22:
145:
2155:
1697:
196:
upon crossing the shock wave so that streamlines behind the shock wave would be parallel to the wedge sides. Such a simple turnover of streamlines is not possible for three-dimensional case. After passing through the shock wave, the streamlines are curved and only asymptotically they approach the
3134:
2597:
1999:
634:
1008:
176:. A conical shock wave can form in this situation, with the vertex of the shock wave lying at the vertex of the solid cone. If it were a two-dimensional problem, i.e., for a supersonic flow past a wedge, then the incoming stream would have deflected through an angle
2943:
2808:
724:
1224:
3339:
Taylor, G. I., & Maccoll, J. W. (1933). The air pressure on a cone moving at high speeds.—I. Proceedings of the Royal
Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 139(838),
2980:
1416:
3352:
Von Karman, T., & Moore, N. B. (1932). Resistance of slender bodies moving with supersonic velocities, with special reference to projectiles. Transactions of the
American Society of Mechanical Engineers, 54(2),
2457:
2411:
2150:{\displaystyle {\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho {\frac {\partial \phi }{\partial \rho }}\right)-\beta ^{2}{\frac {\partial ^{2}\phi }{\partial z^{2}}}=0,\quad \beta ^{2}=M^{2}-1}
3283:
1349:
493:
197:
generators of the cone. The curving of streamlines is accompanied by a gradual increase in density and decrease in velocity, in addition to those increments/decrements effected at the shock wave.
3362:
Maccoll, J. W. (1937). The conical shock wave formed by a cone moving at a high speed. Proceedings of the Royal
Society of London. Series A-Mathematical and Physical Sciences, 159(898), 459-472.
1404:
2819:
1098:
2452:
385:
2691:
814:
323:
264:
2259:
2685:
describes the uniform flow upstream of the shock cone, whereas the non-trivial solution satisfying the boundary condition on the solid surface behind the shock wave is given by
649:
1971:
1876:
485:
1740:
3310:
2199:
2631:
2297:
2663:
2331:
407:
3380:
Landau, L. D., & Lifshitz, E. M. (2013). Fluid mechanics: Landau And
Lifshitz: course of theoretical physics, Volume 6 (Vol. 6). Elsevier. section 123. pages 432-434.
2972:
1766:
825:
354:
2219:
1826:
1247:
447:
765:
3164:
1991:
1896:
1786:
1278:
288:
229:
194:
174:
2683:
1936:
1916:
1806:
427:
328:
Since all incident streamlines intersect the conical shock wave at the same angle, the intensity of the shock wave is constant. This particularly means that
270:). These detached cases are not considered here. The flow immediately behind the oblique conical shock wave is typically supersonic, although however when
1106:
3129:{\displaystyle v_{z}=U-U\chi ^{2}\cosh ^{-1}{\frac {z}{\beta \rho }},\quad v_{\rho }={\frac {U\chi ^{2}}{\rho }}{\sqrt {z^{2}-\beta ^{2}\rho ^{2}}}.}
1692:{\displaystyle \leftF''=(\gamma -1)(1-F^{2})F+{\frac {\gamma -1}{2}}\cot \theta (1-F^{2})F'-\gamma FF'^{2}-{\frac {\gamma -1}{2}}\cot \theta F'^{3}.}
2592:{\displaystyle v_{\rho }/v_{z}=(\partial \phi /\partial \rho )/(U+\partial \phi /\partial z)\approx (1/U)\partial \phi /\partial \rho =\chi }
266:
beyond which shock polar do not provide solution under in which case the conical shock wave will have detached from the solid surface (see
86:
58:
3169:
1836:
When the cone angle is very small, the flow is nearly parallel everywhere in which case, an exact solution can be found, as shown by
65:
39:
629:{\displaystyle \varphi =rf(\theta ),\quad v_{r}=f(\theta ),\quad v_{\theta }=f'(\theta ),\quad v_{\phi }=0,\quad p=g(\theta ).}
1287:
72:
2938:{\displaystyle \varphi =Uz+U\chi ^{2}\left({\sqrt {z^{2}-\beta ^{2}\rho ^{2}}}-z\cosh ^{-1}{\frac {z}{\beta \rho }}\right)}
2339:
1354:
1788:
is the half-angle of shock cone, which must be determined as part of the solution for a given incoming flow Mach number
54:
1021:
200:
The direction and magnitude of the velocity immediately behind the oblique shock wave is given by weak branch of the
105:
2419:
359:
2803:{\displaystyle g(\xi )=U\chi ^{2}\left({\sqrt {1-\beta ^{2}\xi ^{2}}}-\cosh ^{-1}{\frac {1}{\beta \xi }}\right).}
1284:. Introducing this formula into the general Taylor–Maccoll equation and introducing a non-dimensional function
325:, it can be subsonic. The supersonic flow behind the shock wave will become subsonic as it evolves downstream.
