506 PROFESSOR CHRYSTAL 



Availing ourselves of the principle of superposition, as heretofore, we can build up 

 the general solution now required by adding together the contributions to £ due to all 

 the different elements of the lake-surface, and all the different elements of time at each 

 element of surface. 



Setting aside in the meantime the initial seiche motion, and calculating merely the 

 part d?t, of £ due to the disturbance of pressure, corresponding to /*<%<> + d/u and 

 T<£<T + c/T, we see at once from (23) and (24) that 



d' 2 t, = h~Z(2v+ 1)Q',(«>)Q',,(/aK sin «„(T-0/(/x, T)dfidT. 



If, therefore, we suppose the disturbance to last from t = to t = T, we get, if tyT, 

 the following expression for the contribution to £ due to the disturbance of pressure : 



3£ = A2(2v+ l)Q',(w) d/*Q',(/*) dTn p sin n v (T - *)/(/* , T) . . . (31). 



This last equation will give the required disturbance in any particular case. We 

 have merely to give the proper determination to the function /(/ul , t) and carry out the 



two integrations. It may be noted that in general the integral dTn„ sin w„(T — t)f(iu., T) 



will be a function of m. 



If the pressure-disturbance have the form of a wave steady in shape and propagated 

 with a uniform velocity v, then instead of /(/*. , t) we may write the more specialised 

 function f{a{\ + w) — vt). 



10. Special Case of a Sudden Rise of Pressure 3p, propagated ivith uniform 

 velocity v, starting at the negative end of the lake at t = 0, and ceasing all over the 

 lake at t = T. — First suppose vT>2a, so that every point of the lake is sooner or later 

 affected. In this case, it is obvious that after T has reached the value 2a/v, the 

 disturbance contemplated has no longer any effect on the seiche motion, beyond an 

 increase of the pressure everywhere by the amount dp. We may therefore suppose 

 T = 2a /v, and our formulae will be applicable for t^2a/v. 



Since at any particular point w = fn the pressure is undisturbed until t = a(l +n)/v, 

 and thereafter is raised by dp until t = T, the determination oif(iu. , t) in this case is 



f(/ji, t) = for t<a(l + /x)/»; 

 f(fx , t) = dp for tya(l + fi)/v. 



Hence (31) gives 



8^ = ^ S(3v + 1)Q',( W ) dnQ' y (ji) dTn„ sin n v (V-t), 



J-l JaQ.+M/v 



= ^S(2v+l)Q',(w) d/*Q',(/*)[co8 n r {a(l +ii)/v-t}- cob n v (T-t)]. 



