Ursell (1953)'*^ devised the following criterion, the Ursell number U, 

 for applying the Boussinesq equation or special cases with terms omitted. 

 When 



>>1 Nonlinear long wave equations 



U { 6(1) Boussinesq equations (136) 



<<1 Linear wave equations 



where 

 U 



(wave amplitude/depth ratio) ^ max _ max ,, „^. 



(wave steepness)^ (n /L)^ d^ 



max 



The region where U is comparable to unity is often cited as applicable to 

 cnoidal wave theory. 



No waves of permanent form can exist on a sloping beach. By comparison 

 with experiments, Madsen and Mei (1969)'*^ concluded that the Boussinesq 

 equations given by equations (133) and (134) give reasonable shoaling solu- 

 tions for H/d up to about 0.5. Waves break when H/d - 0.8-1.0. As discussed 

 further below, even this limit is approachable when the Boussinesq equations 

 are expanded to include sloping bottom terms and more accurate numerical 

 solution techniques are employed. 



b. Derivation of Boussinesq Equations . The Boussinesq equations are 

 normally derived (Peregrine, 1972) by expanding all dependent variables as 

 polynomials in terms of scaling parameters 



e = ^ (138) 



d 



a = I (139) 



with e = on the order of (6) a^^ substitution in Euler's equation of motion 

 for inviscid flow, and integration over the vertical water column. Retaining 

 all terms up to order a^ (or e) results in equation (13A) for a horizontal 

 bottom. 



'* '^URSELL, F., "The Long-Wave Paradox in the Theory of Gravity Waves," 



Proceedings of Cambridge Philosophical Society, Vol. 39, London, 1953, 



pp. 685-694 (not in bibliography). 

 '^^MADSEN, O.S., and MEI, C.C, "The Transformation of a Solitary Wave Over an 



an Uneven Bottom," Journal of Fluid Mechanics, Vol. 39, Pt. 4, 1969, 



pp. 781-791 (not in bibliography) . 



147 