43:
770:
816:
only. Substituting the above assumed form for the velocity field, into the governing equation, we obtain the
293:
234:
3321:
719:{\displaystyle c^{2}\nabla \cdot \mathbf {v} -\mathbf {v} \cdot (\mathbf {v} \cdot \nabla )\mathbf {v} =0,}
2224:
1406:(the speed of the potential flow when it flows out into a vacuum), we obtain, for the polytropic gas, the
332:
jump across the shock wave is also constant throughout. In this case, the flow behind the shock wave is a
79:
1941:
1843:
452:
1742:(no penetration on the solid surface) and also must correspond to conditions behind the shock wave at
3288:
2163:
2264:
2639:
1837:
133:
2302:
390:
3399:
2951:
1745:
1705:
1003:{\displaystyle (c^{2}-f'^{2})f''+c^{2}\cot \theta f'+(2c^{2}-f'^{2})f=0,\quad c=c(f^{2}+f'^{2}).}
387:. Since the problem do not have any length scale and is clearly axisymmetric, the velocity field
333:
32:
2602:
1828:. The Taylor–Maccoll equation has no known explicit solution and it is integrated numerically.
339:
2204:
2201:
is the Mach number of the incoming flow. We expect the velocity components to depend only on
1811:
1232:
432:
735:
3142:
1976:
1881:
1771:
1256:
273:
207:
179:
159:
1840:
and Norton B. Moore in 1932. The solution is more apparent in the cylindrical coordinates
8:
1281:
1250:
2668:
1921:
1901:
1791:
412:
1219:{\displaystyle c^{2}=(\gamma -1)h_{0}\left(1-{\frac {f^{2}+f'^{2}}{2h_{0}}}\right),}
267:
132:
and J. W. Maccoll, whom described the flow in 1933, guided by an earlier work of
1015:
640:
3393:
129:
730:
204:. This particularly suggests that for each value of incoming Mach number
201:
153:
125:
21:
2636:
In the small-angle approximation, the weak shock cone is given by
329:
2333:
is a self-similar coordinate. The governing equation reduces to
3278:{\displaystyle p_{s}-p_{\infty }=\rho _{\infty }U^{2}\chi ^{2}}
487:
is taken to be located at the vertex). This means that we have
148:
Coordinate system, where the dotted line depicts conical shock
2261:
in cylindrical coordinates, which means that we must have
144:
128:
that is attached to a solid cone. The flow is named after
1344:{\displaystyle F(\theta )=f(\theta )/v_{\mathrm {max} }}
156:
flow past a solid cone that has a semi-vertical angle
3291:
3172:
3145:
2983:
2954:
2822:
2694:
2671:
2642:
2605:
2460:
2422:
2342:
2305:
2267:
2227:
2207:
2166:
2002:
1979:
1944:
1924:
1904:
1884:
1846:
1814:
1794:
1774:
1748:
1708:
1419:
1357:
1290:
1259:
1235:
1109:
1024:
828:
773:
767:
is expressed as a function of the velocity magnitude
738:
652:
496:
455:
435:
415:
393:
362:
342:
296:
276:
237:
210:
182:
162:
2406:{\displaystyle \xi (1-\beta ^{2}\xi ^{2})g''+g'=0.}
1399:{\displaystyle v_{\mathrm {max} }={\sqrt {2h_{0}}}}
46:. Unsourced material may be challenged and removed.
