10 ZEILON, ON TIDAL BOUNDARY-WAVES. 



of the bottom of not too great a wave-length will thus cause standing waves of that 

 length in the common boundary, whereas the free surface will not be sensibly disturbed. 

 Now, let 



y = — h + f(x) 



be the equation of the bottom, broken by a ridge at x = o; f(x) will then be a function, 

 positive and tending towards zero for great values of x. Developing according to 

 Fourier's theorem we at once from above obtain the corresponding displacements: 



00 + OD 



cooe iat i'k(gk — g- coth kh') 7 , C .... . , .. .,. 



\h> — 



Introducing: 



w s Qe iat f kdk /•" .... 



-£rj sinhM.sinhM' 7D i ™ sin W-*W- 



-13 



(,) I; (/. — x) g— ;'/.(/. —x) 



sin k (i — x) — - 



2 i 

 we at once transform these formula?, D being an even function of k: 



+ *> 



2tt J sinh kh.D J ![ ' 



I bis - 



+ 5» +V0 



ca 3 ge iat C kdk 



>,h> = + 



2?r 



'/sHmIf.»/""'"-**' 



Now, as a fact these integrals are indeterminate, owing to the integration contour 

 passing through the real roots of D. This is an ordinary feature of the solutions of this 

 kind of problems, and is physically due to the circumstance that no account has been 

 taken of possible free waves. As suggested by Lord Rayleigh in a similar case, the in- 

 determinateness will be avoided as soon as any frictional forces are introduced, tending 

 to annihilate the free waves. Mathematically this is equivalent to the addition to k of 

 a small imaginary quantity of a definite sign, tending towards zero after the execution 

 of the integration. 



To determine the sign of that imaginary it is, however, necessary that we should use 

 the primary form I, since a negative wave-length has no physical reality. Now we note 

 that the formulae I are composed by terms containing: 



é ;io.-x) ande-**^-*', 



where k is essentially positive, and where for the moment the same may be assumed 

 for x. To affirm this thoroughly, we write: 



