In terms of the scaling parameters it can readily be shown that the 

 normalized Boussinesq equations for horizontal bottom become (Whitham, 1974)- 



A 



Mass 4^ + |- [(1 + en*)u*] + e(ea^,a'') = (140) 



dt oX 



* 



Motion 



+ eu* -— — I- —' — a"^ + eCea-^jO^) = (141) 



9t 9x 9x 3 g^23t 



where the star indicates normalized terms and all higher order terms of 

 order eo^ , o , z , etc. have been neglected. The Ursell parameter simply 

 becomes 



-. _ _e Importance of nonlinear term (^L'}\ 



CT^ Importance of Boussinesq term 



so that when U - 1, the relative importance of both dispersion terms in 

 equation (141) is apparent. 



Other forms for the Boussinesq term have appeared in the literature 

 (e.g., Abbott, 1979) and result from using the linearized mass equation to 

 transform the term into a mixed derivative in h. 



c. Boussinesq-Type Equations for Variable Water Depth . The derivation 

 of Boussinesq-type equations over variable depths remains an active research 

 area. The derivation by Peregrine (1967) and summarized in Peregrine (1972) 

 is for propagation of arbitrary long waves over slowly-varying bathymetry. 

 A new length-scale parameter associated with depth variations is needed and 

 the local bottom slope, B is used. Peregrine assumed a ^ B to be consistent 

 with the order of approximation for surface boundary conditions. He obtained 

 for the one-dimensional motion equation (Peregrine, 1972) 



9u 9u 9h 1 9 r 9 ^ j9u, , 1 or 9 /-^ /O'Jwi 



?F + " "9^ + g ^ = 2 '^ -^^-^^^^"^ - 6^ fi^(^^-9l)>] 



or 



9u , 9u 9n 1 ,2 9^u_ . 1 J 9d,9u ^ "b^u , f^ i n\ 



T— +u— -+g-— = - d^ + T ^ ^"(tT + ^TT;:) (143) 



dt 9x ° 9x 3 „ 2^^ 2 9x 9t 9x9t 



dX dt 



which reduces to equation (134) for a horizontal bottom. Peregrine (1972) 

 states : 



^OlJHITHM, G.B., Linear and l^onlineaT leaves, Wiley Interscience, 



New York, 1974 (not in bibliography) . 

 ^ -^ ABBOTT, M.B., Co?72putationaZ_%draM lies J Pitman, London, 1979, p. 52. 

 ^^PEREGRINE, D.H. , "Long Waves on a Beach," Journal of Fluid Mechanics j 



Vol. 27, 1967, pp. 815-827 (not in bibliography). 



148 



