202 M. K. Zoppritz on some lli/drodynamie Problems 



better to introduce into the problem the progressive motion of 

 the superficial stratum of the ocean (which, indeed, can every- 

 where be easily determined by observation) as a given quantity, 

 as a known function of the time (eventually as a constant). 

 Thereby a much simpler problem is obtained, which can be 

 mathematically deduced from the foregoing, if in this we put 

 p = co . The surface-condition then reads, for x = 0, w=(f>(t); 

 and the transcendental equation is transformed into smmh = 2 

 the roots of which are 



mr 

 h 



Therewith the general expression of equation (8) for p = cn 

 changes into the following : — 



2 °° 

 li 



w=-Ze ^ h ' sm— /(?)sm-^^ 



4- 



1 " «^o 



2 



^i(-l)-'nsin 5*?f V(X>-(t) 3 <'-^. (13) 



For the stationary state we get the same formula as before ; 

 but now io =w v 



In reference to the periodic motion in different depths the 

 same holds which was indicated above for the more general 

 expression. But if the following formula be applied to water 

 of infinite depth, therefore to h = x> , we still get the laws 

 already mentioned. If, namely, in the expression (11) we put 

 p = cc , the second part, above denoted by v, becomes 



v=. — I m sin mx dm \ 6(X)e~ m2a{t ~^d\. 

 17 Jo Jo 



The integral according to m can be completed ; for it is 



I m sin mxe~ ym2 dm = 7-1 cos mxe~ ym2 dm. 



Jo *Og 



But the latter integral is well known. It is, namely, 



cos mxe~ ym2 dm = - \ / — e *y. 

 Accordingly 



- go / — *a 



xVir - — 



I 



m sin mxe~ ym2 d?n = e *y, 



f o Arfi 



Consequently 



v= 



Jo (t-\)i V^J V 4«2 2 / 



2\/«7tJ («—%){ 



