What Can We Learn About Wave Speed from a Vertical Standing Wave?

[arkofnoah]

Homework Statement

A spring is held vertically between two supports separated vertically. That stationary wave set up looks like this:

http://img690.imageshack.us/img690/9417/80122335.jpg

Deduce what you can about the speed of the waves in the spring. (Note that the wave pattern is slightly "fatter" at the bottom than the top)

Homework Equations

The Attempt at a Solution

I have no idea where to start thinking. The ovoid shape definitely means something. I know that the frequency must be the same but what about the wavelength for such asymmetrical oscillation?

 

[Stonebridge]

You know that the speed of the wave in the spring depends on the tension in it.
The spring clearly has mass.
The tension in the spring is not constant/uniform all the way down.

Think:
How does the tension in the spring vary from top to bottom?
How does the speed of the wave vary?

A wave can still have a wavelength even if it is not sinusoidal.

Does this help you get started?

 

[arkofnoah]

oh right. i got it. it's basically the v=\sqrt{\frac{T}{\mu}} (mu is the mass per length) thing right?

is there any intuitive reason why the wavelength of waves decreases down the spring?

 

[Stonebridge]

oh right. i got it. it's basically the v=\sqrt{\frac{T}{\mu}} (mu is the mass per length) thing right?

is there any intuitive reason why the wavelength of waves decreases down the spring?

Yes the speed of the wave depends on the tension and you have the formula.
The next step is to think about the change in the tension in the spring from top to bottom.
If, as you say, the wavelength is getting smaller down the string, what does this say about the speed, given v=frequency x wavelength. (The frequency is a constant)
Do you think the tension in the spring is greater at the top or the bottom?
Your guess at the change in the speed should be consistent with the change in the tension.
You are on the right track.

 

[arkofnoah]

ok i got it. thanks :D