XXXIII. On the Theory of Oscillatory Waves. By G. G. Stokes, M.A., 



Fellow of Pembroke College. 



[Read March 1, 1847.] 



In the Report of the Fourteenth Meeting of the British Association for the Advancement of 

 Science it is stated by Mr. Russell, as a result of his experiments, that the velocity of propagation 

 of a series of oscillatory waves does not depend on the height of the waves'". A series of oscillatory 

 waves, such as that observed by Mr. Russell, does not exactly agree with what it is most convenient, 

 as regards theory, to take as the type of oscillatory waves. The extreme waves of such a series 

 partake in some measure of the character of solitary waves, and their height decreases as they 

 proceed. In fact it will presently appear that it is only an indefinite series of waves which 

 possesses the property of being propagated with a uniform velocity, and without change of form : 

 at least this is the case when the waves are such as can be propagated along the surface of a fluid 

 which was previously at rest. The middle waves, however, of a series such as that observed by 

 Mr. Russell agree very nearly with oscillatory waves of the standard form. Consequently, the 

 velocity of propagation determined by the observation of a number of waves, according to Mr. 

 Russell's method, must be very nearly the same as the velocity of propagation of a series of 

 oscillatory waves of the standard form, and whose length is equal to the mean length of the waves 

 observed, which are supposed to differ from each other but slightly in length. 



On this account I was induced to investigate the motion of oscillatory waves of the above form 

 to a second approximation, that is, supposing the height of the waves finite, though small. I find 

 that the expression for the velocity of propagation is independent of the height of the waves to a 

 second approximation. With respect to the form of the waves, the elevations are no longer similar 

 to the depressions, as is the case to a first approximation, but the elevations are narrower than the 

 hollows, and the height of the former exceeds the depth of the latter. This is in accordance with 

 Mr. Russell's remarks at page 448 of his first Reportf. I have proceeded to a third approximation 

 in the particular case in which the depth of the fluid is very great, so as to find in this case the 

 most important term, depending on the height of the waves, in the expression for the velocity of 

 propagation. Tiiis term gives an increase in the velocity of propagation depending on the square 

 of the ratio of the height of the waves to their length. 



There is one result of a second approximation which may possibly be of practical importance. 

 It appears that the forward motion of the particles is not altogether compensated by their backward 

 motion ; so that, in addition to their motion of oscillation, the particles have a progressive motion in 

 the direction of propagation of the waves. In the case in which the depth of the fluid is very great, 

 this progressive motion decreases rapidly as the depth of the particle considered increases. Now 

 when a ship at sea is overtaken by a storm, and the sky remains overcast, so as to prevent astro- 

 nomical observations, there is nothing to trust to for finding the ship's place but the dead reckoning. 

 Hut the estimated velocity and direction of motion of the ship are her velocity and direction of 

 motion relatively to the water. If then the whole of the water near the surface be moving in the 

 direction of the waves, it is evident that the ship's estimated place will be erroneous. If, however, 

 the velocity of the water can be expressed in terms of the lengtli and height of the waves, both 

 which can be observed approximately from the ship, the motion of the water can be allowed for in 

 the dead reckoning. 



Pa({c 3IHt (nolc), and page M». t /le/mrl.^ nj the llriliKli Associution. Vo\. vi. 



a 1, 2 



