Tsunami word problem (waves practice)

[cmorissette]

Homework Statement

A Tsunami has been generated as a plane wave 4000km off the shore of California. The sea surface displacement is found to be 3 meters and the wavelength of the tsunami is 100km. You can assume a depth of 4000m from the location of wave generation to shore.

a.) Calculate the speed of the wave (assume it is a deep water wave if h/lambda > 1/4 and it is a shallow wave if h/lambda < 1/20) Estimate the time it will take this wave to reach the shore.

b.) calculate the frequency and period of the tsunami

c.) As the wave approcahes the shore its depth decreases from 4000m to 10m. What is it's new wavelength? (the wave period is conserved as the depth changes)

d.) What is the new amplitude of the wave? (you must conserve energy flux)

Homework Equations

shallow water wave velocity Cp = √gh
frequency = Cp/λ
Period = 1/F

The Attempt at a Solution

Part a is pretty easy. IT is a shallow water wave because h/λ < 1/20. The wave velocity is [(g)(h)]^1/2 which equals ~198m/s. Given the offshore distance of 4000000m, it would take a time of (4E6m/197.99m/s)= 20203 s.

PArt b is also straight forward: The frequency is F= Cp/λ which equals (198m/s)/(100000m) = 505.1 Hz. The period T = 1/F or 1.97E-3 s

Part c is what I'm having trouble on. if the wavelength λ=Cp/F and the period is conserved, does that mean the frequency is also conserved? So does this simply become (198m/s)/505.1s = 0.39m? Or do I need to recalculate Cp with a depth of 10m?

Cp= [(g)(10m)]^1/2 = ~10m/s so then wavelength would be (10m/s)(505.1Hz)= 5051m?

Also, I'm not sure about part d. If I were to guess, I would use
E= (1/2)ρga^2
But I can't see how amplitude would change here, because ρ and g are constant!

Help!

 

[Delphi51]

Better check the calc for frequency in (b); looks like you inverted it or something.
Yes, the speed changes as the water gets shallow.
The amplitude is related to the power and energy of the wave. It says energy is conserved, so you must have the same power in the shallow wave as in the ocean wave. If you don't have a formula for water wave power, you might check
http://en.wikipedia.org/wiki/Wave_power
The power depends on the period and amplitude, so if power stays constant amplitude will change when it goes shallow.