Images in Some Problems of Surface Waves. 278 
We shall calculate the value of (30) for & equal to 10, 8, 6, 4, 2,1 and 0-5, 
given in order of increasing velocity. Omitting the intermediary steps for the 
numerical values of the L. and M functions, the following table gives the values 
of A and B, calculated from (24), for these values of & and with h = k/32 in 
each case. 
k | 10 | 8 | 6 | 4 | 2 | 1 | 0-5 
A 0-021 0-064 0-204 0-646 1-805 2-311 1-891 
B —0-418 —0-522 —0-716 | —0-950 —0-596 0-668 1-742 
The simplest form in which to show the difference made by the second approxi- 
mation is to express (30) in each case in the form 
y/a = D sin kp (« + &), (31) 
and compare it with the first approximation 
y/a = C sin koe. (32) 
A comparison of D and C gives the alteration in the amplitude of the waves ; 
further, there is an alteration in phase expressed as a displacement of the crests 
to the rear by an amount €. 
In this form the final numerical values, for f = 2a, are given in the following 
table :— 
c/ /(ga). Cc D t/a. 
0-63 0-212 0-263 0-006 
0-71 0-460 0-568 0-017 
0-82 0-939 1-159 0-050 
1-0 1-701 2-046 0-148 
1-41 2-312 2-396 0-468 
2-0 1-906 1-721 0-669 
2-83 1-223 1-081 0-595 
We see that the second approximation makes a considerable difference in 
the amplitude in this case ; but it should be noted that, in addition to the depth 
being only twice the radius, the velocities are relatively large, the wave-length 
at the lowest velocity being about 1} times the depth. 
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