Sound- Waves of Finite Amplitude, 327 



Paet II. — Spherical Waves. 



10. When plane waves of finite amplitude are propagated 

 through a frictionless compressible fluid, discontinuity must 

 always occur sooner or later, and a moment's consideration 

 will show that there are at least some cases when the motion 

 in spherical waves becomes discontinuous ; the question arises 

 whether in any case it is possible (in the absence of viscosity) 

 for divergent spherical waves to travel outward indefinitely 

 without arriving at a discontinuous state. This question was 

 suggested to me by Mr. Bryan, who at the same time kindly 

 handed me notes of his manner of attacking the problem. 

 His method was to write down the exact kinematical equation 

 for spherical sound-waves, and then to obtain successive 

 approximations to the integral of this equation. If it appears 

 that after any number of approximations the integral would 

 remain convergent for large values of the radius, we may con- 

 clude that our equation holds good throughout, and hence that 

 no discontinuity arises. If, on the other hand, the second or 

 any higher approximation becomes divergent for large values 

 of the radius, it is probable that the motion becomes some- 

 where discontinuous. This method I have not followed out ; 

 but by another method which is, I hope, sufficiently con- 

 clusive, I shall now endeavour to show that discontinuity 

 must always arise. 



The case in which the motion loses its continuity compara- 

 tively early requires no further consideration here ; we have 

 only to concern ourselves with the case in which the initial 

 disturbance has spread out into a spherical shell of very small 

 disturbance whose mean radius is very great compared with 

 the difference between its extreme radii. The equations 

 applicable to the disturbance are then, very approximately, 



u = a^-, (11) 



r 



c\ -I 



u or a-^cc - for a given part of the wave, . (12) 



where p is the mean density, p + 8p the actual density at a 

 point where the velocity is u, and a is the velocity of infinitely 

 feeble sounds in air of density p ; r is as usual the distance of 

 a point from the centre of symmetry. Let us consider two 

 neighbouring points M and N, on the same radius, each being 

 fixed hi a definite part of the ivave, the point M being behind 

 N (i. e. nearer to the origin), and the air-velocity at M ex- 

 ceeding that at N by Aw. Then, as the wave advances, oath 



