On the deformation of periodic long waves over a gently sloping bottom

Two-dimensional time-periodic water waves on a gently sloping bottom are investigated under the classical long-wave assumptions that ε = h ′ 0 /λ′ and δ = H ′/ h ′ 0 are small parameters ( H ′ being the wave height, h ′ 0 a characteristic water depth and δ′ the horizontal scale for the oscillatory motion) and the assumption that δ/ε 2 = O (1) as δ and ε tend to zero. It is shown that for a bottom slope h x for which h x = o (ε 3 ) the governing KdV equation with slowly varying coefficients (derived by Johnson 1973 a ) has a time-periodic solution which in the first approximation is a slowly varying cnoidal wave. The second-order approximation in an asymptotic expansion with respect to the bottom slope represents the deformation of this wave due to the sloping bottom. For h x [Gt ] ε 5 this deformation is larger than the second-order contribution from the basic expansion with respect to wave amplitude which underlies the KdV equation itself; the calculation is then a consistent approximation to physical reality. Numerical results for the deformation are given. Also, the wave profiles are compared with experiments on a plane of slope hₓ = 1/35 and show good agreement even for the large values of H’/h’ appropriate to waves rather close to breaking.