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IV. EFFECTS OF PRESSURE wAES| 22 
4. IMMOVABLE INTERFACE 
pee 2 p2C, C0sd 
DP Pye, C086 + pic, cosd’ [2] 
PY _ Pols cos@ — p,c, c0sd’ (3] 
Ps Pay C080 + p,c, cos’ 
The reflection coefficient is thus 
2 (F) = (22% cos@ — p,c, Sebi C4 
Polo cosé + Pic, cos6" 
The ratio of the total pressure on 
the interface to the incident pressure is 
ee oh pe ae 2pP2ly cos 6 
NS Spi P ~ P2e, cos8 + pyc Cosd (5) 
The pressure measures the rate of transmis- 
Figure 9 
sion of_momentum across a surface (so long as the amplitude of the waves is small); 
hence the value of N is also the ratio of the momentum absorbed by the target to the 
momentum brought up by the incident wave. 
These equations hold so long as Equation [1] can be solved for 6’. If, 
however c,sin6>c,, total reflection occurs, with p” = p and R= 1, N= 2. 
We note that if c, =c,, p, = p., then R= 0 and N= 1, that is, the wave 
merely continues into the target without reflection. If p,c, = 0 (e.g. for vacuum), 
R= 1 and N= 0; also p” =—p. In this case the incident wave is completely re- 
flected with change of phase, compressions becoming rarefactions and vice versa; the 
particle velocity has the same direction in the reflected wave as in the incident 
wave, so that the reflected wave carries all of the incident momentum back into the 
water. If p,c,> co, on the other hand, as for an extremely dense or rigid material, 
N= 2. Although the reflection of energy is again total (R= 1), the motion of the 
water is reversed by the reflection and the momentum given to the target is double 
that brought up by the incident wave. In this latter case the pressure in the water 
at the face of the target is likewise doubled. 
At normal incidence the equations become 
pi_ 2 pce B" _ nlp = pits 
P Pola + Pil, DP Pale + Pic} [6a, b] 
2 
= Poo — Pi 
R= ee + — (7] 
pe 2 Poy an P2Ce— Picy [8] 
Nec pita pecs FRC 
The effects of reflection at normal incidence depend only upon the ratio of the 
acoustic impedances, p,c, and p,c,, in the two mediums. The pressure on the target 
is greater or less than that in the incident wave according as pc, > p,¢c, or 
Pf, < Pe, In Figure 10, R and N are plotted as functions of the ratio p,c,/p,c, , 
