63 
APPENDIX A 
Theory of small~amplitude sound pulses 
Al. Pressure in pulse 
Let time interval after the initiation of the explosive charge 
mass density of water 
velocity of sound in water 
pressure in pulse (additional to the hydrostatic pressure 
existing prior to the arrival of the pulse) 
velocity potential 
distance of point from the centre of the charge 
3S cw atec 
Assuming the pulse is of small amplitude and retaining only the first order 
term, the pressure is related to the velocity potential V by the equation 
= -p ov sete) Melcle || -uicicn || Jotsict Mucle’e’ TP efaie't tts lctenmmirerate (A1) 
The partial differential is used since p and V will vary in general, both 
with time and with space co-ordinates defining the position of any point 
in the water. The velocity potential V can be shown to satisfy the wave 
equation 
2 
vv-=4.-9% eee eee ooo eoe eee eco cece coe (42) 
where in rectangular co-ordinates 
vv = ve q. + q, siere eee eee SOON NOCON = O0C (A3) 
Transforming equation a2 to spherical polar co-ordinates (r, 9, f) 
2 
dgeVe dco) RON tert eety Gets ow 
BeSpe- Pak (SD + pepe ad (sme$D sappre hye oe (aH) 
But for any point distant r from this centre, both p and V will be 
functions of t and r only and equation A4 can be written 
AVav-SOv.. iow 
a ote ar + F ar eee eee eco ere ere eee eve (A5) 
Equation A5 can be written in the form 
2 . \ 
die) . 1, Siew Pun CER rh es 
r t 
One solution of equation A6 is 
I$ 
Se haat scree, ada loeb: he ae ee 
cr 
where F can, in general, be any function. 
