1158 
-4- 
modifications In the reflected wave. The pressure in the incident wave will first be taken as 
: P, = Pe en(t — x/c) (2) 
where x is distance measured perpendicular to the plate in the direction of the oncoming wave. 
The pressure in the reflected wave will necessarily be of the form 
Pe = py (t + 2) (3) 
‘At the surface of the plate itself x = 0 and the total pressure is 
pe oh + o tt) (4) 
° 
where p = py + Pie if c =v dp/do is the velocity of sound, the velocity of the water (or other 
medium of density 9) in the incident wave is p;/pc, while the velocity due to the reflected wave 
is - ¢/pc. if € is the displacement of the plate the equation of continuity at the surface is 
therefore 
eee = ont _ ¢ (5) 
where € is written for d&/dt and ¢ for ¢ (t). 
The motion of the plate is determined by its mass per unit area, m, by the pressure, p, and by 
external constraints such as the supporting framework. For simplicity it will be assumed that 
these constraints are equivalent to a spring which would cause the plate to oscillate freely (i.e. 
when not in contact with water) with a pewiod 277/ 4. The equation of motion is 
£ = é + we (6) 
Eliminating €& the equation for ¢ is 
¢ + Ag + weg = (rts Esp?) ent (7) 
The solution of (7) is 
2 MyinoGha a2 
5 n° + + Ls 
g¢ = a Sits 6 Sate Sp ent (a) 
Nee +u 
where Sy and S, are the roots of 
SA PNIEe Sy et gire io (9) 
The conditions at t = 0 are€ = 0 and é = p/m. When expressed in terms of g¢ these become 
Gomes 
when t = 0 (10) 
(1+ ¢) &+ n+h = 0 
The equations for A and 8 are therefore 
Ae ee cue 1 ] 
™m Pia, ToC y je 
m J 
% NOC + 2 
SNS Gp = 6s = 
1 2 m no —- + mi 
m 
4 Be Foes 10), Be por reno 
Hence, using the relations implicit in (9), namely <= -S,- Sw S455 
A= wevee 
