158 Dr. E. A. Houstoun on the Damping of 



Differentiating (11) with respect to m and integrating with 

 respect to z from to h, neglecting the term divided by k, 

 we find 



f P dz = XB qr ^e-*H\npt cos ?™ sin ^cosh kh. 

 Jo ^ x a a b 



Integrating this with regard to t and at the same time 

 neglecting the variation of e~ Kt with t, we obtain by means 

 ■of (4) 



^XB^e-"* cos ptcos^sin^ cosh kh. . (19) 



pa 



° B r ,/ — coshkh=A Qr , 



1 pa 



and expressing the initial condition, we obtain 



2/A_ cos - — sin -V- • 



<*-!)- 



If we calculate the different values of A qr , we find that the 

 most important ones are as follows : — 



. _ 16ac . 16ac A 16ac 



An — s~> A-31— o5Tl) A 51 — 



3 2 tt 3 ' 51 5V 3 ' 



lQac . 16ac A 16ac 



Al3 ~"""37?' A33 ~""373V' A53 ~~375V' 



A 16ac . 16ac . 16ac 



Ai5_- 5t?'' A35__ 5T3V' ^-"STSV 5 - 



The values with an even suffix vanish. 



For the different terms of (19) g/p is constant and cosh hli 

 increases rapidly with q ; hence ~Bq r must converge more 

 rapidly than Aqr. For a point near the centre of the trough, 

 the terms for which q = l in the expression for u will be at the 

 very least nine times as great as the terms for which ^=3. 

 Hence the rate of damping of the vibrations there will be 

 sensibly that of the first term. 



§ 3. The trough used for the experiments had a slate 

 bottom and plate-glass sides and ends. It was 152*4: cms. 

 long and 20*3 cms. broad inside and the sides were 15'9 cms. 

 high. The waves were excited by raising one end of the 

 trough about 1 cm. by means of a lever. Then the lever was 

 held by a catch until the surface was perfectly still. When 



