Wave Reflections on a String

While reading about the physics of the piano, I got interested in trying to understand exactly why a wave pulse on a string is inverted when it is reflected from a fixed boundary point. Curiously, this is glossed over in every elementary physics book in the library, even Halliday and Resnick's Principles of Physics, and the vast number of books dedicated to wave motion, like Towne's Wave Phenomena and Main's Vibrations and Waves in Physics. I know it falls out of the differential equation with certain boundary conditions; that is not an intuitive explanation. There is a common heuristic, given for example by H. Young in University Physics 8th ed, that the shape of the string is given by the superposition of a equal and opposite virtual pulse entering the string from the boundary material, but let's investigate more deeply.

I want to explore 2 questions here:

  1. How exactly is a wave pulse reflected? What happens in the particles of the string? (after all, there is no virtual pulse traveling the other way!)
  2. How do 2 equal and opposite waves pass through each other without putting the whole string to rest?

So I wrote this applet, which models a flexible string as discrete mass points attatched by elastic bands obeying Hooke's law F = -k y , and the evolving position of the masses is computed numerically by Stoermer's rule. Accurate numerical integration is necessary to be sure that the phenomena of interest is due to the physics of the string and not numerical errors! The equation for the interior points of the string is:

m y''i = -k ( yi - yi+1 ) + -k ( yi - yi-1 )

where I take k=1 and m=1 on the left part of the string.

If the motion is jerky on your computer, try the "slow" speed setting

To change the settings and run again, click on RESET

Applet failure..? Your browser does not support Java???
View source code Change .txt to .java before running javac


Single Pulse going down homogenous string

  1. As one would expect, we see the "force wave" leading, followed by the "velocity wave", followed by the displacement wave, the real wave we see on the string. Curiously, this is no longer apparent after the pulse reflects from the wall.
  2. The pulse moves (to the right, say) because the force profile {F(t)} of the pull of mass k-1 on mass k (mass k's left neighbor) is out of phase with the almost identical, but oppositely signed, force profile of mass k+1 on mass k. See the bottom pictures.
  3. The pulse itself loses energy due to trailing wiggles in the string. I think these wiggles will disappear as the number of mass points increases, making the discretized string more closely approximate a continuous one.
  4. Although it is not displayed here, the total energy of the string is conserved to 5 decimal places, holding at the value 962.11
  5. The shape of the net force profile in the lower pictures is suprising (to me, anyway -- I would have guessed the shape of a sine wave on the interval [0,2pi] ). The red curve is separated into 4 regions by where it crosses the axis. The first region I call the initial lift, followed by the pulldown, followed by the back lift and finally the residual wiggles.

Single Pulse passing into heavier right string

  1. When the right string is more massive, the wavelength of the pulse is compressed when it is transmitted (due to greater inertia of heavier masses; they get moving more slowly). The pulse also propagates more slowly.
  2. Most noticeably, the back lift gets higher (and steeper) as the right string gets heavier. Put the bottom graphs in "overlay" mode to see this clearly.
  3. Curiously, the maximum height of the initial lift also decreases. Why? The residual wiggles on the left string should be the same in all cases...
  4. Part of the pulse is transmitted into the heavier string, and part of it is reflected with an inversion. The 1:40 mass ratio approximates a fixed end point, and you can see it has almost total reflection.

Single Pulse passing into lighter (or nonexistant) right string

  1. When the right string is lighter, the wavelength is elongated. (That is why it is not practical to have the string too light; that would mean having to compute the motion of far too many mass points. Instead, the limiting case of a completely free end point is used.)
  2. Here the portion of the pulse that is reflected is not inverted.
  3. Most noticeably, the pulldown trough gets deeper as the right string gets heavier. Put the bottom graphs in "overlay" mode to see this clearly. Curiously, the back lift also gets higher; by symmetry with the heavier mass case, this is not expected. Note that for the free end point case, the sum of the forces is just the force from the right neighboring mass, and is thus drawn in green.

Two Equal and Opposite pulses colliding

  1. Despite common folklore, the pulses don't actually pass through each other; the point where they meet is held fixed by the equal and opposite impulse streams. Separately, the pulses undergo ordinary inverting reflection from a fixed end point.
  2. By setting the speed to Slow and Freezing the motion when the pulses are beginning to touch, and then Stepping repeatedly, you can find the time when the string is almost perfectly flat. But you can also see that the velocity of certain segments is very high. Their momentum reconstitutes the wave.


Here is the answer to my original question: EXCEPT THAT IS IT WRONG! Since the pulse moves with a constant speed down a homogeneous string, energy flows at the same constant rate. When the boundary of the heavier string is encountered, the heavier string must move more slowly due to inertia, and thus energy accumulates in the extra stretching of last few segments of the lighter string. Since this string has less inertia, the extra stretching causes the lighter string to snap back hard, and make the inverted pulse, which then travels back the way it came.

At the free end point, the last mass point has no right neighbor to restrain it, and moves far from its equilibrium position as it absorbs the energy transmitted down the string. This also causes extra stretching in the last segments of the string, and causing a pulse to be reflected back the way it came, but not inverted.

Further questions, that I am deferring for now:

  1. Do the wiggles really disappear as the string model becomes more continuous?
  2. For a nice symmetric pulse moving through a point in the middle of the homogeneous string, why isn't the impulse stream just one period of a sine wave? Also, to raise a string point and then settle it back to rest, in what sense must the impulse stream from the right neighbor be equal that of the left neighbor?
  3. Reflections from the edges of the picture are sometimes annoying. One way to kill them is to extend the string past what is drawn on the screen, and when the pulse has entered that region, set all the variables to zero. Ideally, I'd like to avoid the overhead of doing this, and just gobble up the pulse at the end point, but how?

Some interesting references are:

Jack Ord's Physics Page.
Lots of good stuff; go specifically here for some applets (earlier versions of which served as a prototype for my own).

Joe Wolfe's discussion of strings at the University of New South Wales.

Robert H. Johns, "Musical String vibrations", The Physics Teacher 15 (1977) p.145-156. Also "Pulse Reflection: Correcting a Common Textbook Error" The Physics Teacher 33 (1995) p.442; That last article was itself corrected in 34 (1996) p.4-5 by John McGervey and Clay Schluchter.

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