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



As connected with this subject, I have also considered the motion of oscillatory waves propagated 

 alone the common surface of two liquids, of which one rests on the other, or along the upper 

 surface of the upper liquid. In this investigation there is no object in going beyond a first 

 approximation. When the specific gravities of the two fluids are nearly equal, the waves at their 

 common surface are propagated so slowly that there is time to observe the motions of the individual 

 particles. The second case affords a means of comparing with theory the velocity of propagation of 

 oscillatory waves in extremely shallow water. For by pouring a little water on the top of the mercury 

 in a trough we can easily procure a sheet of water of a small, and strictly uniform depth, a depth, 

 too, which can be measured with great accuracy by means of the area of the surface and the quantity 

 of water poured in. Of course, the common formula for the velocity of propagation will not apply 

 to this case, since the motion of the mercury must be taken into account. 



1. In the investigations which immediately follow, the fluid is supposed to be homogeneous 

 and incompressible, and its depth uniform. The inertia of the air, and the pressure due to 

 a column of air whose height is comparable with that of the waves are also neglected, so that 

 the pressure at the upper surface of the fluid may be supposed to be zero, provided we afterwards 

 add the atmospheric pressure to the pressure so determined. The waves which it is proposed to 

 investigate are those for which the motion is in two dimensions, and which are propagated with 

 a constant velocity, and without change of form. It will also be supposed that the waves are 

 such as admit of being excited, independently of friction, in a fluid which was previously at rest. 

 It is by these characters of the waves that the problem will be rendered determinate, and not by 

 the initial disturbance of the fluid, supposed to be given. The common theory of fluid motion, 

 in which the pressure is supposed equal in all directions, will also be employed. 



Let the fluid be referred to the rectangular axes of x, y, z, tlie plane xx being horizontal, 

 and coinciding with the surface of the fluid when in equilibrium, the axis of y being directed 

 downwards, and that of ,■» taken in the direction of propagation of the waves, so that the ex- 

 pressions for the pressure, &c. do not contain z. Let p be the pressure, fj the density, t the 

 time, M, V the resolved parts of the velocity in the directions of the axes of x, y ; g the force of 

 gravity h the depth of the fluid when in equilibrium. From the character of the waves which 

 was mentioned last, it follows by a known theorem that iid.v + vdy is an exact differential dtp. 

 The equations by which the motion is to be determined are well known. They are 



^-^^'-/i-mf^it)'] <■" 



a^^.o, P); 



d.v" dy'' 



—L. - 0, when y - h, (3); 



dy 



'^ + ^^+^^ = 0, whenp = 0, (4); 



dt d.v dx dy dy 



wliere 



(3) expresses the condition that the particles in contact with the rigid plane on which the 

 fluid rests remain in contact with it, and (4) expresses the condition that the same surface of 

 particles continues to be the free surface throughout the motion, or, in other words, that there is 

 no generation or destruction of fluid at the free surface. 



