Solving for Wavelength of a Surface Water Wave

[deimos]

If you have a surface water wave (shallow) with equation c = sqrt((gλ / 2 pi) * tanh (2 pi d / λ))
I was wondering how one would go about determining the wavelength. I though it was one continuous wave. Say the wave tank it was in was 60cm long, but it was reflected once (traveled 1.2m). Would the wavelength be 0.6m or 1.2m.. and why?

 

[arildno]

1. First, the equation of the phase velocity c that you have given is correct, to linear order, for all sinusoidal waves where surface tension has been neglected, irrespective of a shallow water condition.
2. It seems you are a bit uncertain of the meaning of wavelength; the wavelength is the length between to adjacent wave tops.
The wave length concept does not depend on the length of the tank; it is one of the intrinsic properties of the wave itself.
3. Note that I used the phrase "sinusoidal wave". A wave signal that looks like a sine/cosine function has the important property that all the wavelengths it has (distances between successive tops) are equal.
For a general wavesignal, the wavelengths will usually vary spatially and in time, there is no single number that can be called the signal's wavelength.
4.Any general signal can, however, be thought of composed of a multitude of constituent sine waves, each of these differing with its own,unique wavelength.
Decomposing and analyzing general wave signals in terms of elementary sine waves is part of what is called Fourier analysis.

5.An important difference between a single sine wave signal and a signal composed of many such, is signal dispersion, that the signal may warp over time becoming unrecognizable from how it looked to begin with.
Suppose you have a wave that can adequately be described by a function A*sin(k*x-c*t). Here k is the inverse wavelength, wavenumber, while the phase velocity c is given by your formula.
How does this signal look like?
First, you can see that a specific point, or value, on the wave propagates with velocity c in the right direction.
Secondly, you see that the wave signal has the same form at all times; the wave signal retains the form it had at t=0, f.ex., and moves steadily along.

Consider now a signal A1*sin(k1*x-c1*t)+A2*sin(k2*x-c2*t):
Here A1, A2, are amplitudes, while you get c1 by plugging in k1 in your formula, and analogously for c2.
How does this signal change in time?
Since c1 and c2 in general are different, the wave changes form over time.
This phenomenon is called dispersion; in this particular case, wavelength dispersion.

6. In the shallow water case, i.e. when the ratio d/(lambda) is small, you find that
the phase velocity can written as c=sqrt(gd).
Since this phase velocity is independent of wavelength, the phenomenon of wavelength dispersion will not occur.

 

[W.A. McCormick]

Waves

So far I have been unable to start any new thread, but this looks like a good place to ask my question.

Tidal waves are almost imperceptible when over the deep ocean, but they pile up to fabulous heights when they run up on and inclining ocean floor. The typical description is that the wave "drags it's feet" and piles up. My question is this: What happens when a tidal wave hits a vertical cliff such as can be seen in some places in Hawaii if this cliff extends to the depth of the ocean at that location? Does the wave dissipate without being noticed, or does it suddenly explode with horrendous force and without warning. What is the effect of the particular degree of incline in the ocean floor?

Thanks. You wouldn't imagine how many people I have contacted about this and have gotten absolutely no answer.

W.A. McCormick

 

[kurious]

The tidal wave gets taller if it passes through a channel that gets narrower and narrower.If it hits a vertical cliff it will rise high because reflected water meets
more incoming water at high speed and causes a surge.

 

[deimos]

I see, well that’s essentially what I was wondering, how can you measure its wavelength, if it was only one continuous wave?

-can this be found using the function A*sin(k*x-c*t) if so let me just clarify the terms, is it amplitude, K(1/wavelength), wavenumber(assuming this is one), and phase velocity c, and time.

 

[W.A. McCormick]

Thanks Kurious.

W.A. McCormick