Cases of Three-dimensional Fluid Motion. 357 
in terms of a simple harmonic wave in rear of the line x cos $¢+y sin ¢ = 0, 
together with a disturbance symmetrical with respect to this line. We 
might continue the discussion for general types of pressure distribution, 
provided the functions are such that certain transformations may be used ; 
however, it is simpler to study in detail a few cases for which the conditions 
are all satisfied. 
4. At this stage it is convenient to specify the system 
p= F(o) = As/(PP+ay", - (12) 
where A and f arecunstants. Using (3) for this case, we have f («) = Ae—*/. 
Returning to the expression (11) for the elevation, we consider the element 
making an angle @ with Oz. We change to axes Oz’, Oy’ given by 
“x=xcospd+y sin d, 7’ =y cos $¢—«# sin p. The integral with respect to « 
then becomes 
C) tex! —ikz! 
A Sey a oes fl . (13 
[- ee K—Ky Sec? f +7 (u/c) sec ar K—ky bec? P—2 (w/c) sec p Ko) 
This integral can be transformed by contour integration; as it is of a 
type familiar in plane wave problems* we write down the results when these 
have been simplified by making p zero after the transformation has been 
carried out. We have for the value of (13), 
4arnoA sec? pe-F 8" sin (xgx’ sec? p) 
492A [ Ky Sec? d cos fm + mi sin fm , 
mz! 5 GH : 
: GPae eeTO mdm, for x’ <0; 
2A | BIBER O08 UIST eri tone) SOMA) 
0 m* + Ko" sec* 
From (11) and (14) we could now write down the elevation €as the sum 
of the constituents for all values of ¢ in the range from —7/2 to 7/2. The 
first term in (14) represents simple waves in the rear of the corresponding 
wave front z cos ¢+y sin ¢=0; hence the integration of this term 
would only extend over elements for which the assigned point (a, y) was in 
the rear of the wave front. The other terms in (14) represent a disturbance 
symmetrical with respect to the wave front, and diminishing with increasing 
distance from it. We shall not write down the expressions, as we do not 
intend to examine the wave pattern in detail. From the definition in (7), 
it follows that we can evaluate the wave resistance R by considering first a 
simple constituent of the elevation and then summing with respect to ¢. 
Since the pressure system is symmetrical with respect to the origin, the 
symmetrical local disturbance in (14) gives no resultant contribution to R: 
* Compare, for example, ‘ Roy. Soc. Proc.,’ A, vol. 93, p. 524 (1917). 
149 
