NO. 15.1 RESISTANCE OF VESSELS. 35 



The waves following a vessel in rapid motion consist, as we are accus- 

 tomed to seeing, of two different systems, which are both easily seen in the 

 photographic illustration Fig. 3, PI. XII. One set of waves, the transverse 

 waves (best seen on the left-hand side of the figure), stretch right across the 

 wake, bending slightly aftwards; the other, the diverging waves (best seen 

 on the right-hand side) stretch out from the stem as wedges each one more 

 extensive than the preceding one. 



The shape of these waves has been calculated mathematically under the 

 simplifying supposition that the vessel producing them is infinitely small 1 . 

 The error caused by this simplification is, as far as interests us at present, 

 without appreciable influence on the shape of the waves at a distance from 

 the vessel. Fig. 3, PI. V shows the crest-lines of both systems of waves cal- 

 culated in this way, in the case of deep water. The vessel must be imagined 

 to be at the point at the extreme left of the figure. Theoretically the diverg- 

 ing waves should fill the whole space to both sides of the middle line, but in 

 the proximity of this middle line there would be an infinite number of infinitely 

 short waves which cannot be drawn on the figure. As a matter of fact, the 

 diverging waves are never perceptible in the middle of the wake. 



Really, the stem and the stern of a vessel each produces both these sy- 

 stems of waves; but owing to the shape of the vessel, the transverse bow- 

 waves and the diverging stern-waves are very low and are easily overlooked. 



The wave-length depends upon the velocity v, of the vessel and upon the 

 direction of the crest-line of the wave considered : in order that ship and wave- 

 system may keep the same position relative to one another, the velocity c of 

 a wave must be in a fixed ratio to v. — The transverse waves, for instance 

 (or more properly the central part of them) must have the same velocity v as 

 the vessel; the diverging waves, because moving in different directions to the 

 vessel, must have smaller velocities. If a be the angle between the wave- 

 crest and the course of the vessel, the velocity of the wave must evidently 

 be c = v sin a (see Fig. 3, PI. V). — On the other hand, c is a function of 

 the wave-length, varying, as the square root of the latter, in deep water. The 

 transverse waves are consequently longer than the diverging waves (longer, 

 the greater is a), and the lengths of all the waves increase if the velocity of 



See Lamb, Hydrodynamics, Art. 228. 



