Mr Pocklington, Standing waves parallel to a plane beach 309 



The image of €>2 with respect to the upper boundary may be 

 taken to be 



% = cot ^72 . exp {- Xr sin (j8' + d)] cos {q'- 7r/2 + Xr sin (/S' + 6)) 



(and it will be found on trial that this is the only value of <I>3 that 

 has the same general form as that given and satisfies the conditions 

 stated above). 



4. Let 



Oi = exp {—Xr sin 6} cos {{n — 1) 7t/4 + Xr cos 6} 



= /{0,(n- 1)77/4,^} 



and add to it the first n — I images taken alternately with respect 

 to the lower and upper boundaries, so that 



O =/{0, {n - 1) 77/4, 6} +f{2a, {n - 1) 77/4, - 6} 



+ cot a . / {2a, {n - 3) 77/4, dj + cota.f {4a, {n - 3) 77/4, - 6} 



+ cot a cot 2a ./{4a, {n — 5) 77/4, 9} + etc., 



the last term being 



cot a cot 2a ... cot (n — 1) a/2 .f{{n — 1) a, 0, 6} 



if 01 is odd and 



cot a cot 2a ... cot (n — 2) a/2 .f{na, ir/i, — 6} 

 if n is even. 



Each term satisfies (i). Pairing the terms starting from the 

 beginning we see that each pair satisfies (ii) and that if w is odd the 

 odd term at the end also does so. Pairing the second term with 

 the third, the fourth with the fifth and so on we see that each pair 

 satisfies (iii) and that the odd term at the beginning does so, as 

 does the odd term at the end when n is even. Also ii ^ 6 1^ a 

 the sine under the exponential sign is positive for each term except 

 the first so that these terms vanish exponentially at the infinite 

 part of the fluid. The first term has the correct form for standing 

 waves. Hence O satisfies all the conditions and A^ cos {pt + e) is 

 the velocity potential required. 



5. The ampHtudes at various points of the upper boundary 

 are proportional to the values of O there taken positively. The 

 value of O at ^ = 0, r = 00 varies from + 1 to — 1. Hence p the 

 amplitude at the origin divided by the maximum amplitude at 

 infinity is equal to the value of <|) at the origin. 



Firstly, let n be odd. Pairing the terms starting from the 

 beginning we have 



/> = 2 cos (n — 1 ) 77/4 + 2 cos (n — 3) 77/4 . cot a 



+ 2 cos {n — 5) 77/4 . cot a cot 2a + etc. 



