = 0'8316, 111 = 0 1968 (see equatin (37)) s that 8 1, the value f 8 at the shck wave is given by Hence 8 1 = 0'056, s that r 1 R = 1 056 and 111 = 0 1968 = a-2{1-2(0 056)}, s that a = 2 12. The apprximate value f y' is 1-3ya28 2 r 0 94. These values may be cmpared with thse given in table 1 which were fund by numericalintegratinfthe full equatins, namely, r1r = 1 060, a = 2'103, y' = 0 93.
The air wave surrunding an expanding sphere 287 COMPARSON WTH SOUND WAVE SOLUTON FOR LOW RATES OF EXPANSON Fr small values f X the sund wave slutin f equatins (8) and (9) may be expected t affrd a gd apprximatin t the true mtin. The 10 09 24f-- 1 06 YSOLUllON BYTEORV OF SOUND. '221--- O 07 Q. 20 O 06 1 = 0 2 '16 0.0+ '1+ 0 " '12 0 SOUJTON BY ") THEORY OF SOUND. Q 10 t:-... 0 d: c 0 2 '0. t--... 0 '01 '0 SOLLlTlON BY TiEORV OF 50U " 0 0 1 0'4 O S 0 6 0'7 e 09 10 'a.t FGURE 7. Cmparisn between velcity and,pressure distributins calculated by the thery f sund and by the cmplete equatins. velcity and pressure distributins calculated frm the cmplete equatins and frm the apprximate equatins f the thery f sund are cmpared in figure 7 fr the case X = 0 2. S far as the velcity distributin is
288 G.. Taylr cncerned the agreement is gd but the pressure distributins are distinctly different near the sphere. The true pressure distributin is in fact parablic near the sphere and initially dpdr = O. Accrding t the sund wave equa- tin (9) the value f dpdr clse t the sphere is -1-a 2pa 2 (a 2 ) 2 R The reasn fr the discrepancy is evidently that it is nt justifiable t apply the equatins f the thery f sund in the neighburhd f the sphere wing t the neglect f the term u auar in the equatin f mtin in cmparisn with auat. n the crrect equatin u auax is equal t - auat at the sphere. Apa.rt frm this difference at the inner bundary the chief cntrast between the sund wave slutin and the true slutin fr values f a greater than 0 5 lies in the fact that the frmer invlves n shck wave at the uter bundary. The true slutin fr a = 0 2 appears in figure 7 t resemble the sund wave slutin in this respect. f this resemblance were true then sme limiting value f a wuld exist belw which n shck wave wuld be prduced. Assuming that such a limit exists the authr's rugh attempts t determine its value placed it between a = 0 4 and a = 0 5. The matter was, hwever, examined later by Dr J. W. Maccll using mre accurate methds f numerical slutin, and he came t the cnclusin nt nly that a shck wave is frmed when a is less than 0 4 but that n lwer limit f the assumed type wuld be fund. Subsequent analysis shws that this predictin is crrect. FORM OF SOLUTON NEAR (;=0, 1J= ) Near the pint (;= 0, 7J= ) equatin (19) takes the frm d +(y+1)g d;= g where = 7J - 1. The slutin f equatin (43) is = (y+ ) glg (Ag), where A is the cnstant f integratin. As g -+ 0, -+ 0 but is negative when g < la. Multiplying bth sides f (44) by A it will be seen that A is a functin f Ag s that the shape f the (g,) curve des nt depend n the cnstant f integratin thugh its scale is prprtinal t la. n the neighburhd f (;1 =0, 7J1 = ) the relatinship between ;1 and 7J1 atashckwaveisfund bytakingy-1 as small in equatins (26) and (31). Hence Lt 7J1-1 = _ 3-y = -0,797. (45) ko ;1 2 (43) (44)
The air wave surrunding an expanding sphere 289 Cmparing equatins (45) with (44) the shck wave cnditin is satisfied if 3-y lgeag = - 2(y+ 1) = -0 3316. (46) Hence Ag = +0 718, } A = A(1J-l) = -0 572. (47) +3'5,---,--..,.---,---, -O!iH----+-"'<::V- FGURE 8 Figure 8 shws the frm f the slutin near ( = 0, '1' = 1) and the shck wave line intersecting the (,1J) curve at the pint (+0 718, -0 572). t may be cncluded that prvided the numerical slutin brings the (1J) curve int the neighburhd f the pint (g=o., '1'= 1) the expanding air must be bunded by a shck wave. This indeed prves t be the case, but, as will be seen later, the shck wave is f very small intensity when a is less than 0'5." FTTNG THE NUMERCAL SOLUTON TO THE SOLUTON VALD NEAR"g = 0 Writing = '1'-1 equatin (19) becmes d 2(1+)+(y+l)g-yg2 dg = g 2+3+2g-g2' (48) Vl. 186. A. 19