3304:
3277:
3158:
3128:
2966:
2937:
2802:
2677:
2657:
2625:
2591:
2446:
2405:
2325:
2291:
2253:
2213:
2193:
2149:
1985:
1965:
1930:
1910:
1890:
1870:
1820:
1800:
1780:
1760:
1734:
1691:
1398:
1343:
1272:
1241:
1218:
1092:
1002:
808:
759:
718:
628:
479:
441:
421:
401:
379:
348:
317:
282:
258:
223:
188:
168:
1938:is the speed of the incoming flow, then we write
429:will be turn out to functions of the polar angle
3391:
1093:{\displaystyle c^{2}=(\gamma -1)(h_{0}-v^{2}/2)}
336:. Hence we can introduce the velocity potential
449:only (the origin of the spherical coordinates
1702:The equation must satisfy the condition that
2447:{\displaystyle \xi =\tan \chi \approx \chi }
380:{\displaystyle \mathbf {v} =\nabla \varphi }
139:
124:refers to the steady flow behind a conical
1014:The equation is simplified greatly for a
106:Learn how and when to remove this message
3376:
3374:
3372:
3370:
3368:
3348:
3346:
3139:The pressure on the surface of the cone
1831:
809:{\displaystyle v^{2}=(\nabla \phi )^{2}}
143:
2948:exhibiting a logarthmic singularity as
3392:
318:{\displaystyle \chi _{\mathrm {max} }}
259:{\displaystyle \chi _{\mathrm {max} }}
3365:
3343:
3312:is the density of the incoming gas).
2974:The velocity components are given by
1898:here is the radial distance from the
2254:{\displaystyle \rho /z=\tan \theta }
1993:is a small correction and satisfies
44:adding citations to reliable sources
15:
13:
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2078:
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2039:
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2015:
1370:
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1335:
1332:
1329:
790:
696:
663:
371:
309:
306:
303:
250:
247:
244:
231:, there exists a maximum value of
14:
3411:
1966:{\displaystyle \varphi =Uz+\phi }
1871:{\displaystyle (\rho ,\varpi ,z)}
480:{\displaystyle (r,\theta ,\phi )}
1918:-axis, and not the density). If
703:
689:
678:
670:
395:
364:
118:Flow behind a conical shock wave
20:
3305:{\displaystyle \rho _{\infty }}
3050:
2194:{\displaystyle M=U/c_{\infty }}
2117:
953:
818:general Taylor–Maccoll equation
604:
584:
550:
521:
31:needs additional citations for
3356:
3333:
3272:
3255:
3238:
3229:
2958:
2704:
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2543:
2517:
2509:
2489:
2375:
2346:
2292:{\displaystyle \phi =zg(\xi )}
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1:
3327:
2658:{\displaystyle z=\beta \rho }
2326:{\displaystyle \xi =\rho /z}
643:is governed by the equation
402:{\displaystyle \mathbf {v} }
7:
3315:
2967:{\displaystyle \rho \to 0.}
2665:. The trivial solution for
2416:On the surface of the cone
1761:{\displaystyle \chi =\psi }
1735:{\displaystyle F'(\chi )=0}
10:
3416:
2626:{\displaystyle g'=U\chi }
349:{\displaystyle \varphi }
140:Mathematical description
2214:{\displaystyle \theta }
1821:{\displaystyle \gamma }
1408:Taylor–Maccoll equation
1242:{\displaystyle \gamma }
442:{\displaystyle \theta }
409:and the pressure field
3306:
3279:
3160:
3130:
2968:
2939:
2804:
2679:
2659:
2627:
2593:
2448:
2407:
2327:
2293:
2255:
2215:
2195:
2151:
1987:
1967:
1932:
1912:
1892:
1872:
1822:
1802:
1782:
1762:
1736:
1693:
1400:
1345:
1274:
1243:
1220:
1094:
1004:
810:
761:
760:{\displaystyle c=c(v)}
720:
630:
481:
443:
423:
403:
381:
350:
319:
284:
260:
225:
190:
170:
149:
3307:
3280:
3161:
3159:{\displaystyle p_{s}}
3131:
2969:
2940:
2805:
2680:
2660:
