B Traveling wave solution notation

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The discussion focuses on the notation for traveling wave solutions to the wave equation, specifically the forms f(kx - ωt) and f(-kx + ωt) for right-traveling waves. It clarifies that both notations can represent right-traveling waves, but the choice depends on the evenness or oddness of the function f. A misunderstanding arose from an algebraic mistake in deriving the reflection and transmission equations, which initially led to incorrect conclusions about the wave behavior. Ultimately, both forms can yield the same physical interpretation for even functions, although they may differ in shape. The key takeaway is that notation preference can influence the representation but does not change the fundamental wave direction.
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Is there a difference between writing ## f(-kx + \omega*t) ## and ## f(kx - \omega*t) ## for right traveling waves?
This is probably kind of dumb, but it's really bothering me for some reason. I originally saw traveling wave solutions to the wave equation as ##f(kx−\omega t)## for right traveling (as t gets bigger, x needs to be bigger to "match" it's previous value) and ##f(kx+\omega t)## for left-traveling waves. And that all made sense to me. Then I saw some people writing ##f(−kx+\omega t)## for right-travelling waves. I'm pretty sure it's the same thing right? Like this also says that as t gets larger x needs to be larger to match it's original value? Is it just a notation preference when choosing between the two?

My confusion sort of stemmed from seeing the derivation of the reflection and transmittance for a traveling wave at a boundary:

IMG_20210326_104726998.jpg


where ##f_i## is the incident pulse, ##f_r## is the reflected, and ##f_t## is the transmitted.

The derivation for the reflectance and transmittance is like:

Because the wave function has to be continuous at the boundary, x = 0:
## f_i(\omega t) + f_r(\omega t) = f_t(\omega t) ##

Because it's space derivative has to be continuous at x = 0 so it doesn't have infinite acceleration:
## -k_1 f'_i(\omega t) + k_1 f'_r(\omega t) = -k_2 f'_t(\omega t) ##

And then integrating both sides of the second equation with respect to t
## \frac {-k_1} {\omega} f_i(\omega t) + \frac {k_1} {\omega} f_i(\omega t) = \frac {-k_2} {\omega} f_t(\omega t) ##

And then by the dispersion relation ## \omega = v*k ## so ## \frac {k} {\omega} = \frac {1} {v} ## ...
## \frac {-1} {v_1} f_i(\omega t) + \frac {1} {v_1} f_r(\omega t) = \frac {-1} {v_2} f_t(\omega t) ##

which gives ## -v_2 (f_i(\omega t) - f_r(\omega t)) = -v_1 (f_t(\omega t)) ##

and then we can solve this equation and the continuity of f and get

## f_r(\omega t) = \frac {v_2 - v_1} {v_1 + v_2} f_i(\omega t) = R f_i(\omega t) ##
## f_t(\omega t) = \frac {2v_2} {v_1 + v_2} f_i(\omega t) = T f_i(\omega t) ##

The problem is is that if I do this whole thing by saying the incident, right traveling wave is ##f_i(k_1x - \omega t)##, the reflected left traveling wave is ##f_r(k_1 + \omega t) ##, and the transmitted wave is ##f_t(k_2x - \omega t)##

I get these two equations to solve (using the same methods as above)

##f_i(\omega t) + f_r(\omega t) = f_t(\omega t)##
##v_2(f_i(\omega t) + f_r(\omega t)) = v_1 f_t(\omega t)##

which gives me ##f_r = \frac {v_2 - v_1} {v_1 - v_2} f_i = -f_i## and ##f_t = 0##.

So why is it necessary to write the right traveling waves as ##f_i(-kx + \omega t) ## and why does ##f_i(kx - \omega t) ## not work instead?
 

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baseballfan_ny said:
Summary:: Is there a difference between writing ## f(-kx + \omega*t) ## and ## f(kx - \omega*t) ## for right traveling waves?
It depends if ##f## is even or not. If ##f## is even then there is no difference. Otherwise there is a difference.
 
Dale said:
It depends if ##f## is even or not. If ##f## is even then there is no difference. Otherwise there is a difference.

Ok. I'm not sure I 100% understand the difference. I drew some odd waves and they still seem to shift right with both forms.

The wave I was dealing with in post 1 is even, so I based on what you wrote I would expect both forms to give the same R and T. Turns out I made a stupid algebra mistake in this line of post 1:

baseballfan_ny said:
fi(ωt)+fr(ωt)=ft(ωt)

The ##f_i## and ##f_t## should have a negative argument, and then I follow through with the rest of the algebra and get the same answer -- as they're both even.
 
baseballfan_ny said:
I drew some odd waves and they still seem to shift right with both forms.
Yes, they will still shift right, but they will have a different shape with the two forms.
 
baseballfan_ny said:
Summary:: Is there a difference between writing ## f(-kx + \omega*t) ## and ## f(kx - \omega*t) ## for right traveling waves?
They both represent a wave traveling to the right. Obviously, for a given wave, you'd use different ##f##s in the two cases.
 
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