U8 Mr. stokes, ON THE THEORY OF OSCILLATORY WAVES. 



10. The waves produced by the action of the wind on the surface of the sea do not probably 

 differ very widely from those which have just been considered, and which may be regarded as 

 the typical form of oscillatory waves. On this supposition the particles, in addition to their 

 motion of oscillation, will have a progressive motion in the direction of propagation of the waves, 

 and consequently in the direction of the wind, supposing it not to have recently shifted, and this 

 proo-ressive motion will decrease rapidly as the depth of the particle considered increases. If the 

 pressure of the air on the posterior parts of the waves is greater than on the anterior parts, 

 in consequence of the wind, as unquestionably it must be, it is easy to see that some such pro- 

 o-ressive motion must be produced. If then the waves are not breaking, it is probable that equation 

 (23), which is applicable to deep water, may give approximately the mean horizontal velocity 

 of the particles ; but it is difficult to say how far the result may be modified by friction. If 

 then we regard a ship as a mere particle, in the first instance, for the sake of simplicity, and put 

 Uo for the value of U when y = 0, it is easy to see that after sailing for a time t, the ship 

 must be a distance UJ to the lee of her estimated place. It will not however be sufficient to 

 regard the ship as a mere particle, on account of the variation of the factor e'^"'", as y varies from 

 to the greatest depth of the ship below the surface of the water. Let ^ be this depth, or rather 

 a depth something less, in order to allow for the narrowing of the ship towards the keel, and suppose 

 the effect of the progressive motion of the water on the motion of the ship to be the same as 

 if the water were moving with a velocity the same at all depths, and equal to the mean value 

 of the velocity U from y = to y = S. If f/, be this mean velocity, 



1 /-s ma"c I 



On this supposition, if a ship be steered so as to sail in a direction making an angle with the 

 direction of the wind, supposing the water to have no current, and if V be the velocity with which 

 tlie ship moves through tlie water, her actual velocity will be the resultant of a velocity V in 

 the direction just mentioned, which, for shortness, I shall call the direction of steering, and of 

 a velocity f/, in the direction of the wind. But the ship's velocity as estimated by the log-line 

 is her velocity relatively to the water at the surface, and is therefore the resultant of a velocity V in 

 the direction of steering, and a velocity U„ — f/, in a direction opposite to that in which the wind 

 is blowing. If then E be the estimated velocity, and if we neglect {/^ 



E= V - (Uo- U,)cose. 



But the ship's velocity is really the resultant of a velocity V + U, cos 6 in the direction of steering, 

 and a velocity U^ sin 6 in the perpendicular direction, while her estimated velocity is E in the 

 direction of steering. Hence, after a time t, the ship will be a distance f/,,/ cos 6 ahead of 

 her estimated place, and a distance t/^, ^sinfl aside of it, the latter distance being measured in a 

 direction perpendicular to the direction of steering, and on the side towards which the wind is 

 blowing. 



I do not suppose that the preceding formula can be employed in practice ; but I think it 

 may not be altogether useless to call attention to the importance of having regard to the magnitude 

 and direction of propagation of the waves, as well as to the wind, in making the allowance for 

 lee-way. 



11. The formul;e of Art. 6 are perfectly general as regards the ratio of the length of tlie waves 

 to the depth of the fluid, the only restriction being that the height of the waves must be sufficiently 

 small to allow the series to be rapidly convergent. Consequently, they must apply to the limiting 

 case, in which the waves are supposed to be extremely long. Hence long waves, of the kind 

 considered, are propagated without change of form, and the velocity of propagation is independent 

 of the height of the waves to a second approximation. These conclusions might seem, at first sight, 