2628:
2594:
2449:
2408:
2328:
2294:
2256:
2216:
2196:
2152:
1988:
1986:{\displaystyle \phi }
1968:
1933:
1913:
1893:
1891:{\displaystyle \rho }
1873:
1832:Kármán–Moore solution
1823:
1803:
1783:
1781:{\displaystyle \psi }
1763:
1737:
1694:
1401:
1346:
1275:
1273:{\displaystyle h_{0}}
1244:
1221:
1095:
1005:
811:
762:
721:
631:
482:
444:
424:
404:
382:
351:
320:
285:
283:{\displaystyle \chi }
261:
226:
224:{\displaystyle M_{1}}
191:
189:{\displaystyle \chi }
171:
169:{\displaystyle \chi }
147:
55:"Taylor–Maccoll flow"
3289:
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3143:
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1977:
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1902:
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1844:
1812:
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1772:
1746:
1706:
1417:
1355:
1288:
1257:
1233:
1107:
1022:
826:
771:
736:
650:
494:
453:
433:
413:
391:
360:
340:
294:
274:
235:
208:
180:
160:
40:improve this article
3322:Kármán–Moore theory
1838:Theodore von Kármán
1282:stagnation enthalpy
1251:specific heat ratio
134:Theodore von Kármán
122:Taylor–Maccoll flow
3302:
3285:(in this formula,
3275:
3156:
3126:
2964:
2935:
2813:We therefore have
2800:
2675:
2655:
2623:
2599:and conesequently
2589:
2444:
2403:
2323:
2289:
2251:
2211:
2191:
2147:
1983:
1963:
1928:
1908:
1888:
1868:
1818:
1798:
1778:
1758:
1732:
1689:
1396:
1341:
1270:
1239:
1216:
1090:
1000:
806:
757:
716:
626:
477:
439:
419:
399:
377:
346:
315:
280:
256:
221:
186:
166:
152:Consider a steady
150:
3121:
3084:
3045:
2928:
2891:
2790:
2756:
2678:{\displaystyle g}
2106:
2054:
2026:
2011:
1931:{\displaystyle U}
1911:{\displaystyle z}
1801:{\displaystyle M}
1660:
1576:
1477:
1441:
1394:
1206:
422:{\displaystyle p}
116:
115:
108:
90:
3407:
3381:
3378:
3363:
3360:
3354:
3350:
3341:
3337:
3311:
3309:
3308:
3303:
3301:
3300:
3284:
3282:
3281:
3276:
3268:
3248:
3228:
3227:
3218:
3217:
3208:
3207:
3195:
3194:
3182:
3181:
3165:
3163:
3162:
3157:
3155:
3154:
3135:
3133:
3132:
3127:
3122:
3120:
3119:
3110:
3109:
3097:
3096:
3087:
3085:
3080:
3079:
3078:
3065:
3060:
3059:
3046:
3044:
3033:
3028:
3027:
3015:
3014:
2993:
2992:
2973:
2971:
2970:
2965:
2944:
2942:
2941:
2936:
2934:
2930:
2929:
2927:
2916:
2911:
2910:
2892:
2890:
2889:
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2857:
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2849:
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2807:
2806:
2801:
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2778:
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2772:
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2744:
2729:
2722:
2721:
2684:
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2613:
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2516:
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2404:
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2220:
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2198:
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2180:
2156:
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2153:
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2140:
2139:
2127:
2126:
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2090:
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2060:
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2045:
2037:
2027:
2025:
2014:
2012:
2004:
1992:
