Wave Motion: Examining Displacement of Particles

[chaoseverlasting]

Homework Statement

The displacement of particles in a string stretched in the x direction is represented by y. Which of the following expressions for y describe wave motion:

1: cos kx sin wt
2:$$k^2x^2-w^2t^2$$
3:$$cos^2(kx+wt)$$

Homework Equations

Equation of a progressive wave is of the form $$y=f(t-\frac{x}{v})$$

The Attempt at a Solution

The first equation represents a standing wave. The second obviously can't be it (its not of the form f(t-x/v) ). But I thought the third one would represent the equation of a progressive wave. It doesn't though. Why?

 

[christianjb]

x'=-x
wt+kx=w(t-k/w x')=w(t-x'/v)
(using v=w/k)

 

[chaoseverlasting]

Yeah. I got that. But it isn't the answer. The third equation doesn't represent a wave. Why?

 

[christianjb]

Why not?

cos^2 (wt-kx')=1/2+1/2cos(2[wt-kx'])
w'=2w
k'=2k

so the fn=A+Bcos(w't-k'x')

Looks like a sinusoidal wave to me.

What's wrong with using a standing wave? Isn't that what you'd get in a stretched string?

 

[chaoseverlasting]

Yeah, you get a standing wave cause of superpoosition in a string. The thing is, that only one answer is correct. Maybe the language points to a standing wave?

 

[christianjb]

You get a standing wave, because the string is stretched- so it must be held at both ends.

 

[chaoseverlasting]

If a string is stretched, it doesn't necessairily mean that you would have to get a standing wave.