64 
From equations Ai and A7 the pressure is given by 
Bia oS (bi Spin stuk'so she vans, cate PEERME RSE MR oe (a8) 
where - f denotes the derivative of F with respect to t as defined by 
d 
f(t) hs au Sr («)} eee eee eee eee eee eee (A9) 
42 Particle Velocity. The particle velocity u can be derived from 
the velocity potential V by the relation 
Hy = oy So) ISAS Sna oct Mos ooo. oon) | gsc (A410) 
From equations A7 and A10 
4 1 
u=anf (t - &) - pe F(t - &) na soo, ace (A11) 
where f is defined as before by equation A9. 
The first term in equation Ai1 is similar to the expression for the 
pressure in equation A8 and decreases inversely as the distance, whereas 
the second term decreases more rapidly as the inverse square of the 
distance. At large distances this second term, sometimes called the 
afterflow, becomes negligible compared with the first term and to a first 
approximation 
i fe 
u=c ve ((( - a) eee ove eee eee eee eee eco (412) 
Using the approximate relation of equation A12 and equation A8 the pressure 
and particle velocity are related by 8 
ye Pee coe eee eee eee eco eee coe eco (A13) 
Ad. repulss per unit area. The impulse per unit area I transmitted by 
the pulse across the spherical surface at radius r is given by the area of 
the pressure/time curve at any distance. If the origin of time be taken 
from the instant at which the sound pulse starts from the centre, so that 
it arrives at radius r at time r/c then from equation A1 
oo 
Ms ne 3 r 
we ye Sensi E(t = 4) dt ce see cee ane (Atk) 
r Tv 
ra) Ci 
Therefore 
oo 
= P 1 t a c 
T= < [f (t') at!, where t' =(t - 5) ce cee ene (A15) 
re) 
Since the shape of f(t') is independent of distance, the impulse per unit 
area varies as the inverse of the distance. 
