198 
39 APPENDIX I 
1. SMALL-AMPLITUDE WAVES 
is the integral of the excess pressure with respect to the time, or the impulse, at 
the point in question from time t, up to the time t to which wu and prefer. Since p 
itself falls off as 1/ras the wave moves outward, the additional velocity represented 
by the second term on the right in [3] varies from point to point as 1/r?. This term 
is thus of importance only near the source of the waves. 
The name afterflow will be given to the part of the velocity represented by 
the second term on the right in Equation [3]. Each part of the pressure wave, as it 
passes outward from the center, makes a contribution to the afterflow whose magnitude 
is proportional to 1/7’. 
5. ENERGY AND MOMENTUM 
In plane progressive waves of small amplitude the energy at any point is 
half kinetic and half potential. If HF is the energy density or energy per unit volume, 
and Mis the momentum per unit volume, 
2 
E = put = 2 i pu -£ [4a,b] 
If I is the intensity of the wave, or the energy transferred across unit area per sec- 
ond as the wave advances, 
I = cE = pew = Be [5] 
In sea water, if J;, denotes J expressed in foot-pounds per square inch per second, 
and if P., denotes p expressed in pounds per square inch, 
1 
lin = 68.2 Pin [6] 
In small waves the energy transferred equals the work done by the pressure p on the 
water moving with speed u =p/pc. 
| APPENDIX I 
2. REFLECTION OF WAVES 
II. REFLECTION OF SMALL-AMPLITUDE WAVES 
When a plane wave encounters a plane surface at which the nature of the med- 
ium changes abruptly, the wave divides into two waves, one of which travels into the 
second medium as a transmitted wave, while the other returns into the first medium as 
a reflected wave. The conditions to be satisfied at the interface are that the net 
pressure and particle velocity must be the same on both sides of the interface. 
Let the incidence be normal, and let p, p’,p” denote the excess pressure 
(above normal) in the incident, transmitted, and reflected waves, respectively (Figure 
14). Let the particle velocity be measured positively in the direction of propagation 
of the incident wave. Then, if u, u’,u” denote the corresponding particle velocities, 
and if p,, c, and 2, c2 denote density and speed of sound in the first and second 
mediums, respectively, we have p = 01c,uU, Pp’ = paceu’, p” = - prc, u'(the negative sign 
because of the reversed direction of propagation). At the interface 
