Derivation of the wave dispersion equation

[freja]

My homework question is:

If the ocean surface is disturbed by a wave, η=η_0 cos(ωt-kx), show that the dispersion equation is given by ω^2=gHk^2+f^2.

I have looked every where and while there is a lot of sites with this on they tend to just jump from one to the other saying 'substitue and solve to get...' and don't show the steps in between. I can't find the steps in any of the recommended course books either. Could some one please help me understand how to move from one to the other?
Thank you.

 

[member 553083]

The dispersion equation is derived from the linearized shallow water equations, and can be derived using the following steps:1. Begin with the linearized form of the shallow water equations:u_t + gu_x = -fv_xv_t + gv_x = fu_x2. Substitute the wave solution for u and v into the equations:η_t + gu_x = -fv_x-η_0ωcos(ωt-kx) + gu_x = -f(-η_0ωsin(ωt-kx))3. Simplify the equations by multiplying both sides by a factor of 2 and collecting like terms:2η_0ωcos(ωt-kx) +2gu_x = 2fη_0ωsin(ωt-kx)4. Solve for u_x:u_x = η_0ω[sin(ωt-kx)/(2g) - cos(ωt-kx)/(2f)]5. Apply the chain rule to find d/dt of u_x:(d/dt)u_x = η_0ω[cos(ωt-kx)(-ω)/(2g) - sin(ωt-kx)(-ω)/(2f)]6. Rearrange the equation to solve for ω^2:ω^2 = (2g/η_0)[cos(ωt-kx)/sin(ωt-kx)] + (2f/η_0)7. Simplify the equation by replacing cos(ωt-kx)/sin(ωt-kx) with its tangent form:ω^2 = gHk^2 + f^2 where H is the wave height.