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Two general questions about wave functions

by fazio93
Tags: functions, wave
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fazio93
#1
Apr17-13, 04:11 PM
P: 3
In my Physics I class, we started learning about wave functions in the form:

y(x,t) = sin(kx ωt ∅) or y(x,t) = cos(kx ωt ∅)


1) I saw a question where the wave function was structured as:

y(x,t) = sin(ωt - kx + ∅)

and the answers for the direction of the wave was in the +x direction.

I thought I could rewrite the equation as y(x,t) = sin(-kx + ωt + ∅), meaning the direction is in the -x direction, as the symbol preceding the "ω" is positive. Obviously that was wrong, so how does it actually work?

--------

2) Also, just a random question I was wondering:

If the derivative with respect to t (holding x constant) of the equations above give you the speed of a particle in the wave, what does the derivative with respect to x give you?

Thanks
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stevendaryl
#2
Apr17-13, 05:59 PM
Sci Advisor
P: 2,210
Quote Quote by fazio93 View Post
In my Physics I class, we started learning about wave functions in the form:

y(x,t) = sin(kx ωt ∅) or y(x,t) = cos(kx ωt ∅)


1) I saw a question where the wave function was structured as:

y(x,t) = sin(ωt - kx + ∅)

and the answers for the direction of the wave was in the +x direction.

I thought I could rewrite the equation as y(x,t) = sin(-kx + ωt + ∅), meaning the direction is in the -x direction, as the symbol preceding the "ω" is positive. Obviously that was wrong, so how does it actually work?
Propagation velocity is related to [itex]k[/itex] and [itex]\omega[/itex] through:
[itex]v = \omega/k[/itex]. So if you switch the sign of both [itex]\omega[/itex] and [itex]k[/itex], the sign of the velocity remains the same.
Philip Wood
#3
Apr18-13, 08:41 AM
PF Gold
P: 963
A subsidiary question: I've always preferred the 't first' version, y = A sin (wt - kx + phi), which is so clearly an oscillation (wrt time), with a phase that lags further and further behind with distance travelled by the wave. Yet most writers seem to prefer the 'x first' version. Why is this?

fazio93
#4
Apr18-13, 08:54 AM
P: 3
Two general questions about wave functions

Quote Quote by stevendaryl View Post
Propagation velocity is related to [itex]k[/itex] and [itex]\omega[/itex] through:
[itex]v = \omega/k[/itex]. So if you switch the sign of both [itex]\omega[/itex] and [itex]k[/itex], the sign of the velocity remains the same.
oh, ok.
so basically if either the k or ω is negative that would make it +x direction, so:

y(x,t) = sin(ωt - kx + ∅) == y(x,t) = sin(kx - ωt + ∅)
stevendaryl
#5
Apr18-13, 09:20 AM
Sci Advisor
P: 2,210
Quote Quote by fazio93 View Post
oh, ok.
so basically if either the k or ω is negative that would make it +x direction, so:

y(x,t) = sin(ωt - kx + ∅) == y(x,t) = sin(kx - ωt + ∅)
Yeah, except that
[itex] sin(\omega t - k x + \Phi) = sin(k x - \omega t + \Phi')[/itex]
where [itex]\Phi' = \pi - \Phi[/itex].
fazio93
#6
Apr19-13, 05:35 PM
P: 3
ok, i got it.
thanks :)


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