To make progress, we linearise the system by assuming the amplitude of the waves

Question

To make progress, we linearise the system by assuming the amplitude of the waves A is small compared to the depth H. That is, we set e = A/H and write the solution out as a perturbation series

?(x, y) = Ux + e ?1(x, y) + O(e2 ),
?(x) = H + e ?1(x) + O(e2 ).
By substituting these expressions into the governing equations, we derive the linear system
?2?1 = 0, 0 < y < H, (5)
??1/?y = U d?1/dx on y = H, (6)
U ??1/?x + g?1 = 0 on y = H, (7)
??1/?y = 0 on y = 0. (8)

Use a separation of variables argument to write ?1 = F(y)G(x) to derive general solutions for F and G. Assuming the solution has wavelength ?, apply the boundary conditions (7)-(8) to show that
? = Ux + g?A 2?U cos (2?x ?) cosh(2?y/?)/cosh(2?H/?) + O(e2 )
?(x) = H + A sin ( 2?x ? ) + O(e2 ).

Explanation / Answer

The question is not clear, as most of the places the sign " ? " is given.This does not give any idea.

if you provide a right one , then I will try to the best of my knowledge. Thnks.

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