Standing Waves: Exploring Asin(kx)sin(wt) and Interfering Waves

[erisedk]

Homework Statement

Why does Asin(kx)sin(wt) also represent a standing wave? Which two interfering waves may superpose to make it?
Acos(kx+wt) and Acos(kx-wt) could if we were subtracting them, but we're adding so that doesn't make much sense? Also, is there something like a phase shift in the time term?

Homework Equations

The Attempt at a Solution

 

[Quantum Defect]

Homework Statement

Why does Asin(kx)sin(wt) also represent a standing wave? Which two interfering waves may superpose to make it?
Acos(kx+wt) and Acos(kx-wt) could if we were subtracting them, but we're adding so that doesn't make much sense? Also, is there something like a phase shift in the time term?

Homework Equations

The Attempt at a Solution

If you think about situations where you see a standing wave, what kinds of waves are you dealing with?

Ex: Waves in a jump rope --> you wiggling one end of the rope, the other end tied to the wall. Think about what happens when you launch a single impulse down the rope.

 

[BvU]

Remember the rules for $\cos(\alpha+\beta)$ ? Those are the ones to write the standing waves (a product) as propagating waves (sums).

[edit] You used those already, I realize.

If a sign is in the way, you can always add a phase to change it ... something to do with $\cos(\alpha+\pi)$

 

[erisedk]

Remember the rules for $\cos(\alpha+\beta)$ ? Those are the ones to write the standing waves (a product) as propagating waves (sums).

[edit] You used those already, I realize.

If a sign is in the way, you can always add a phase to change it ... something to do with $\cos(\alpha+\pi)$

^^That's precisely the problem. If I change the phase, how will it still remain a standing wave?

 

[haruspex]

Not sure exactly what you're asking. You're looking at two general ways of writing standing waves, one as a trig product and one as a trig sum, but seem to be complaining that a specific example of one form does not turn into a specific example of the other form.
If you start with $\cos(kx+\omega t)+\cos(kx-\omega t)$ then in general it can be turned into $A\sin(kx+\alpha)\sin(\omega t+\beta)$. That will be a standing wave, regardless of the values of the three constants introduced. Specifically, you will get A = 2, $\alpha = \beta = \pi/2$ (or something like that).

 

[BvU]

^^That's precisely the problem. If I change the phase, how will it still remain a standing wave?

If you change the phase it will just stand in a shifted location ! One of the two waves that travel in opposite directions is shifted.

 

[erisedk]

Ok fine. That makes sense.