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1972:
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1964:
1937:
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1934:
1929:
1917:
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1914:
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1827:
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1402:
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1339:
1338:
1322:
1279:
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1271:
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1268:
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1225:
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1222:
1217:
1212:
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1204:
1203:
1190:
1189:
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1119:
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1099:
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1065:
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1009:
1007:
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1001:
993:
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991:
975:
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919:
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900:
883:
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841:
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815:
813:
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804:
783:
782:
766:
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758:
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723:
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706:
692:
681:
673:
662:
661:
635:
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594:
593:
571:
560:
559:
531:
530:
486:
484:
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478:
448:
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440:
428:
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367:
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324:
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321:
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289:
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265:
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230:
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167:
111:
104:
100:
97:
91:
89:
48:
24:
16:
3415:
3414:
3410:
3409:
3408:
3406:
3405:
3404:
3390:
3389:
3384:
3379:
3366:
3361:
3357:
3351:
3344:
3338:
3334:
3330:
3318:
3296:
3292:
3290:
3287:
3286:
3264:
3244:
3223:
3219:
3213:
3209:
3203:
3199:
3190:
3186:
3177:
3173:
3171:
3168:
3167:
3166:is found to be
3150:
3146:
3144:
3141:
3140:
3115:
3111:
3105:
3101:
3092:
3088:
3086:
3074:
3070:
3066:
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3055:
3051:
3037:
3032:
3020:
3016:
3010:
3006:
2988:
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2982:
2979:
2978:
2953:
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2949:
2920:
2915:
2903:
2899:
2885:
2881:
2875:
2871:
2862:
2858:
2856:
2855:
2851:
2845:
2841:
2821:
2818:
2817:
2782:
2777:
2765:
2761:
2750:
2746:
2740:
2736:
2728:
2727:
2723:
2717:
2713:
2693:
2690:
2689:
2670:
2667:
2666:
2641:
2638:
2637:
2606:
2604:
2601:
2600:
2572:
2555:
2532:
2512:
2498:
2480:
2476:
2471:
2465:
2461:
2459:
2456:
2455:
2454:, we must have
2421:
2418:
2417:
2389:
2378:
2369:
2365:
2359:
2355:
2341:
2338:
2337:
2315:
2304:
2301:
2300:
2266:
2263:
2262:
2231:
2226:
2223:
2222:
2206:
