1904 - 5 .] Lord Kelvin on Deep Water Ship-Waves . 563 
The generator may be a ship floating on the water, or a submarine 
ship or a fish moving at uniform speed below the surface; or, 
as suggested by Kayleigh, an electrified body moving above the 
surface. For canal ship-waves, if the motion of the water close 
to the source is to be two-dimensional, the ship or submarine 
must be a pontoon having its sides (or a submerged bar having 
its ends) plane and fitting to the sides of the canal, with 
freedom to move horizontally. The submerged surface must be 
cylindric with generating lines perpendicular to the sides. 
§ 35. The case of a circular cylindric bar of diameter small com- 
pared with its depth below the surface, moving horizontally at a 
constant speed, is a mathematical problem which presents interest- 
ing difficulties, wrnrthy of serious work for anyone who may care 
to undertake it. The case of a floating pontoon is much more 
difficult, because of the discontinuity between free surface of 
water and water-surface pressed by a rigid body of given shape, 
displacing the water. 
§ 36. Choosing a much easier problem than either of those, I 
take as wave generator a forcive * consisting of a given continuous 
distribution of pressure at the surface, travelling over the surface 
at a given speed. To understand the relation of this to the 
pontoon problem, imagine the rigid surface of the pontoon to 
become flexible ; and imagine applied to it, a given distribution II 
of pressure, everywhere perpendicular to it. Take 0, any point at 
a distance h above the undisturbed water-level, draw O X parallel 
to the length of the canal and 0 Z vertically downwards. Let 
£, £ be the displacement-components of any particle of the water 
whose undisturbed position is ( x , z). We suppose the disturbance 
infinitesimal ; by which we mean that the change of distance 
between any two particles of water is infinitely small in comparison 
with their undisturbed distance ; and that the line joining them 
experiences changes of direction which are infinitely small in 
comparison with the radian. For liberal interpretation of this 
condition see § 61 below. Water being assumed frictionless, its 
motion, started primarily from rest by pressure applied to the 
* “Forcive” is a very useful word introduced, after careful consultation 
with literary authorities, by my brother the late Prof. James Thomson, to 
denote any system of force. 
