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VII. On the Damping of Long Waves in a Rectangular- 

 Trough. By Robert A. Houstoun, Ph.D., D.Sc, 

 Lecturer on Physical Optics in the University of Glasgow*.. 



IN this paper an expression is derived for the rate of 

 damping of stationary long waves in a rectangular 

 trough, and the result is compared with the results of 

 experiment. 



§ 1. We shall begin by obtaining the differential equation 

 for long waves, taking viscosity into consideration. In the 

 ordinary treatment of long waves without friction (cf. Lamb's 

 4 Hydrodynamics,' p. 239) the horizontal velocity is the same 

 at all points in the same vertical ; the bottom is perfectly 

 smooth. We shall suppose that there is no slipping at the 

 bottom and that the velocity there is zero. 



Let the motion be in one horizontal dimension in a medium 

 of uniform depth h. Take the axis of x horizontal and in the 

 direction of motion and the axis of z vertically upwards. 

 Let the bottom of the medium be given by z = 0, and the 

 surface in its undisturbed state be given by z=h. 



The ordinate of the free surface corresponding to the 

 abscissa x at time t will be denoted by 7i + ?. Then, on the 

 usual assumption that the vertical acceleration of the fluid 

 particles may be neglected, p the pressure at any point (x, y, z) 

 is given by 



P=Po + 9pQ l +£-z)> C 1 ) 



where p is the external pressure and p the density of the 

 medium. Hence 



m7c=wm« • • (2) 



To obtain the equation of motion, consider the element 

 Bx By Bz. The difference of the pressure-thrusts on the ends- 

 is 



— ~Bx Bu Bz. 

 ?>x J 



The difference of the tangential stresses on the upper and, 

 lower faces is 



and on the sides is 



d 2? ' j s x 

 fi^-jBxbybz, 



"d 2 U cs cv o, 



* Communicated by Professor A. Gray, F.R.S. 