2203:
2202:
2185:
2181:
2176:
2165:
2162:
2161:
2135:
2131:
2122:
2118:
2099:
2095:
2091:
2081:
2077:
2076:
2074:
2068:
2064:
2046:
2038:
2036:
2032:
2028:
2018:
2013:
2003:
2001:
1998:
1997:
1978:
1975:
1974:
1943:
1940:
1939:
1923:
1920:
1919:
1903:
1900:
1899:
1883:
1880:
1879:
1845:
1842:
1841:
1834:
1813:
1810:
1809:
1793:
1790:
1789:
1773:
1770:
1769:
1747:
1744:
1743:
1709:
1707:
1704:
1703:
1679:
1675:
1671:
1646:
1644:
1634:
1630:
1626:
1609:
1600:
1596:
1562:
1560:
1545:
1541:
1506:
1492:
1488:
1463:
1461:
1451:
1447:
1443:
1427:
1425:
1424:
1420:
1418:
1415:
1414:
1388:
1384:
1379:
1363:
1362:
1358:
1356:
1353:
1352:
1328:
1327:
1323:
1318:
1289:
1286:
1285:
1264:
1260:
1258:
1255:
1254:
1234:
1231:
1230:
1199:
1195:
1191:
1183:
1179:
1175:
1166:
1162:
1161:
1159:
1152:
1148:
1142:
1138:
1114:
1110:
1108:
1105:
1104:
1079:
1073:
1069:
1060:
1056:
1029:
1025:
1023:
1020:
1019:
1013:
987:
983:
979:
970:
966:
931:
927:
923:
914:
910:
893:
878:
874:
863:
853:
849:
845:
836:
832:
827:
824:
823:
800:
796:
778:
774:
772:
769:
768:
737:
734:
733:
702:
688:
677:
669:
657:
653:
651:
648:
647:
589:
585:
564:
555:
551:
526:
522:
495:
492:
491:
454:
451:
450:
434:
431:
430:
414:
411:
410:
394:
392:
389:
388:
363:
361:
358:
357:
341:
338:
337:
302:
301:
297:
295:
292:
291:
275:
272:
271:
268:Mach reflection
243:
242:
238:
236:
233:
232:
215:
211:
209:
206:
205:
181:
178:
177:
161:
158:
157:
142:
119:
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
3413:
3403:
3402:
3400:Fluid dynamics
3388:
3387:
3383:
3382:
3364:
3355:
3342:
3331:
3329:
3326:
3325:
3324:
3317:
3314:
3299:
3295:
3274:
3271:
3267:
3263:
3260:
3257:
3254:
3251:
3247:
3243:
3240:
3237:
3234:
3231:
3226:
3222:
3216:
3212:
3206:
3202:
3198:
3193:
3189:
3185:
3180:
3176:
3153:
3149:
3137:
3136:
3125:
3118:
3114:
3108:
3104:
3100:
3095:
3091:
3083:
3077:
3073:
3069:
3063:
3058:
3054:
3049:
3043:
3040:
3036:
3031:
3026:
3023:
3019:
3013:
3009:
3005:
3002:
2999:
2996:
2991:
2987:
2963:
2960:
2957:
2946:
2945:
2933:
2926:
2923:
2919:
2914:
2909:
2906:
2902:
2898:
2895:
2888:
2884:
2878:
2874:
2870:
2865:
2861:
2854:
2848:
2844:
2840:
2837:
2834:
2831:
2828:
2825:
2811:
2810:
2799:
2795:
2788:
2785:
2781:
2776:
2771:
2768:
2764:
2760:
2753:
2749:
2743:
2739:
2735:
2732:
2726:
2720:
2716:
2712:
2709:
2706:
2703:
2700:
2697:
2674:
2654:
2651:
2648:
2645:
2622:
2619:
2616:
2612:
2609:
2588:
2585:
2582:
2579:
2575:
2571:
2568:
2565:
2562:
2558:
2554:
2551:
2548:
2545:
2542:
2539:
2535:
2531:
2528:
2525:
2522:
2519:
2515:
2511:
2508:
2505:
2501:
2497:
2494:
2491:
2488:
2483:
2479:
2474:
2468:
2464:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2414:
2413:
2402:
2399:
2395:
2392:
2388:
2384:
2381:
2377:
2372:
2368:
2362:
2358:
2354:
2351:
2348:
2345:
2322:
2318:
2314:
2311:
2308:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2250:
2247:
2244:
2241:
2238:
2234:
2230:
2210:
2188:
2184:
2179:
2175:
2172:
2169:
2158:
2157:
2146:
2143:
2138:
2134:
2130:
2125:
2121:
2116:
2113:
2110:
2102:
2098:
2094:
2089:
2084:
2080:
2071:
2067:
2063:
2059:
2052:
2049:
2044:
2041:
2035:
2031:
2024:
2021:
2017:
2010:
2007:
1982:
1962:
1959:
1956:
1953:
1950:
1947:
1927:
1907:
1887:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1833:
1830:
1817:
1797:
1777:
1757:
1754:
1751:
1731:
1728:
1725:
1722:
1719:
1715:
1712:
1700:
1699:
1688:
1682:
1678:
1674:
1670:
1667:
1664:
1659:
1655:
1652:
1649:
1643:
