CALCULATIONS ILLUSTRATING THE EFFECT OF BOUNDARY LAYER ON WAVE RESISTANCE 
as an illustration of the possible effect of boundary 
layer on wave resistance. The problem is attacked in- 
directly by taking a simple form and making small modi- 
fications of the lines near the stern so as to obtain the 
required kind of change in the calculated resistance 
curve; one may then consider whether such modifica- 
tions can reasonably be ascribed to boundary layer 
displacement. We consider first the ideal case of a 
thin plank, with some incidental remarks on wave- 
making in experiments with planks. Then we consider 
a form with simple parabolic lines and with vertical 
sides: in the first place of infinite draught, and then 
of finite draught. Finally calculations are made for a 
form which is unsymmetrical fore and aft, in order to 
show the difference in resistance between motion with 
bow leading and motion with stern leading. 
Wave Resistance of Planks 
We begin with the ideal case of a plank of negligible 
thickness. Assuming the boundary layer to be turbulent, 
we take for its thickness 6 at a distance x from the leading 
edge the expression 
S = Os OD ae o 5 3, 6 (@) 
where » is the velocity. In the present problem it is the 
displacement thickness 5, with which we are concerned, 
as this gives a measure of the outward deflection of the 
streamlines; in general, 5, is defined by 
Oy HICh = DAM so 6 oo @M) 
where uw is the fluid velocity at a point in the boundary 
layer, u, the velocity at the outer limit of the layer, and 
the integral is taken along a normal through the layer. 
Assuming the usual velocity distribution we have 
8, = 46. At the rear end of a plank of length L, the 
displacement thickness 5, has a value*b given by 
E10, 0462 5¢Ruic yin aon (3) 
R being Reynolds number. Some values for a plank 
16 ft. long are given in the following table for various 
values of the Froude number f= /,/(gL); taking 
vy = 1-228 x 10-5, and with 6 in inches, we have 
We suppose the plank to be immersed in water and cut- 
ting the free surface. We might devise a source dis- 
tribution which would give this displacement boundary 
as a streamline. Knowing the surface waves produced 
by the source distribution and hence the energy required 
to maintain the system, we can deduce the corresponding 
wave resistance. Of course, for any body with form the 
wave resistance is associated with the normal pressures, 
and the skin friction with the tangential forces; there 
must always be some interaction between these, but the 
usual practice of treating them as entirely independent is 
a valid approximation in most cases. It is obvious that 
in the present hypothetical case any wave resistance 
must be associated with a change in the tangential 
frictional forces with, no doubt, a consequent disturbance 
of the conditions in the boundary layer. However, 
leaving aside this interaction for the time being, we may 
attempt to find some numerical value for a possible wave 
resistance. It is clear, from the values of b given above, 
that on any assumption it will be very small compared 
with the usual skin friction; but a rough estimate may 
be made. 
We quote now expressions for the wave resistance of 
a given source distribution.> We take the origin O in 
the free surface, Ox in the direction of motion, O z 
vertically downwards, and Oy transversely. If the 
source distribution is in the z x-plane and is of amount o 
per unit area at any point, the corresponding wave 
resistance is 
rr ]2 
R= 16 7x3p | P+ Q) sect 0d 0 . 4) 
0 
with P+iQ=|[oermnetiaseega dz (5) 
where xo = g/v?, and the latter integration is taken over 
the distribution. 
We shall be concerned with streamline forms which 
are narrow compared with their length, and if the form 
is given by an equation for y as a function of x and z, 
we use the approximation 
Ge ee et (0) 
Further, in the present paper, all the forms have vertical 
sides; so that, if dis the draught, we have 
1/2 
qy2 
Res =o [a — e~ Kodsec?®)2 (2 + J2)cos0d@ . (7) 
0 
cy Tas oy ikoxsec 6 
with r+is=| oe EX eA (8) 
Returning to the immediate problem, we simplify it by 
replacing the displaced streamline by a simple parabolic 
curve which starts from the front edge of the plank and 
leaves the rear end parallel to the plank and at a distance b 
from it; the resulting integrals are familiar in this work 
and can be readily computed and the approximation will 
serve for the immediate purpose. Taking the origin at 
the rear end of the plank, we use (7) and_(8) with 
peb@=si) 2. sic 6 ©) 
and we obtain 
r/2 
R 16 p pi 2.2 ; 
pean) ol Oe) Pear ape’. 
2 
mie cos 7) cos? 6d 0 (10) 
with 
Yo = & LIv?; Bo = g dfv?; y = yo Sec 8; P = Bo sec? 8. 
