208 



The problem is on; of axial symmetry ^nd wc take r, &, to Dt ths usj».1 sph;ric?.l-Dol ?.r 

 co-orflindtes ref'?rrc-a to the centre of the sphert ^s orijin, the lines joining this orijin to the 

 point scarce Deinj 8^0. 



The pressure puis- aiv-Tsing from the ooi if source is taken to be 

 p Xj J ct - r' + (X - i) :, 



Where r' is distance from the point source, c is wave velocity, p^ is maximum pressure in pulse at 

 the distance Xa correspond inj to centre of sphere, and t is time measured from the arrival of the 

 incilcnt w3V.= at the ne/irest point of the sphere; the non-dimensional function t will thus be zero 

 for negative values of its srjum.nt. 



In Appendix a it is shown that the folloK/inj relation holds at tin* tj for any solution p 

 of the wave-equation. 



1 cjr B p + 2 Br 3g_ 



r 3n c'Bt 



r 3n cBt 



♦ 1 B^o + 2p ^r + 1 ^ 



Cos 9 dS 



cBnBt 



1 2e. 



r cBt 



~7 



r^ Bn 



t=t J- r/c 



Bn 



t= t j^-r/ c 



Sini^M dS = 

 B n 



(3) 



where the summation extends over all surfaces bounding the region within which the wave equation 

 holds and n is the normal to the surface drawn into this region. 



We apply equation (3) to the present problem oy considering the region bounded externally 

 by a sphere of very large radius f) -• oc and internally by the sphere r = a and a sphere of very snail 

 radius e "• surrounding the point source. The contribution to equation (3) from the large sphere 

 becomes zero as i' — oc since for finite t^ the integrand is than to De evaluated for negative times 

 (t, - R/c) prior to the setting up of the pressure pulse froin the point source. 



The contribution to equation (3) from the sphere r = a comes solely from the first term, 

 since B 0/Bn = 0, and is:- 



Cn 



ITT a. 



;i?' 



a cBt 



cBrBt 



a 



1 3£ 



a Br 



t=tj-a/c 



Cos 5\n 6 ad 

 («) 



The contribution to equation (3) from the small sphere surrounding the point source will, 

 as e -* 0, arise solely from the terms in 1/fi due to the incident pressure pulse. Hence 

 substituting from equation (2) inequation (3) and letting £— we find for this final contribution, 



Ct, - a 



*" P. 



IntroOucing tne non-dimensional time T defined Dy 



ct, - a 



T - Si 



a 



(5) 



(«) 



then from equations (3) (t) and (5) we obtain 



2 77 



a 



^ ♦ 2|e ♦ aJffi + 2p ♦ a ^1 CosflSinSd^ 



3t^ Bt Bt Br '^'' J r = 



I f (T) ♦ i f {T)"j 



U 77 p 



(7) 



