SURFACE AND OTHER EFFECTS 4II 



If we consider an infinite boundary surface A in the water the velocity 

 at any point of which is given by u{t', t), where r' is the vector displace- 

 ment from a suitable origin, it can be shown that the velocity potential 

 at any point r on the surface is given by the diffraction integral 



(10.12) .(r, ^ - - ff^ ^ |,_,.|" ^ ''^' 



where ^^n(^^ is the component of velocity normal to the surface at r' 

 and directed away from the water. This equation can be interpreted 

 as giving the effect, at a point r, of spherical waves emitted from points 

 t' over the surface. Each such diffracted wave, w^hich travels with a 

 velocity Co in the water, requires a time \r—r'\/co to reach the point r from 

 r', since the absolute value |r — r'l is the distance between the points. 

 The effect at r and time t is thus associated with the velocity Un Sit i' 

 for an earlier, retarded time t = t — \t — t'\/co. The diffraction integral 

 thus bears a close relation to Huyghens' principle for light waves, by 

 which perturbing effects of boundaries are represented by diffracted 

 spherical waves, and is derived by similar mathematical methods. 



The value for the present purpose of the velocity potential formu- 

 lation is the fact that it permits calculation of the pressure at any point 

 on the surface A. In order to do this, it is convenient first to express 

 the perturbed potential (p in terms of the unperturbed value (po which 

 would exist if the boundary A had no effect and the normal component 

 of velocity were the value Uon in the unperturbed incident wave. The 

 value of (Po is then given by 



(10.13) ^„(r, i) = -1-jf "^^^^ dA' 



In the linear approximation, velocity potentials are additive, and tak- 

 ing the difference of Eqs. (10.12) and (10.13) gives 



<p(r, t) = (Pair, t) --- \ '^ V ,1 ^ ' ^ dAo 



^-^ J J ^ k - r'l 



The pressure at any point on A is given by P = p^ d<p/dt, and if Pi is 

 the pressure in the incident wave, we have 



(10.14) 



P(r, = PKr. i) - l ^^^ ^ [I «„(r', t) - | uUr', t)\ iA' 



the subscript r indicating evaluation at time t = t — \t — r'\/co. 



