Long Waves in Canals and Standing Waves in Closed Basins 151 



and separate the terms with co%a{t—q>) and s\na(t— cp), which by themselves must be zero, we 

 obtain the relations 



and 



(hb)' r] <p' + 2hbrj' (p' + hbr\ a (p" = . 



From the last equation follows hbrf <p' = constant = C; so that 



d l drj \ lb C-' \ 



sr^) + (r*sr:) < "" = - (VL30) 



. . dx 

 If the variable velocity, according to Kelland, were \'(gh), then q> = J —, ; then the second 



term at the left of the equation (VI. 30) will vanish provided that 



r? = prop b-^h- 1 '* . (VI.31) 



The equation (VI. 30) is therefore satisfied assuming that the first term can be neglected. But 

 (VI.31) is the law of amplitude by Green which has as a condition that 



d l dr\\ bo 2 



-r[hb^-\<—r h . (VI. 31a) 



dx \ dx J g 



11 1 1 lb' h' IlTtV 



It can be shown that - (b 'lb), - (b lb) 2 , and - (h jh), — (h jhf and <| — respecti- 



2 4 4 16 2 b h \A J 



vely. In order that, for instance 



- A' 2 <§ 64jr 2 , 



h 2 



dh/dx must be small compared to h\X. The same applies for the other terms i(b'lb), i(b'/b) 2 , 

 WlhX^ih'/h) and \{b'h'\bh) < (2tt/A) 2 respectively. (VI. 3 la) is equivalent to the condition that 

 /.(dbldx), and ?Adhjdx) are small compared with b and h respectively. In other words, it is assu- 

 med that the transverse dimensions of the canal vary only by small fractions of themselves within 

 the limits of a wave length. Therefore, it is not to be expected that the law of Green applies to 

 long waves, such as tidal waves and sea-quake waves, (see p. 237), whereas it can be used, for 

 instance, for shorter shallow water waves. 



It was customary to substitute the mean or average depth h for the exact 

 depth h is the equation c = } (g/h) (when the depth changes in the direction 

 of the propagation of the wave). Sometimes the average depth of a sea was 

 computed from the duration of travel of a long wave. The bad results 

 (see p. 239) obtained have shown that the use of the simple formula c = \{gh) 

 is not permissible. Du Boys (see Forel, 1895) has tried to avoid these dif- 

 ficulties in assuming for the travel time of a long wave 



(VI. 32) 



y[gh(x)]' 



Green's investigations have shown that this formula can only be used under 

 certain conditions, which are not satisfied for all long waves in nature. 

 Thorade (1926 a, b, p. 217), investigated the behaviour of a long wave, 



