KUNGL. SV. VKT. A KADKM I KNS II ANDLI N(JAK. HAND 47. N:0 4. <J 



t hus: 



gk(a' sinh kh' I- /?' cosh M') = — c/ 2 («' cosh fcA' + /?' sinh kh 1 ) 

 or: 



5) ff'(grÄ;sinhÄ;/i' — « 2 coshA;Ä') + /?' (gk cosh k h' — o* sinh kh') = 0. 



Collecting our results, wc then have four linear equations to determine the four un- 

 known quantities a, [i, «', fi' in terms of the amplitudc /of the bottom corrugat.ion. The 

 determinant of that system is found to be (but for the factor sinh kh): 



D = a* {q coth kh coth k h' + o') — o 2 gko(coth kh + coth kh') + (o — o') g- k-; 



The solution fails for D = o. This of course just expresses the known fact that the roots 

 of D determine the wave-lengths of free waves of period " ' . ' 

 The quantities of interest for our purposes are />' and 



a 1 sinh kh' + [i' cosh kh', 



which latter expression has the same relation to the amplitude of the free-surface elevation 

 rjh> as /? has to the amplitude /?, a relation given by 3). Resolving we find: 



,_ yo 2 q(gk — a 2 coth kh') 

 sinh kh . D 



a' sinh kh' + /?' cosh kh' = § . Ö, 



where: 



sinh &A' (g k — a 2 coth &A') 



is the known expression 2 for the ratio of the amplitudes at the lower and upper 

 boundaries. 



Thus to the bottom-corrugation: 



y = — h + y cos kx 



will eorrespond the displacements of the common boundary and of the free surface: 



yakg(gk — a 2 coth &/*') int . , 

 ' i . sinh &A . D 



— c . o 3 koy t ■ i 



V-' = w '/o = • sinh kh sinh kh , D e l " ^nkx. 



Tlie discussion of & shows that for a small value of <>, and provided k be not too 

 small, the amplitude of /,/,- will be quite insensible compared to that of •/;„. A corrugation 



1 Lamb, loc. cit. Art. 231. 



2 Lamb, loc. cit. 



K. Sv. Vet. Akad. Handl. Iland 47. N:o 4. 



