Here 6 = B/L is the beam-length ratio 



£ 



V) = 2/L \y 



H) h 



and K Q (6) satisfies the equation 



K H sec 2 6 tanh K Q H = 



F 2 



It should be noted that /, and /„ are, respectively, even and odd functions of £ and that they 

 both are even functions of 77. Therefore since 



P n 2 



it is sufficient to evaluate the integrals for only positive values of their arguments. 



SHALLOW-WATER APPROXIMATION 



The limiting value of the pressure ratio p/p as the depth approaches zero is referred 

 to here as the shallow-water approximation. From Equation [2] it can be seen that if H -» 0, 

 K n H -0, and F -» °°, then 



p 4 



~0 "* rr 2 



77/2 



sec tf esc 



/ 



dK 



sin (A cos 6) sin (A" /3 sin 



cos (K f cos 0) cos (A" 77 sin 



= — [p7.£ 77) + ?"(-£ 1?) + p (£ -7?) + /T(-£ -17)] 

 4 



where 



r<& »?) = 



« 2 Jo 



- [ 



* 2 Jo 



dd sec esc d 



f dK 



I sin 



J K 



dd sec esc 6 In 



X + tan B 

 X - tan 6 



[K(l - £) cos 6] sin [ff(/3 - tj) sin d] 



sgn [(1- £)(£-,)] 



