CALCULATIONS ILLUSTRATING THE EFFECT OF BOUNDARY LAYER ON 
WAVE RESISTANCE 
By Professor T. H. Havetock, M.A., D.Sc., F.R.S., Honorary Member (Associate Member of Council) 
Summary 
The main object of the paper is to examine the possible 
effect of the boundary layer in producing a virtual modifi- 
cation of the lines of the ship near the stern. This is re- 
garded as a deflection of the streamlines due to increased 
displacement thickness of the boundary layer in this 
region. By superposing a source distribution to produce 
this additional deflection, expressions can be obtained 
for the modified wave resistance. No attempt is made to 
attack the problem directly for actual ship forms. Instead, 
an indirect method is taken of considering some ideal 
simple forms and assuming small modifications of the 
lines near the stern such as might reasonably be ascribed 
to boundary layer effects. It is shown that such variations 
suffice to eliminate the humps and hollows on resistance 
curves at low speeds while making relatively much less 
difference at high speeds, a result which would improve 
the general comparison between calculated and measured 
wave resistances. The paper also includes some remarks 
on experiments with plank-like forms whichare not wholly 
submerged, and an attempt is made to assess numerically 
the wave-making resistance in such experiments on skin 
friction. 
Introduction 
The theory of wave resistance in a frictionless liquid 
leads to a resistance curve Which oscillates rapidly and 
excessively at low speeds, and such oscillations do not 
Occur in resistance curves derived from experimental 
results. This is commonly ascribed to the wave making 
at low speeds being mainly due to the bow of the ship; 
and an obvious explanation is that the effect of viscosity 
has been to render the stern relatively ineffective in wave 
production at low speeds. Some years ago! the author 
considered the matter from the point of view that the 
effect of the friction belt surrounding the ship is equiva- 
lent to smoothing out the lines in the rear portion and 
some calculations were made to show that this would lead 
to a diminution of interference effects at low speeds; 
however, the calculations were too complicated to pursue 
in any detail at that time. Later? the direct assumption 
of a reduction factor for the rear half of the model was 
made; the assumption was as simple as possible so as to 
make calculations practicable, the wave-making proper- 
ties of the whole of the rear half being reduced by an 
arbitrary factor less than unity. Subsequently this idea 
of a reduction factor was largely extended and examined 
in detail by Wigley. In particular, Wigley compared 
theoretical and experimental resistance curves for a 
large number of models, deducing the necessary re- 
duction factor to give reasonable agreement and obtain- 
528 
ing an empirical formula for the variation of the factor 
with the speed. In this work also the factor was applied 
to the whole of the rear half of the model and it was found 
to vary in value from zero at the lowest speeds, where 
only the front half is effective, to unity at the highest 
speeds, where front and rear are equally effective. This 
extension and analysis by Wigley is very useful in giving 
a practicable way of modifying theoretical resistance 
curves, but, admittedly, it leaves much to be desired from 
a theoretical point of view. In particular, the variation 
of the factor from zero to unity seems rather para- 
doxical; no doubt viscous effects vary with the velocity, 
but not to such an extent as is implied by that range of 
values. I believe an explanation can be found in the 
fact that boundary layer effects on wave formation are 
appreciable over only a small length of the model near 
the stern; just as one has a similar comparison between 
actual normal pressures and those calculated for a 
frictionless liquid. It is well known that for a friction- 
less liquid the wave-making effect of bow and stern 
angles is predominant at low speeds, while at high speeds 
this is not the case. Hence if the modification of the 
form is confined to a region near the stern, and even if 
that modification does not vary much with the speed, it 
will automatically have greater effect at the lower speeds 
than at the higher. The present paper is an attempt to 
find out how far this is the case. 
The general point of view so far as the friction belt is 
concerned has been well expressed by Baker* in the 
remark: “In the after body two things take place, first 
the contraction of the virtual body, round which the free 
flow is taking place, which includes the slow-moving 
portion of the friction belt—a rather indefinite extension 
of the real form—causes an expansion of all the stream 
tubes and of the frictional belt, and second expanding 
stream lines are never very stable and do not adhere 
to the form from midships to stern post.” It must be 
admitted that this “rather indefinite extension” of the 
form still remains undefined. In principle, if we know 
the thickness of the boundary layer and can deduce its 
displacement thickness, we know by how much the 
streamlines of the outer flow are deflected. We can 
then, in theory, superpose on the original form a source 
distribution which would produce the required extra 
defiecticn and hence calculate the modified wave re- 
sistance. It may be said at once that the necessary data 
are not available, and in any case the calculation would 
be almost impracticable. The scope of the present paper 
is much less ambitious, and the work may be described 
