Bessel- Clifford Function, and its applications. 511 



Tidal waves in an estuary. 



10. Consider too the theory of a long flat tidal wave in 

 an estuary or channel where the cross section A and the 

 surface breadth b are treated as slowly variable with 

 the length x. 



The equation of continuity becomes, for a horizontal 

 displacement £ and vertical elevation rj of a liquid particle, 



and the dynamical equation, on the usual theory that the 

 pressure head is the depth below the free snriace, is 



d 2 % _ dij _ d ( 1 dAf 



gdt 2 



d 2 V ^ _\d L ( 



gdt 2 b dx 



These become for the wave, synchronizing with the beat 

 of an equivalent pendulum of length /, and 



-JaW = ~T' • ; • • • ( 4 ) 



dx~ dx\b dx J' K [ 



\ dt 2 J b dx \ dx) K ; 



reducible at once to the Bessel- Clifford equation when 

 A and b are taken to vary each as some power of x, 

 say x q and x v \ leading to differential equations for r/ and f 

 of the form of (9) and (11) § 9, with a simple solution by 

 the Clifford function. 



In a V-shaped estuary, of uniform depth h feet, put 

 <? = !, m = 1 ; then 



I / dr)\ XT] - d tldxP\ f /V lM 



x\ dx) lh dx \x dx ' I 



and with y = 42\ %=x£, 



d 2 r) dri /v d 2 t (It u ., _ x 



^-T ■+- r / + ^ = 0, £ _>4--2/ + ?=0, • . 7) 

 dz- dz dz* dz 



v = bC '>(Jj],) =l '- h ' v \ii l )- f = "' e 'u'//, or Ji t5a)- ,8) 



Denoting by T the complete period of the wave, 



