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WickedSymphony
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- TL;DR Summary
- Why is a backwards traveling wave expressed as exp[i (kx + wt) ] instead of exp[ i ( -kx -wt ) ], since k is the term that carries directional information?
I've been having an issue with understanding the convention of wave direction notation, here is my current understanding where I am at currently:
A 3D harmonic solution to the differential wave equation can be given as:
If we make some assumptions about the wave, that its amplitude is 1, its traveling along the positive x-axis, and that the phase shift is 0, we get the 1D wave equation traveling along the x-axis.
Up until this point I feel as though my understanding is fine. However, when I try to describe a wave that's identical to this one except traveling in the opposite direction is where I get confused. From both some classes I've taken and things I've read, an often used convention for describing a wave traveling backwards is:
And to me this doesn't make sense. If I'm not mistaken (which I very well might be) the frequency of a wave isn't negative, its always greater than 0, and time is not running backwards. So I don't understand why the w*t term would change sign. In my mind since the propagation vector, k, is the term that carries direction, it should be what becomes negative in the expression. So I think the (or an) expression "should" be:
and
So there is clearly a difference between the convention I've been taught and read about, and the expression that, to me at the very least, seems like it should be the proper one.
One thing I've noticed, is that if you use euler's formula on both of these expressions, and then use the even and oddness of the cosine and sine functions you get:
and
Which have the same real part but different imaginary parts. Often times the real part of the wave is the only part that we care about, in which case the conventional method and "my" method actually agree. However, if I am not mistaken the imaginary part of the expression actually carries information about the phase of the wave, so may indicate that there's only a phase difference between the 2 expressions. That said, since they are both trying to describe the same wave, they should have the same phase.
If anyone could lend insight into this issue, as to where I may have gone wrong or to a convention issue or anything else I may have missed it would be much appreciated.
A 3D harmonic solution to the differential wave equation can be given as:
If we make some assumptions about the wave, that its amplitude is 1, its traveling along the positive x-axis, and that the phase shift is 0, we get the 1D wave equation traveling along the x-axis.
Up until this point I feel as though my understanding is fine. However, when I try to describe a wave that's identical to this one except traveling in the opposite direction is where I get confused. From both some classes I've taken and things I've read, an often used convention for describing a wave traveling backwards is:
And to me this doesn't make sense. If I'm not mistaken (which I very well might be) the frequency of a wave isn't negative, its always greater than 0, and time is not running backwards. So I don't understand why the w*t term would change sign. In my mind since the propagation vector, k, is the term that carries direction, it should be what becomes negative in the expression. So I think the (or an) expression "should" be:
and
So there is clearly a difference between the convention I've been taught and read about, and the expression that, to me at the very least, seems like it should be the proper one.
One thing I've noticed, is that if you use euler's formula on both of these expressions, and then use the even and oddness of the cosine and sine functions you get:
and
Which have the same real part but different imaginary parts. Often times the real part of the wave is the only part that we care about, in which case the conventional method and "my" method actually agree. However, if I am not mistaken the imaginary part of the expression actually carries information about the phase of the wave, so may indicate that there's only a phase difference between the 2 expressions. That said, since they are both trying to describe the same wave, they should have the same phase.
If anyone could lend insight into this issue, as to where I may have gone wrong or to a convention issue or anything else I may have missed it would be much appreciated.