Does Point A Oscillate in Simple Harmonic Motion with a Wave?

[smithj1990]

you have a transverse sinusoidal wave on a string, and say there's a point A on that string.
does point A oscillate in simple harmonic motion with a frequency that's equal to the frequency of the wave?

 

[Chi Meson]

That is a good question. What do you think?

(Please review forum guidelines. We are not here to do your homework.)

What are the characteristics of simple harmonic motion?

 

[smithj1990]

sorry, its not a homework question. I am simply curious if it is true. I am pretty positive the point would be in SHM. i just don't know how to justify that its SHM frequency of the point would be the frequency of the wave..
im not really sure what equation would be useful here.

 

[Chi Meson]

Well it is true. If the wave moves at a continuous speed, and the y-position of the wave follows a sine function of the x-position, then each particle of the medium will need to moved transversely as a sine function of time. Every point on the wave will need to be at maximum amplitude at exactly the moment that the crest of the wave passes by that point, for every cycle. Therefore, both must have the same frequency.

Since the nature of SHM is sinusoidal, by definition any back-and-forth oscillation that is sinusoidal with time will be SHM.

 

[sungod]

If the transverse wave is moving assuming it's not a standing wave - wouldn't really matter if it was, all that happens is that the amplitude is now a function of x - the wave obeys:
y = A*sin(wt $$\pm$$ kx)

Choosing the origin, it doesn't matter where you look along the wave, the wave equation is now:
y = A*sin(wt)

In other words, a sinusoidal variation with the same frequency as the traveling wave. If you looked at another point along the wave, it'd carry out the same sinusoidal variation, but phase shifted.

If the phase at some x1 and x2 is
phi_1 = kx1 - wt, phi_2 = kx2 - wt,
the phase difference is phi_1 - phi_2 is k(x1 - x2)