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.

 

[Nate810]

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.

 

[ogb p]

I understand the frustration of not being able to find clear and detailed steps for a derivation. So, let me walk you through the derivation of the wave dispersion equation from the given wave equation.

First, let's start with the wave equation: η=η_0 cos(ωt-kx).

This equation represents a wave on the ocean surface where η is the displacement of the surface from its equilibrium position, η_0 is the amplitude of the wave, ω is the angular frequency, t is time, k is the wavenumber, and x is the horizontal distance.

To derive the dispersion equation, we need to use the following equation for the wave speed: c=ω/k.

This equation relates the angular frequency (ω) to the wavenumber (k) and represents the speed at which the wave is propagating.

Now, let's differentiate the wave equation with respect to time (t):

dη/dt = -ωη_0 sin(ωt-kx)

Next, we differentiate the wave equation with respect to horizontal distance (x):

dη/dx = kη_0 sin(ωt-kx)

Now, using the wave speed equation (c=ω/k), we can rewrite these derivatives as:

dη/dt = -c sin(ωt-kx)

dη/dx = c sin(ωt-kx)

Next, we can substitute these derivatives into the wave equation and solve for ω^2:

ω^2 = (dη/dt)^2 + (dη/dx)^2

ω^2 = (-c sin(ωt-kx))^2 + (c sin(ωt-kx))^2

ω^2 = c^2(sin^2(ωt-kx) + cos^2(ωt-kx))

ω^2 = c^2

Therefore, ω^2 = gHk^2+f^2, where g is the acceleration due to gravity and H is the water depth.

This is the dispersion equation, which relates the angular frequency (ω) to the wavenumber (k). It shows that the wave speed (c) is dependent on both the water depth (H) and the Coriolis parameter (f).

I hope this explanation helps you understand the steps involved in deriving the wave dispersion equation. If you have any further questions, please do not hesitate to ask.