NO. IB.] RESISTANCE OF VESSELS. 37 



vessel, and fixed in space; for simplicity let I be a multiple of the wave-length 

 of the transverse waves. The energy of the transverse waves between AA' 

 and BB' may be I X E x • As the height of the waves and their position relative to 

 the vessel are invariably the same, the energy of transverse waves ahead of BB' 

 is increased by I X -E\> when the vessel has moved a length = I. In the 

 same time nearly half this energy 1 is carried by transverse waves across BB', 

 and consequently the rest or about PX^i must be supplied by the vessel. 

 Secondly consider the diverging waves. The energy of them between the 

 lines A A' and BB' may be I X E 2 . The average velocity c of the diverging 

 waves is c = v sin a, and its component in the direction of the vessel's keel- 

 line is v sin 2 «, v being the velocity of the vessel, and a some mean value 

 of the angle between the diverging wave-ridges and the keel-line of the 

 vessel. When the vessel moves forward I units of length (= AB), the energy 

 \ E 2 X & sm2 a is therefore carried across BB' by diverging waves; the energy 

 of diverging waves ahead of BB' is increased by I X E 2 , because the motion is 

 steady, and consequently the energy £ I X E 2 (2— sin 2 «) must be supplied by Ihe 

 vessel. Thus, when the vessel moves a distance I, the wave-energy to be cre- 

 ated by her is, in all, about £ I X E t -\-\l\E 2 (2— sin 2 a), and the value of the 

 wave-making resistance consequently approximates to | E x -\- \ E 2 (2 — sin 2 a). 

 Greater accuracy could be obtained by further dividing the wave systems into 

 smaller parts, so that within each of them, 1 the extreme values of a differ less 

 from each other. The analysis above, gives, however, an idea of the degree 

 in which the waves contribute to the resistance of a vessel ; it shows that diver- 

 ging waves cause comparatively more resistance than transverse waves, because 

 the energy of the former is in a smaller degree propagated with the vessel. 



It is well known that a vessel moving at slow speed, creates but very low 

 waves, and as the energy of waves varies as the square of their heights, the 

 wave-making resistance is in this case inconsiderable. At moderate speeds the 

 wave-making resistance increases, in deep water, approximately as the square 

 of the velocity, but at higher speeds it increases as the cube — or even ac- 

 cording to an higher power. 



It would be exactly one half, if the transverse waves were perpendicular to the keel- 

 line of the vessel throughout their whole length. 



