25 T. H. Havelock. 
comparison with an equivalent surface pressure ; it consists in calculating the 
rate of dissipation of energy by a certain integral taken over the free surface 
when, as is usual in these problems, a small frictional force has heen introduced 
into the equations of motion of the fluid. Lamb, however, deals only with a 
single doublet, to which a submerged body is equivalent to a first approxima- 
tion, and so does not obtain the interference effects which arise from an extended 
distribution of doublets ; further, he carries out the necessary calculation by 
analysing first the surface distribution of velocity potential, or in effect analysing 
the wave pattern. In the following paper it is shown that this intermediate 
analysis may be avoided by a direct application of the Fourier double integral 
theorem in two dimensions. This step simplifies the extension of the calcula- 
tion to any distribution of doublets in any positions and directions ; various 
cases, which it is hoped to use later, are given. in some detail for deep water, 
and one case of a single doublet in a stream of finite depth. 
Two-dimensional Motion. 
2. The results for a two-dimensional doublet are well-known, but there are 
one or two points of interest in the calculation. We shall suppose the liquid 
to be at rest, and the doublet to be moving with uniform velocity c. Let the 
doublet be of moment M, with its axis horizontal, at a depth f. Take the origin 
in the free surface, with Oz in the direction of motion and Oz vertically upwards. 
If ¢ is the surface elevation, and if there is a frictional force proportional to 
velocity, the pressure condition at the free surface gives 
2 — gC + u’'d = constant, (1) 
¢ being the velocity potential. Since, at the free surface oC /ot = — dd/oz, 
we have for the steady motion relative to the moving axes, 
od | od od 0, (2) 
Coa eo ca 
to be satisfied at z= 0, with xy = q/c? and = p’/c. The conditions of the 
problem are satisfied by 
NE a a 
a+i(z+f) x+i%(—f) 
eS) erk® — k(f —2z) 
2eeyM | ————__ dx, 3 
= UK 9 == Sp ( ) 
$ 
where the real part is to be taken, 
279 
