Spherical Waves And Shocks Part I a General Theory of the Propagation of Weak

Gerald B Whitham

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- X . , ^O cF ~ u+c ~ C '^'^ '~T~^ ?~2, Q. Dt _ 1 ^. 1 p F(r) ^ o c^r o ;^^X*^ ^'f-, k)h■^^^y%■•i. •S '■^ •. 7, y. -, '. , ':*i:rt ■I •■-, -» . . -, x: j, ;, 'V V- >-^ ^ C V'i 'J ■, - V-" '- -■J, . Vjhere k = ^{y+l)/, -ii rj«, '••. : * 8 l^. O The shocks The solution obtained In the last section is not yet com- plete. The characteristics now depend on the solution; where P'(t^) they run together and eventually overlap. The extreme cases of "centered" expansion or compression, with a fan of cha...racteristics from a single point on r = a(t), occur when ?{T), due to a discontinuity in a»(t), is discontinuous (since in that case a whole range of values of P, and hence a fan of characteristics (10), is found for the sarae value of r ) • If the characteristics were allowed to overlap, the solution would cease to be single-valued and there would be a fold in the flow plane; a shock must be fitted in such regions so that the ch'iracteristlcs meet the shock and are cut off by it before they meet each other.

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