The Propagation of Weak Spherical Shocks in Stars

Cover The Propagation of Weak Spherical Shocks in Stars
The Propagation of Weak Spherical Shocks in Stars
Gerald B Whitham
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It is easy to show that the velocity of the v/avelet normal to its^^lf in the (r, z) plane is a - ZT / Vr. '2+T^2 . Xhe vel'city of the oarticlos of fluid r z nor;nal to the wavelet is (uTT + \nir) / V^+T^ J tnerefore the char- acteristic condition, that thi. - v. . L, 'city of the vjavolet differs from ty b;\ the soun: \ speed, gi + wrr ^ 1 n ~ - + a. R z . R z or (^^) (-- -. U^^ + w^-)2 = a^fc^+TT^). Since expressions for u, w and a in terms of x: are known, (33) ^-ives an equatio:-] to deter...mine "c" . Njvj, al though TTdiffers from t-a by -n amount which becomes large (this being one of the crucial points in -13- the •-liscussi m), the :Ufforenci; is still small coraparud to a. Hence, xire may Sc;t IZ = t - ^(r, z) + ix(i', z), where [i/^ is small. Th' n, subs ti tutinr- In (33) ■'^nl neglecting terms of second order in small quantities, we obtain ua + v:a, (xising (21)). From (30), (31) and (32), the right hand side of (3lj. ) is ^( Y+l)XFtC)/A^. Hence, [i may be taken equal to p(r, z)P(^30, vrhere (35) ^ (3 + a S = -ilti -4-, from (2l(.

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