4 THEORY OF SHIP WAVES AND WAVE RESISTANCE. 
its main purpose of being a general account which may be of value 
to the student of this aspect of the science and at the same time be of 
interest to those who have not the opportunity of studying the 
mathematical theory for themselves. 
Some formule and references will be found in notes appended to 
this lecture, and I am indebted to the Royal Society for permission 
to reproduce the diagrams. 
Tue GENERAL PROBLEM. 
We wish to know completely the fluid motion produced in the water 
when a ship is towed along at constant speed, and the first step is 
to see what information is necessary before we can attempt to find a 
solution. We can group this under three heads: (1) the laws of motion 
of the fluid, (2) the forces acting throughout, and (3) the conditions 
at the boundaries of the fluid. We are faced at the outset with the 
difficulty of saying what are the laws of motion of an actual liquid 
such as water. We know that water is viscous and we can write down 
equations taking the viscosity into account; and we can also solve the 
equations in simple cases if the velocities are not too large. But we 
also know, unfortunately, that those solutions break down completely 
when the motion becomes eddying or turbulent. It is not my intention 
to discuss here whether the difficulties arise because the solutions of 
the equations of viscous motion are inadequate or because the equations 
themselves are incomplete; in either case the inclusion of fluid friction 
in our problem would complicate it so much as to make progress 
almost impossible at present. 
We are therefore compelled to assume the liquid to be frictionless. 
This is no doubt a serious limitation, but perhaps not so important 
if we confine ourselves meantime to qualitative and comparative con- 
clusions from our results. Moreover the direct influence of viscosity 
upon the wave motion is comparatively small, and indirect effects might 
possibly be allowed for later by some adjustment of the effective form 
of the ship. However that may be, we can only make any advance 
by separating frictional resistance from wave resistance, and we there- 
fore assume the information required under the first head to be the 
laws of motion of a frictionless liquid; these are equations connecting 
pressure, velocity, and acting forces, and their rates of change 
throughout the liquid. We may dispose of the second head by simply 
taking the acting forces to be those due to gravity. Under the third 
head, the conditions at the boundaries are of two kinds; at the free 
upper surface of the water the pressure must be the atmospheric 
pressure, while at the wetted surface of the ship the condition is simply 
that the water must remain in contact with the ship or that the com- 
ponent velocity of the water at right angles to the wetted surface must 
equal at each point the component of the ship’s velocity in that 
direction. 
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