Difference between a normal wave function and a standing-wave function

[Nikitin]

Hello! So as you all know the wave-function can be expressed as:

$$y(x,t) = A\cos(kx-\omega t)$$

However, this can be interpreted as both a standing and moving wave. So when do you interpret it as either of those? Are there any special conditions that should be written along with the wave function?

 

[Vanadium 50]

How can this be interpreted as a standing wave?

 

[jtbell]

For the difference between traveling waves and standing waves, see e.g. here:

http://www.physics.buffalo.edu/claw/Page15/ProjectCLAW-P15.html

This page uses sines instead of cosines, but the idea is the same in either case.

 

[Nikitin]

How can this be interpreted as a standing wave?

You could just use a identity to transform it into

$$A\cos(kx)\cos(\omega t)- A\sin(kx)\sin(\omega t)$$

Which is a sum of two clearly standing waves.

Am I wrong in saying that sum is a standing wave too? I have difficulties in graphing trig expressions in my mind.

jtbell: ahh, thanks I'll check it out now

 

[Nikitin]

OK after checking out jtbell's excellent website I see that I am an idiot for thinking that the wave function can be used to model a standing wave.

But still, what's up with a moving wave being the sum of two standing waves? Is this a random math-quirk or is there some physics behind this?

EDIT: Also, isn't it a bit weird that a standing-sound wave in a pipe open-ended on both sides is able to survive and stay standing for a long time? You'd think the two waves making up the standing wave would just dissipate out the ends the moment they reach them, but for some reason the waves get reflected back?

 

[jtbell]

You could just use a identity to transform it into

$$A\cos(kx)\cos(\omega t)- A\sin(kx)\sin(\omega t)$$

Which is a sum of two clearly standing waves.

Yes, you can decompose a traveling wave into a sum of two standing waves. You can also decompose a standing wave into a sum of two traveling waves, which has a natural physical interpretation in terms of a single wave "bouncing back and forth" between the fixed ends of the string (or whatever the medium is).

 

[AlephZero]

At risk of stating the obvious, your OP said "a (one) standing wave" which is not quite the same as "the sum of TWO standing waves".

If you want to make this more fun, think about a vibration pattern in a rotating disk, when the pattern and the disk are rotating at a different angular velocities: (1) in a coordinate system fixed to the earth, (2) in a system fixed to the disk.

(Hint: the frequency you "see" looking from the Earth is not the same as the frequency you see (or feel?) if you are sitting on the disk. This is similar to the Doppler effect.)