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...

 

[asteriatic]

I would approach this problem by first understanding the concept of wave propagation and how the depth of water affects it. In this case, the waves are sinusoidal, meaning they have a constant frequency and wavelength. However, as the depth of water decreases near the shore, the velocity of the waves will also decrease due to the change in water depth. This will result in a decrease in wavelength near the shore.

To find the wavelength near the shore, we can use the equation v=λf, where v is the velocity, λ is the wavelength, and f is the frequency. We are given the velocity at a depth of 5 m and the corresponding wavelength. Using this information, we can calculate the frequency at 5 m depth.

Next, we can use the equation v=\sqrt{gh} to calculate the velocity at a depth of 0.5 m. Since the frequency remains constant, we can use this new velocity and the previously calculated frequency to find the wavelength near the shore using the equation v=λf. This will give us the separation between wave crests near the shore.

However, it is important to note that this is a simplified approach and does not take into account factors such as wave refraction and shoaling, which can also affect the separation of wave crests near the shore. These factors may need to be considered for a more accurate calculation.