Subject: Re: Pi phase change on reflection?

Date: Sun May 17 03:37:36 1998
Posted by Ricky J. Sethi
Position: PhD

That's a very good question. I'm going to assume that you're talking about classical, mechanical waves in my answer. You've asked why the reflected wave (when it's reflected off a denser medium) suffers a 180degree change of phase compared to the incident wave. And why this doesn't happen when the incident wave is reflected by a less dense medium.

Before we approach this problem for waves, let's construct an analogy to particle reflections by dense mediums; i.e., a ball bouncing off a wall! Okay, when you toss a ball at a wall it bounces back. The ball has a certain energy and momentum and when it's reflected from the wall, energy and momentum are conserved. It hits the wall and, according to Mr. Newton's third law, the wall hits back and tosses it right back at us. Of course, this is assuming a perfectly elastic collision. What if the collision is inelastic (say, we throw a ball of clay, rather than a ball of rubber, at the wall)? In this case, either the wall or the clay absorbs the incident energy and momentum. End result: the clay sticks to the wall instead of bouncing back to us.

All well and good but what's this got to do with waves and the media they encounter? Well, waves also have energy and momentum. And, when they encounter boundaries (either hard, like a wall, or soft), something has to happen to the incident energy and momentum, just like for our ball (of rubber or clay). To see what happens to waves reflecting from a more dense medium (a hard boundary), let's imagine a wave produced on a string that's tied to a pole. Since this string is tied tight to a pole the knot delineates a fixed end (boundary)... this is our wave encountering a (much more) dense medium.

Now say you move the string up and down quickly, thus generating a wave pulse. As the wave travels down the length of the string (see Fig. 1, such as it is, below), towards the knot, the forces on the string are upward at the forefront of the wave (this is what allows the wave to move along). When the pulse reaches the fixed end (the knot), the wave transfers these upward forces to the wall. And, as our friend Mr. Newton dictated, the wall pushes back on the string with an equal force but in the opposite direction. This reaction force creates a new wave which propagates backward (this reflected wave travels from left to right) with the same wavelength and amplitude but with the opposite polarity. This is the induced phase shift of 180degrees. All this was because the end (at the knot) was fixed and couldn't move (i.e., displacement = 0).

Figure (humour me here) 1:

__
/  \----> Wave Pulse
/    \         *
_____/      \________@ --> String Knot
*

Great, now what about the less dense medium? Well, you can simulate this situation with the same string and pole... but this time, instead of tying it in a knot, tie the string around a ring and slip the ring through the pole. Now the wave that travels down the string is free to move up and down. Displacement is no longer fixed at zero but varies with the wave. Since the net vertical force at the ring end (the free end) must be zero, this is transmitted back to the string. This sets up a reflected wave in the string that goes backwards (again, left to right whereas the original, incident wave was directed forward, right to left), only this time with the same polarity (and, of course, the same velocity, amplitude, etc.). That is, as the incident wave arrives at the free end, it exerts a force on the part of the string that's there. This element of string is accelerated and exerts a reaction force on the string. This is what generates the reflected pulse.

Both these cases were, of course, highly idealized but the principles derived are valid for extension to a wide variety of waves. If you didn't like the idea of the knot you can just as well imagine a varying string density (higher density area being equivalent to a hard boundary or fixed end). Of course, this is also a great simplification and this topic has been explored much more rigorously. A good start is Fishbane, Gasiorowicz, and Thornton's Physics for Scientists and Engineers (especially Ch. 15 which does a much better job of drawing Figure 1). Also, I should mention that the incident wave isn't always reflected totally... if the new medium is neither hard nor soft, part of the wave is reflected and part of it is transmitted. Finally, wave reflection from a boundary can also be treated via wave interference. Fishbane, et. al. also do this topic (and many others) justice in their well-illustrated book. However, if you can't find that particular book, I encourage you to check out any book on waves in your local library; in my experience, it's always best to get many different perspectives to maximize understanding. I hope that helped. If you'd like something clarified further, please don't hesitate to drop me a line.

Best regards,

Rick.