Wave Pattern of a Doublet in a Stream. 517 
The last term in (2) gives the regular waves to the rear, and the remaining 
terms the local disturbance which is symmetrical before and behind the origin. 
The integral in (1) is the real part of 
2 4 (a+ipiu 
| Ee is (3) 
o ute 
where « =xkyr, 8 =«x,f. Asymptotic expansions may be obtained for large 
values of the parameters, or the integral may easily be evaluated directly by 
numerical methods when « is not small. For small or moderate values of « 
and (, (3) may be calculated from 
— (A + 7B) e 8, 
A=y+lor+% i ; cos 28, 
1n.n! 
ee ee Ge a ; sin n0, 
1Un.N: 
r—(e2-+ 2) tanO@=a/8, y= 0-5772. (4) 
Consider the surface elevation at the origin (r = 0). Since we have 
“wcos Bu — sin Bu 7 8 1 (eB 5 
[, 1 + uv c a 
for 8 > 0, where li is the logarithmic integral, we have at the origin 
_ 2M 
a a {1 — Be-* li (e*)}. (6) 
Using tables of these functions, we find that 7 is zero when 8 is approximately 
1-35, or when c = 0-864/(gf); when c is less, the value of (6) is negative, 
while at greater speeds it is positive. To illustrate this point, the surface 
elevation has heen calculated from the complete expressions (1) and (2) for 
two different cases, «9f—=4 and x)f= 0-5. The graphs are shown in 
fig. 1, A being for the smaller value of xf and B for the larger. 
The ordinates are to the same scale assuming M and f constant and c to be 
the variable ; the abscisse are in wave-lengths, or more strictly the values of 
Kot. 
3. Consider now the three-dimensional problem. Take Ox and Oy in the 
surface of the stream, the current being in the negative direction of Oz and 
take Oz vertically upwards. lor a horizontal doublet of moment M at the 
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