458 F. W. Boggs and N. Tokita 
y"'( C) 
Cw'(f) 
1/3 
C = -(a,aR) y. 
H(C) 
(3.16) 
FG) =(GCase) Y,, + ia¥,G(a,C) Y,, 
22 
aan) iyY,, + aCY,, - =— 
se! 
o 
4. SOME GENERAL PROPERTIES OF A SURFACE COMPLIANCE 
Rayleigh was the first to show that solid bodies are subject to waves propagating over 
their surface. Such waves will exist in the case of a turbulent boundary layer in contact 
with a flexible surface and will play an important part in the nature of the compliance. 
Basically, there are two factors to be considered; the deformation which occurs at the point 
where the pressure is applied and the way this deformation propagates along the surface. 
We will consider the effect of both of these reactions of the surface. Let us suppose that 
we have a pressure of the type mentioned previously, and for the sake of simplicity, let us 
suppose that it varies periodically with the time so that it may be represented by the product 
of a complex exponential in the time and a function of x. In this case we may represent the 
pressure by an expression similar to 
el Pash = =| PGa) eo dae 7 (4.1) 
This will cause periodic displacements of the surface velocity in the x and y direction which 
may be represented by 
4 SHewiGie chu a(n) 
-iBt 
e 
S 
| 
+0 — 
| Y,,(0,8) P(i)e da 
(oo) 
(4.2) 
v.=-ipe ”* € (x) 
+0 ts 
= = POX YL Ae 2 Bie 
| Vos) P(a) e dae 
fo) 
In Eq. (4.2) the Y’s are the surface compliances for a wave having a wave number a. If we 
use the Fourier inversion theorem for Eq. (4.1) and Eq. (4.2), we will obtain 
+ 00 
PCa) | FGx) en) ax 
iH 
he 
ex) 
PAID) MOB) yee Es) eu apex (4.3) 
