Find Wave Crest Separation Near Shore

[AdkinsJr]

Homework Statement

I have sinusoidal waves traveling along the surface of the water. I'm given the relation $$v=\sqrt{gh}$$ where h is the depth of the water. I'm given that when the water is 5 m deep the wavelengt of the waves is 1.4 m. I"m asked how far apart the wave crests are near the shore.

Homework Equations

$$v=λf$$

The Attempt at a Solution

Basically the wave starts somewhere in the middle of the pond and travels towards the shore, I need to find the wavelength as it approaches the shore, near the shore the water depth is only .5 m, so depth is decreasing, but not continuously, just see the pic, the upper right hand corner represents the shore.

The velocity therefore declines two times before it reaches the shore. I don't know how to find the wavelength when it's at the .5 m depth.

I've attempted to use the bolded data point to relate the wavelength to the velocity:

$$λ=\frac{1}{1.4 Hz}\sqrt{gh}$$

But this relationship is almost certainly wrong because it assumes constant frequency.

 

[th4450]

May I ask why is the frequency not a constant?

 

[AdkinsJr]

May I ask why is the frequency not a constant?

I don't see any reason to assume that it is constant. The frequency is the inverse of the time between successive crests. If the speed of the waves increases it seems reasonable that the frequency and period could increase/decrease.

 

[AdkinsJr]

Apparently the frequency is a constant; my instructor discussed this problem in class earlier, so I know what to do now...