454 F. W. Boggs and N. Tokita 
rate of deformation of the wall at the point 7 + E(r) will be given by €(r). This must be 
identical with the flow velocity at the position 7+ &. Hence, we will have 
Xr) =Urté)= Wr) + We Etres, (3.1): 
Since we are considering only small perturbations, we can neglect the effect of € on ¥ and 
identify the rate of deformation of the surface with the velocity of flow at every point along 
the boundary. This will lead to the approximate relationship 
2r) Ca) (3.2) 
If the motion of the wall is due exclusively to the forces in the fluid boundary layer, 
we must express the surface deformation in terms of these forces. To completely establish 
the boundary conditions, we must, therefore, express the surface forces in terms of the flow. 
When this has been done Eq. (3.2) will become a differential expression in the stream func- 
tion which must be satisfied on the boundary. 
If the response of the surface to external forces is linear, we can always express both 
the forces and the corresponding response of the surface in terms of progressive waves, just 
as we did in the case of perturbation of the flow velocity. If €, and €, are the amplitude of 
a wave of wave number © and frequency £ traveling along the surface and if P(a, 8) and 
T(d, 8) are the corresponding expressions for the amplitude of the wave of pressure and 
tangential force, we can define a set of compliances such that the following equations are 
satisfied: 
UNL: 
x 
I 
Ma T + Moyea 
(3.3) 
UN: 
< 
TT 
Ve Spiny. Pe 
The constants Y,. are dependent on the parameters @ and Bf and are determined by the 
specific nature of the wall. At a later date we hope to present detailed treatments in which 
these constants are calculated from the structure of the coating. In this paper we will con- 
fine ourselves to their general properties. 
To complete the calculation of the boundary conditions, we must express the surface 
forces T and P in terms of the stream function. From the perturbation of the steady-state 
solution of the Navier-Stokes equation, we obtain 
2 
='vpVivi: (3.4) 
We assume that Vs vanishes on the boundary and that its derivative at this point is given by 
a constant depending on the nature of the boundary layer profile. In the case of Blasius 
profile this is equal to yU,,/5 where y is a dimensionless constant and the same for all 
boundary layers of this type. If we express v, and v, as was done in Kgs. (2.1) and (2.2) 
and if we use the dimensionless variable defined in the previous section as the independent 
variable and if we use the same dimensionless parameters, it is readily shown that the pres- 
sure assumes the form 