1637:
1633:
1629:
1625:
1622:
1619:
1615:
1612:
1608:
1603:
1599:
1595:
1592:
1589:
1586:
1583:
1580:
1575:
1571:
1568:
1565:
1559:
1556:
1553:
1548:
1544:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1512:
1509:
1504:
1500:
1495:
1491:
1487:
1484:
1481:
1476:
1472:
1469:
1466:
1460:
1454:
1450:
1446:
1440:
1436:
1433:
1430:
1423:
1391:
1387:
1383:
1378:
1372:
1369:
1366:
1361:
1337:
1334:
1331:
1326:
1321:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1267:
1263:
1238:
1227:
1226:
1215:
1211:
1202:
1198:
1194:
1186:
1182:
1178:
1174:
1169:
1165:
1158:
1155:
1151:
1145:
1141:
1137:
1134:
1131:
1128:
1125:
1122:
1117:
1113:
1089:
1086:
1082:
1076:
1072:
1068:
1063:
1059:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1032:
1028:
1016:polytropic gas
1011:
1010:
999:
996:
990:
986:
982:
978:
973:
969:
965:
962:
959:
956:
952:
949:
946:
943:
940:
934:
930:
926:
922:
917:
913:
909:
906:
903:
899:
896:
892:
889:
886:
881:
877:
873:
869:
866:
862:
856:
852:
848:
844:
839:
835:
831:
803:
799:
795:
792:
789:
786:
781:
777:
756:
753:
750:
747:
744:
741:
727:
726:
715:
712:
709:
705:
701:
698:
695:
691:
687:
684:
680:
676:
672:
668:
665:
660:
656:
641:potential flow
637:
636:
625:
622:
619:
616:
613:
610:
607:
603:
600:
597:
592:
588:
583:
580:
577:
574:
570:
567:
563:
558:
554:
549:
546:
543:
540:
537:
534:
529:
525:
520:
517:
514:
511:
508:
505:
502:
499:
476:
473:
470:
467:
464:
461:
458:
438:
418:
397:
376:
373:
370:
366:
345:
334:potential flow
311:
308:
305:
300:
279:
252:
249:
246:
241:
218:
214:
185:
165:
141:
138:
117:
114:
113:
28:
26:
19:
9:
6:
4:
3:
2:
3412:
3401:
3398:
3397:
3395:
3386:
3385:
3377:
3375:
3373:
3371:
3369:
3359:
3349:
3347:
3336:
3332:
3323:
3320:
3319:
3313:
3293:
3269:
3265:
3261:
3258:
3252:
3249:
3245:
3241:
3235:
3232:
3224:
3220:
3214:
3210:
3200:
3196:
3187:
3183:
3178:
3174:
3151:
3147:
3123:
3116:
3112:
3106:
3102:
3098:
3093:
3089:
3081:
3075:
3071:
3067:
3061:
3056:
3052:
3047:
3041:
3038:
3034:
3029:
3024:
3021:
3017:
3011:
3007:
3003:
3000:
2997:
2994:
2989:
2985:
2977:
2976:
2975:
2961:
2955:
2931:
2924:
2921:
2917:
2912:
2907:
2904:
2900:
2896:
2893:
2886:
2882:
2876:
2872:
2868:
2863:
2859:
2852:
2846:
2842:
2838:
2835:
2832:
2829:
2826:
2823:
2816:
2815:
2814:
2797:
2793:
2786:
2783:
2779:
2774:
2769:
2766:
2762:
2758:
2751:
2747:
2741:
2737:
2733:
2730:
2724:
2718:
2714:
2710:
2707:
2701:
2695:
2688:
2687:
2686:
2672:
2652:
2649:
2646:
2643:
2634:
2620:
2617:
2614:
2610:
2607:
2586:
2583:
2580:
2573:
2569:
2560:
2556:
2552:
2546:
2540:
2533:
2529:
2523:
2520:
2513:
2506:
2499:
2495:
2486:
2481:
2477:
2472:
2466:
2462:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2400:
2397:
2393:
2390:
2386:
2382:
2379:
2370:
2366:
2360:
2356:
2352:
2349:
2343:
2336:
2335:
2334:
2320:
2316:
2312:
2309:
2306:
2283:
2277:
2274:
2271:
2268:
2248:
2245:
2242:
2239:
2236:
2232:
2228:
2208:
2182:
2177:
2173:
2170:
2167:
2144:
2141:
2136:
2132:
2128:
2123:
2119:
2114:
2111:
2108:
2100:
2096:
2087:
2082:
2069:
2065:
2061:
2057:
2050:
2042:
2033:
2029:
2022:
2008:
2005:
1996:
1995:
1994:
1980:
1960:
1957:
1954:
1951:
1948:
1945:
1925:
1905:
1885:
1862:
1859:
1856:
1853:
1850:
1839:
1829:
1815:
1795:
1775:
1755:
1752:
1749:
1729:
1726:
1720:
1713:
1710:
1686:
1680:
1676:
1672:
1668:
1665:
1662:
1657:
1653:
1650:
1647:
1641:
1635:
1631:
1627:
1623:
1620:
1617:
1613:
1610:
1601:
1597:
1593:
1590:
1584:
1581:
1578:
1573:
