WAVE PATTERNS AND WAVE RESISTANCE. 
By Professor T. H. Havetocg, F.R.S. 
[Read at the Summer Meetings of the Seventy-fifth Session of the Institution of Naval Architects, 
July 12, 1934.] 
INTRODUCTION. 
1. It is not my intention to discuss in this paper practical problems of ship resistance, 
but rather to review briefly certain points in the mathematical theory of ship waves and 
wave resistance. In doing so, I shall not attempt to give the derivation of formule or 
any mathematical analysis of them; my main object is to give a descriptive or qualitative 
account of some of the mathematical expressions and to show how in some cases deductions 
may be drawn from an inspection of them. 
The wave pattern made by a ship is familiar both from observation and as a subject 
of mathematical study, and it is equally fascinating from both points of view. Perhaps the 
earliest theoretical account is that given by Kelvin in 1887 in his well-known lecture on 
ship waves to the Institution of Mechanical Engineers. That lecture was based on mathe- 
matical work of which a later improved version was published by Kelvin in 1904,* and 
it is this later work which is usually quoted now in the text-books. The ship, in that 
work, is idealized to a point disturbance travelling over the water and at the same time 
sending out groups of waves which combine in such a way as to produce the characteristic 
pattern of transverse and diverging waves. The early history of this idea of wave groups 
and group velocity is also of some interest. In a letter written to Stokes in 1873, William 
Froude describes the motion of a group of waves, how the group as a whole advances with 
a less velocity than that of the waves composing it, wave crests advancing through the 
group in its motion and appearing to die away at the front while new ones are formed 
at the rear; he writes, in his letter from Torquay, “In my long experimental tank or 
canal here, I have frequent opportunity of noticing this in the propagation of artificially 
generated waves. I have not, indeed, yet investigated it quantitatively, because my hands 
are full: but at a later date when experiments on the oscillation of models will be the work 
in hand, I shall have to establish regular appliances for the generation of waves, and the 
investigation to which I refer will be comparatively easy.” It was in 1876 that Stokes 
gave the kinematical explanation of group velocity, a more general account being given 
shortly after by Rayleigh. This was followed in 1877 by Osborne Reynolds’ dynamical 
theory of group velocity, connecting the flow of energy and the rate of work of the fluid 
pressure in a train of waves; it is this latter point of view which is of fundamental 
importance in the theory of wave resistance. 
Much work has been done since then, both on the detailed structure of wave patterns 
* Edin. Roy. Soc. Proc., Vol. XXV. (i), ““On Deep Water Two-dimensional Waves produced by any 
given Initiating Disturbance”; “On the Front and Rear of a Free Procession of Waves in Deep Water’’; 
and “Deep Water Ship Wayes,” 
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