290 G.. Taylr n the apprximate slutin (44) t; decreases as decreases. n fitting the apprximate slutin t the numerical slutin a pair f values f and t; may be taken and the value f A fund by inserting these in equatin (44). T find the errr cmmitted in using equatin (43) instead f (48) suppse that the slutins are jined where t; = B the true value f dt;d at this pint is expanding this expressin in pwers f the first tw terms are [J = (B+Y+l){l-(l+!B+- Y --)}. B B+y+l The value f [dt;djb in the apprximate slutin is B +Y+ 1. The prprtinal errr is therefre 6 = (1 +!B+ B++ l) and since in the apprximate slutin the errr in dt;d is given by B = (y + 1) lge (A), A6=(1+.lB+ Y )e(y+l)b. 2 B+y+l Values f A6 fr a series f values f B are given in table 3. (49) TABLE 3 B -1 0 1 2 3 4 5 6 7 10 Ae 0 96 1'58 2 9 5 5 9 6 16 9 29 4 50 2 84 4 387 0 a = 0 4. When a = 0 4 values f and t; given in lines 1 and 2 f table 4 were calculated numerically. Values f A calculated frm equatin (44) are given in line 3, and the prprtinal errr calculated frm equatin (49) in line 5. The errr 6 is less than 5 % in the first three pints. TABLE 4. a=0 4 ; 0 0081 0 0122 0 160 0 0231 0 0306 0 0384 0 0446 0 0536 0 0651 S 0 020 0 044 0 065 0'1l5 0 170 0 226 0 287 0 351 A 342 367 337 342 327 301 325 248 227 B 2 46 3 61 4 0 5 3 5 5 5 9 6 4 6 3 6 5 e 0 021 0 038 0 046 0 105 0'12 0 16 0 19 0 24 0 29
The air wave surrunding an expanding sphere 291 Taking A = 340 the (,1]) curve shwn in figure 9 was calculated. The first three values f 1], namely, thse crrespnding with = 0 0081, 0 0122 and 0 160 are shwn in figure 9. t will be seen that thugh the curve calculated using equatin (44) passes very nearly thrugh these three pints the existence f a prtin f the ( 1]) curve fr which S is negative wuld nt have been suspected frm simple inspectin f the apparent trend f the curve calculated step by step ' '0...--.----. thrugh 99 % f the range = 1 t O. This pint may perhaps be appreciated mre clearly if the area cvered by figure 9 is cmpared with the same area (marked with brken line) n the much smaller scale f figure 1. t appears frm equatin (47) that when X = 0 4 the expanding air is bunded by a shck wave fr which 1 = 0 718340 = 0 0021, 1]1-1 = -0 572340 = -0,0017. The crrespnding change in pressure at the shck wave is y-1 = 1' 1 = 0 003 f an atmsphere. X = 0'5: When X = 0 5 the fur lwest values f 1] calculated numerically give A = 24, 21, 21 and 21. Taking A = 21 the shck wave crrespnds with 1 = 0'71821 = 0 034, s that 1]1-1 = - 0 57221 = - 0'027, Y - 1 = 0 0341' = 0'048, which agrees well with the value y = 1 05 determined graphically by means f figure 1. X = 0 2. When X = 0 2 the lwest values f S cal 'Julated numerically are 0 0002 0 0003 0 0005 s +0'0175 +0'026 +0'043 '.Q9---f---il, OS----!---H f-07----!---f-1 1,061---+--1'-1.-51----!-+--1 "04-1---++--1 fj ' 031----'---1 "02--";i-j---! 0 9S'----'---... 0'01 0'02 e FGURE 9. g, 1] curve near f== 0,1] = 1, when a= 0 4. When inserted in equatin (44) these give values f and 1] 'at the shck wave f rder 10-19 S far as the equatins f a nn-viscus fluid are cncerned this shck wave seems t be real enugh in spite f its extreme smallness but frm the physical pint f view such a minute shck wave has n meaning. The effect f viscsity and cnductivity wuld in fact becme
292 G.. Taylr appreciable lng befre a shck wave with pressure change 10-19 atm. culd be frmed. Nevertheless it is curius that there is this definite mathematical differenc between a wave f finite intensity and the equivalent sund wave, t is especially curius that the pint where the slutin f the cmplete equtin differs frm the apprximate slutin f the thery f sund is in the regin f very small velcities and pressure changes, the regin in fact where the thery f sund might be expected t be mst accurate. n cnclusin wish t express my thanks t Mrs H. Glauert wh carried ut sme f the calculatins and t Dr J. W. MacCll fr sme valuable suggestins. SUMMARY The nly case in which the mtin f a gas at high speed in three dimensins has s far been discussed mathematically is that f the disturbance prduced by a cne mving with velcity greater than that f sund. n the present wrk anther case is analysed, namely, the radial utward flw prduced by a unifrmly expanding sphere. The regin f expanding air is bunded by a shck wave utside which the air is undisturbed. As the radial velcity f the sphere increases the thickness f the layer f disturbed air decreases till at infinite rate f expansin it is nly 6 % f the radius f the sphere. The distributins f velcity and pressure are given fr a range f rates f expansin. When the radial velcity f the sphere is small an apprxima.te analysis based n the thery f sund yields results which are inaccurate near the sphere and als at the shck wave which frms the uter bundary f the expanding air. REFERENCES Hugnit, 1889 J. Ec. plyt. Paris. Rankine, 1870 Phil. Trans. 160, 277.