1569:
1566:
1563:
1557:
1554:
1546:
1542:
1538:
1535:
1526:
1523:
1520:
1514:
1510:
1507:
1502:
1493:
1489:
1485:
1482:
1474:
1470:
1467:
1464:
1458:
1452:
1448:
1444:
1438:
1434:
1431:
1428:
1421:
1413:
1412:
1411:
1409:
1389:
1385:
1381:
1376:
1359:
1324:
1319:
1312:
1306:
1303:
1297:
1291:
1283:
1265:
1261:
1252:
1236:
1213:
1209:
1200:
1196:
1192:
1184:
1180:
1176:
1172:
1167:
1163:
1156:
1153:
1149:
1143:
1139:
1132:
1129:
1126:
1120:
1115:
1111:
1103:
1102:
1101:
1084:
1080:
1074:
1070:
1066:
1061:
1057:
1047:
1044:
1041:
1035:
1030:
1026:
1017:
997:
988:
984:
980:
976:
971:
967:
960:
957:
954:
950:
947:
944:
941:
932:
928:
924:
920:
915:
911:
907:
901:
897:
894:
890:
887:
884:
879:
875:
871:
867:
864:
854:
850:
846:
842:
837:
833:
822:
821:
820:
819:
801:
793:
784:
779:
775:
751:
745:
742:
739:
732:
713:
710:
707:
693:
682:
674:
666:
658:
654:
646:
645:
644:
642:
623:
617:
611:
608:
605:
601:
598:
595:
590:
586:
581:
575:
568:
565:
561:
556:
552:
547:
541:
535:
532:
527:
523:
518:
512:
506:
503:
500:
497:
490:
489:
488:
471:
468:
465:
462:
459:
436:
416:
374:
368:
343:
335:
331:
326:
298:
277:
269:
239:
216:
212:
203:
198:
183:
163:
155:
146:
137:
135:
131:
127:
123:
110:
107:
99:
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
3358:
3335:
3138:
2947:
2812:
2635:
2415:
2159:
1835:
1701:
1407:
1228:
1012:
817:
728:
638:
327:
290:is close to
199:
151:
130:G. I. Taylor
121:
120:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
731:sound speed
639:The steady
202:shock polar
3328:References
1018:for which
729:where the
356:such that
154:supersonic
126:shock wave
96:March 2024
66:newspapers
3298:∞
3294:ρ
3259:−
3253:χ
3250:β
3236:
3221:χ
3205:∞
3201:ρ
3192:∞
3184:−
3113:ρ
3103:β
3099:−
3082:ρ
3072:χ
3057:ρ
3042:ρ
3039:β
3030:
3022:−
3008:χ
3001:−
2959:→
2956:ρ
2925:ρ
2922:β
2913:
2905:−
2894:−
2883:ρ
2873:β
2869:−
2843:χ
2824:φ
2787:ξ
2784:β
2775:
2767:−
2759:−
2748:ξ
2738:β
2734:−
2715:χ
2702:ξ
2653:ρ
2650:β
2621:χ
2587:χ
2581:ρ
2578:∂
2570:ϕ
2567:∂
2547:≈
2538:∂
2530:ϕ
2527:∂
2507:ρ
2504:∂
2496:ϕ
2493:∂
2467:ρ
2442:χ
2439:≈
2436:χ
2433:
2424:ξ
2367:ξ
2357:β
2353:−
2344:ξ
2313:ρ
2307:ξ
2284:ξ
2269:ϕ
2249:θ
2246:
2229:ρ
2209:θ
2187:∞
2142:−
2120:β
2093:∂
2088:ϕ
2079:∂
2066:β
2062:−
2051:ρ
2048:∂
2043:ϕ
2040:∂
2034:ρ
2023:ρ
2020:∂
2016:∂
2009:ρ
1981:ϕ
1961:ϕ
1946:φ
1886:ρ
1857:ϖ
1851:ρ
1816:γ
1776:ψ
1756:ψ
1750:χ
1721:χ
1669:θ
1666:
1651:−
1648:γ
1642:−
1621:γ
1618:−
1594:−
1585:θ
1582:
1567:−
1564:γ
1539:−
1524:−
1521:γ
1486:−
1468:−
1465:γ
1459:−
1429:γ
1313:θ
1298:θ
1237:γ
1157:−
1130:−
1127:γ
1067:−
1045:−
1042:γ
921:−
891:θ
888:
843:−
794:ϕ
791:∇
697:∇
694:⋅
683:⋅
675:−
667:⋅
664:∇
618:θ
591:ϕ
576:θ
557:θ
542:θ
513:θ
498:φ
472:ϕ
466:θ
437:θ
375:φ
372:∇
344:φ
299:χ
278:χ
240:χ
184:χ
164:χ
3394:Category
3353:303-310.
3340:278-297.
3316:See also
2611:′
2394:′
2383:″
2299:, where
2221:, i.e.,
1973:, where
1768:, where
1714:′
1677:′
1632:′
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1511:″
1449:′
1351:, where
1181:′
1100:, i.e.,
985:′
929:′
898:′
868:″
851:′
569:′
1280:is the
1249:is the
330:entropy
80:scholar
2160:where
1229:where
82:
75:
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61:
53:
1878:(the
87:JSTOR
73:books
3018:cosh
2901:cosh
2763:cosh
1808:and
1253:and
59:news
2430:tan
2243:tan
1663:cot
1579:cot
885:cot
42:by
3396::
3367:^
3345:^
3233:ln
2962:0.
2633:.
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