184 
IV. EFFECTS OF PRESSURE WAVES 24 
6. INTERNAL INTERFACES 
will then occur at these interfaces. The reflected waves thus produced, returning 
to the outer surface, will be partly transmitted there end partly re-reflected back 
into the target; in part they will again be reflected at the internal interfaces; and 
so on. If the various interfaces are close together, however, as in a ship's skin, 
the interplay by repeated reflection goes on so rapidly that the various waves quick- 
ly blend together. Then other methods of analysis become sufficiently accurate and 
are more convenient. 
Even if the target contains laterally dispersed structures, such as braces, 
the analysis in terms of waves is still applicable, but it becomes much more compli- 
cated. 
IV. EFFECTS OF PRESSURE WAVES 
7. ON A FREE PLATE 
7. IMPACT OF A PLANE PRESSURE WAVE ON A FREE THIN UNIFORM PLATE 
By a thin plate is meant one so thin that the time required for an elastic 
wave to traverse the thickness of the plate is much less than the time required for 
the pressure in the incident wave to change appreciably. This condition may not be 
satisfied at the very front of the wave, but the small error so caused will be ig- 
nored. Under these circumstances it is sufficiently accurate to treat the plate as 
a rigid body. 
As before, let p, wu denote excess pressure (above hydrostatic) and particle 
velocity of the water in the incident wave, and p”, u” the same quantities in the re- 
flected wave. Let m denote mass per unit area of the plate, and xz its position mea- 
sured from any convenient origin in the direction of propagation of the wave, which 
we suppose to be perpendicular to the face of the plate (Figure 11). Then the equa- 
tion of motion of the plate under the influence of the water pressure is 
dz _ . 
p,u y x Air pressure on the opposite side of the plate is sup- 
y) posed to balance the hydrostatic pressure. Since the 
y m per plate and water remain in contact, we have also 
oom y unit dx =u4+tu= (2—# = ’) 
p,u y) area dt pe 
ae 
y) where p and c denote density and velocity of sound in 
Y the water; for p=pcu, p”=—pcu"(Appendix I, Section 
1, Equation (2]). Eliminating p”: 
Figure 11 dx dx 
mage + Pc a, = oP {10] 
Here p is a function of the time which may be denoted by p(t). The equation can be 
solved for xz when p(t) is known. We shall consider in detail only a simple type of 
