Re: [Phys-L] phase of a classical wavefunction

[Carl Mungan <mungan@usna.edu>]

> From: Carl Mungan <mungan@usna.edu>
> Date: January 20, 2015 10:04:28 PM EST
> To: phys-l@phys-l.org
> Subject: [Phys-L] phase of a classical wavefunction
>
>
> Suppose you write down the phase for a wave on a string traveling in the +x
> direction as (kx-wt). Do you write the phase for a wave traveling in the -x
> direction as (kx+wt) or as (-kx-wt), ie. which sign do you reverse: the one
> in front of kx or the one in front of wt?
>
> One might say it doesn't matter. But in some contexts, it can. For example,
> suppose I describe an incident sinusoidal wave as A sin(kx-wt) and the
> reflected wave as A sin(kx+wt) off a free end of the string at x=0. That
> reflected wave has the wrong sign: at x=0 the incident wave has positive
> sign at t=0+ but the reflected wave has negative sign. If we write the
> reflected wave instead as A sin(-kx-wt) then it has the expected positive
> sign for an uninverted reflection.
>
> One person suggested to me that it makes sense to change the sign of k
> because that's proportional to momentum, whereas it doesn't make sense to
> change the sign of w because that's a frequency. For a complex wave, he
> suggested we like to keep the sign of the exp(iwt) term alone and only
> fiddle with the sign of the exp(ikx) piece.
>
> Do you agree with that suggestion? Is there another way to look at this
> situation?
>
> From: John Denker <jsd@av8n.com>
> Date: January 20, 2015 11:08:08 PM EST
> To: Phys-L@Phys-L.org
> Subject: Re: [Phys-L] phase of a classical wavefunction
>
>
> On 01/20/2015 08:04 PM, Carl Mungan wrote:
> > One person suggested to me that it makes sense to change the sign of k
> > because that's proportional to momentum, whereas it doesn't make sense to
> > change the sign of w because that's a frequency. For a complex wave, he
> > suggested we like to keep the sign of the exp(iwt) term alone and only
> > fiddle with the sign of the exp(ikx) piece.
> >
> > Do you agree with that suggestion?
>
> In this case, yes, absolutely.  A reflection reverses k.
>
> > Is there another way to look at this situation?
>
> No, but there are other situations where the answer might
> be different.  It depends on the physics.  If the physics
> is 1D reflection, then k gets reversed.  In higher dimensions,
> k gets mirrored, which changes some components of k
> differently from others.
>
> More generally, asking about "the phase" isn't the optimal
> question, because physical significance attaches to k and ω 
> on their own, not just to the combination we call phase.


Okay. I guess what caught me off guard is if one looks at the wave chapter in a typical intro physics text (anyone reading this message is invited to pause and open up whatever text they use to that chapter right now), one sees something like the following:

"A wave traveling in the +x direction can be described by a function A sin(kx - wt + phi) where phi is a phase constant (let's choose it to be zero), and a wave traveling in the -x direction by A sin(kx + wt)."

Few intro texts that I am aware say we should have flipped the sign of the kx term instead of the wt term.

Again this comes into play when we consider reflections. Specifically the derivation of the Fresnel equations for waves on two strings joined together. Here's the kind of derivation I have in mind:

http://astrowww.phys.uvic.ca/~tatum/physopt/physopt2.pdf

Note that the sign of x not of t was indeed reversed in that PDF. For the case of a cosine wavefunction (rather than the sine I used above), it doesn't actually matter whether you reverse the sign of x or t, because cos is an even function. However, for sin, you will not get the conventional sign for the reflection coefficient if you reverse the sign of t instead of x. (Granted, since it's just a convention, we can correctly reinterpret the opposite sign.)

Of course, one might also argue that the reflection coefficient really should have an absolute value in it. If it's really a ratio of amplitudes, it can never be negative. But I think we know what's intended.

-----
Carl E Mungan, Assoc Prof of Physics  410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363 
mailto:mungan@usna.edu     http://usna.edu/Users/physics/mungan/