DISPERSION RELATIONS

In linearized approximations for water waves over a flat bottom at z = 0, the pressure equation is Dp = 0. For a two-dimensional wave where



pxx + pzz = 0 (1)


and
 pH = 0 at z = H(x,t) = h + z sin(kx-wt), (2)

the pressure is

p = p0 + p1z + (p2cosh kz + p3sinh kz) sin(kx-wt) . (3)


Then

 wB = B t = 0 ® pzB + g = 0 ® p3 = 0 and p1 = -g , (4)



pH = 0 ® p0 = gh and gz = p2cosh kh , (5)


and

 wH = H t ® H t t + pzH + g = 0 ® w2z = kp2sinh kh . (6)

Thus Airy's dispersion relation, w2 = gk tanh kh, follows from what appears to be very little information about the flow. This kind of derivation of the result has been around since Poincaré's work or before, but it is seldom seen, except in treatments of rotating flows, where the ever-popular assumption of potential flow is not an option.