[Reprinted from the PROCEEDINGS OF THE RoyaL Socrety, A. Vol. 100] 
The Effect of Shallow Water on Wave Resistance, 
By T. H. Havetock, F.R.S, 
(Received October 28, 1921.) 
1. The general character of experimental results dealing with the effect of 
shallow water on ship resistance may be stated briefly as follows:—At low 
velocities the resistance in shallow water is greater than in deep water, the 
speed at which the excess is first appreciable varying with the type of vessel. 
As the speed increases, the excess resistance increases up to a maximum at a 
certain critical velocity, and then diminishes. With still further increase of 
speed, the resistance in shallow water ultimately becomes, and remaias, less 
than that in deep water at the same speed. The maximum effect is the more 
pronounced the shallower the water. For further details and references one 
may refer to standard treatises, but one quotation may be made in regard to 
the critical velocity: “This maximum appears to be at about a speed such 
that a trochoidal wave travelling at this speed in water of the same depth is 
about 1} times as long as the vessel. . . . It was at one time supposed 
that the speed for maximum increase in resistance was that of the wave of 
translation. This, however, holds only for water whose depth is less than 
0:2 times the length of the vessel. For greater depths the speed of the 
wave of translation rapidly becomes greater than the speed of maximum 
increase of resistance.’* In a recent analysis of the data, H. M. Weitbrechtt 
expresses a similar conclusion by stating that for each depth of water there is 
a critical velocity, but that the critical velocity does not vary as the square 
root of the corresponding depth. 
It should be noted that experimental results are for the total resistance. 
If we assume that this can be separated into three terms, which are simply 
additive, namely, eddy, frictional, and wave-making resistance, it must be 
admitted that probably all are affected by limited depth of water. However, 
the main differences are due to the altered wave-making, and the general 
explanation is to be found in the fact that there is a limiting velocity, ,/(g/) 
for simple straight-crested waves on water of depth h. 
Leaving aside the difficult problem of a solid body towed or driven through 
the water, we may study the allied problem of a given distribution of surface 
pressure aud the associated wave resistance. Previous calculations of wave 
resistance have been limited to a line distribution of pressure, involving 
*D.W. Taylor, ‘Speed and Power of Ships,’ vol. 1, p. 114; also G.S. Baker, 
‘Ship Form, Resistance and Screw Propulsion,’ p. 134. 
THM. Weitbrecht, ‘Jahrbuch d. Schiffbautech. Gesell.,’ vol. 22, p. 122 (1921). 
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