CALCULATIONS ILLUSTRATING THE EFFECT OF BOUNDARY LAYER ON WAVE RESISTANCE 
Computation was made for a plank 16 ft. long, with a 
draught of 1 ft. at a speed of 8 ft. per sec., or a value of 
f= 0-3535. Further, 6 as given by (3) was taken to be 
0-03 ft. The result was R = 0-003 lb., compared with 
the usual skin friction of about 5-861b. The point of 
this calculation is simply to confirm that the effect of 
the plank boundary layer may be taken as inappreciable. 
It will be still less relatively when we consider a form of 
finite beam with any appreciable wave resistance. When 
we deal with such forms we shall therefore simplify the 
work by neglecting that part of the boundary layer 
which is the same as that for a plank, and shall consider 
only the region near the stern where the boundary layer 
becomes appreciably and rapidly thicker on account of 
the curvature of the form. 
Before proceeding, a few remarks may be made on 
skin friction experiments with planks. Actual planks 
have thickness and form, and if the upper edge is above 
the free surface there will be wave disturbance due to 
the form, modified to some extent by the boundary layer. 
Reference has often been made to the possibility of wave 
resistance being included in some measurements of skin 
friction, but usually only in the form of a caution; there 
do not seem to have been any attempts to give a numerical 
estimate of its value. Perring® refers to the possibility 
of having to make allowance for wave-making in experi- 
ments with plank-like forms, and Schoenherr,’ in re- 
ferring to his experiments with 3-ft. planks, remarks that 
the speed should not exceed about 2:7 ft./sec. on account 
of appreciable wave-making; it may be noted that this 
is a Froude number f of about 0-27, but other experi- 
ments with partially submerged planks have been made 
up to f= 0-4 or even higher. 
Knowing the form of the plank it would be possible to 
calculate the wave resistance from the usual formulae, 
but such results would be of doubtful value at low speeds 
because of the viscous effects which are now under dis- 
cussion. However, wave resistance theory suggests 
another line of attack. According to the formulae, for 
models with the same mathematical lines and with con- 
stant length and draught, the wave resistance varies as 
the square of the beam. This relation was examined by 
Wigley® for a series of three models satisfying these con- 
ditions. The residuary resistance, deduced from the 
total resistance by the usual method, did not quite obey 
this law; but the divergence was attributed to the neglect 
of form effect in estimating the skin friction, and small 
increases in this part of the resistance would give a wave 
resistance approximately obeying the theoretical relation. 
It may be remarked in passing that form effect is not 
easy to estimate for these narrow models because it is 
not sufficiently greater than the possible experimental 
errors in measuring resistance and velocity at low speeds, 
where in addition there may be the complication of 
laminar flow. For a general discussion of the relevant 
data for form effect reference may be made to Todd.? 
For our present purpose we choose Model 1970B, an 
experimental model used at the N.P.L. by Wigley.!° 
The model lines are given by 
530 
Ge ae x4 2 
y=b(1-F)(1tan tae) (I >) . (11) 
with a, = 0-4375; a, = — 0-4375; length = 2/= 16ft.; 
beam = 2b = 1-5 ft.; draught = d = 1 ft. 
The skin friction has now been calculated from the 
standard plank formula corrected for temperature.!! 
Form effect has been allowed for by adding a constant 
amount 0-05 to the corresponding © values, which is 
equivalent to increasing the skin friction by an amount 
ranging from 5% to 6%. The skin friction so increased 
was subtracted from the total measured resistance, and 
the residue was taken to be pure wave resistance. We 
now reduce these values according to the square of the 
beam for a plank-like form, with lines given by (11), 
length 16 ft., draught 1 ft., and beam 3 in.; this value 
of the beam gives.an angle of entrance (to the middle 
line) of 1:75°. The wave resistance for this plank-like 
form, so estimated, is given as R,, in the following Table; 
R, is the skin friction derived from the standard plank 
formula. 
v, ft/sec 
0 
2 
4 
6 
8 
0 
2 
4 
6 
8 
0 
OSDKRNANAAWORON 
BWrANDKHONOH 
0-2 
0:2 
0-2: 
0:2 
0:2 
0-3 
0-3 
0:3 
0:3 
0:3 
0-4 
OMAAININDDUAUNAL 
It appears that the wave resistance is about | per cent of 
the total resistance at f = 0-24 and rises to about 3 per 
cent at f= 0-4. Direct comparison with planks used in 
skin friction experiments is not intended, because these 
have different forms and, in particular, their smaller area of 
vertical cross sections would diminish the wave-making. 
Further, it is clear, from the various steps in the above 
calculation, that the results cannot be more than a rough 
approximation; nevertheless they will serve as an indi- 
cation of the possible numerical magnitude of wave- 
making resistance in plank experiments: 
Parabolic Form of Infinite Draught 
Returning to the main probiem, we consider a model 
with parabolic lines and make small modifications near 
the stern. We suppose at first that the draught is 
infinite, as the integrals can then be expressed in 
terms of functions for which tabulated values are avail- 
able, and the general character of the results is not 
much affected by the draught. . 
Putting y= by and x = /é, and the model having 
vertical sides, the form is given by 
Helo e. 5 oe) 
