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Introduction to Waves
John Denker

## 1  Examples and Counterexamples

### 1.1  Waves

1.
Waves on the surface of the ocean.

Figure 1: Ocean Wave

2.
Sound waves.

Loosely speaking, sound is a mechanical disturbance in some medium. There is an interplay between inertia (involving the density of the medium) and a restoring force (involving the elastic properties of the medium, i.e. pressure and/or shear stress). Coming up with a more precise definition would not be worth the trouble, for reasons discussed in reference 1.

The medium can be solid, liquid, gaseous, or whatever.

The disturbance can be longitudinal or transverse, although in liquids and especially in gases a transverse sound wave is usually so grossly overdamped as to be not worth worrying about.

Gravity waves on the surface of the ocean are generally not considered sound waves. Also capillary waves on the surface are generally not considered sound waves, although it is hard to define “sound” in such a way as to exclude them. In contrast, solids have surface acoustic waves which behave differently from acoustic waves in the bulk.

Acoustic wave is synonymous with sound wave.

Sound is not limited to the range of human hearing. The dictionary defines ultrasound to be a form of sound. Cats can hear sounds up to 70 kHz or thereabouts. Dolphins can hear sounds up to 150 kHz or thereabouts.

3.
Electromagnetic waves, including radio waves, light waves, et cetera.

4.
A transverse pulse running along a rope or string. This includes piano strings.

5.
People in a stadium doing “the wave”.

Figure 2: Doing The Wave

6.
Sand dunes and ripples on the desert

Figure 3: Sand Ripples

7.
Atomic wavefunctions and other quantum-mechanical wavefunctions, as discussed in reference 2.

Figure 4: Standing Wave : |2px⟩ Orbital

### 1.2  Non-Waves

1. The floral stencil pattern in figure 5.

Figure 5: Non-Wave: Floral Stencil Pattern
2. The ridges and furrows in a farmer’s field.

Figure 6: Non-Wave: Ridges and Furrows

## 2  What is a Wave?

### 2.1  Basic Idea

We define a wave to be something that exists as the solution to a wave equation. It is best to focus attention on the wave equation rather than the wave itself. See section 3 for a discussion of wave equations.

To say the same thing the other way: You cannot tell whether something is a wave or not just by looking at it, if you don’t understand the physics that produced it. In particular, the sand ripples in figure 3 are a wave, whereas the ridges and furrows in figure 6 are not, although you could never tell by looking at the pictures. Sometimes a movie will reveal the physics in situations where a snapshot would not, but even a movie is not necessarily sufficient, e.g. in the case of a standing wave.

Just because it is repetitive does not make it a wave, as we see from the examples in section 1.2.

On the other side of the same coin, a wave does not need to be repetitive. It does not need to have any zero-crossings. Example: A single pulse propagating along a string. The pulse in figure 7 repeats every few seconds (to make it easier to see), but you can imagine a pulse that occurs only once.

Figure 7: Running Wave : Pulse Only

Sometimes a wave can be partially repetitive and partially not. It can have a finite number of zero crossings. Example: A carrier modulated by an envelope, as in figure 8.

Figure 8: Running Wave : Carrier plus Modulation

Again, the central idea is that the wavefunction is a solution to a wave equation. You may be accustomed to solving an equation to find a number, but you can perfectly well solve an equation to find a function. A function is more complicated than a scalar, just as a movie is more complicated than a poster. However, it is possible to go shopping for a movie instead of a poster. Most functions are not solutions to the wave equations, just as most movies are not the one you want to buy.

Here are some examples of solving an equation to find a function:

• An exponential function of time is a solution to the equation that says the number of bacteria doubles every hour.
• A parabolic function is a solution to the equations of motion for a particle moving in a uniform gravitational field.
• An elliptical function is the solution to the equations of motion for a planet going around the sun.

### 2.2  Waves in Spacetime

Here is another way to visualize what is going on:

Figure 9: Spacetime Diagram of Running Pulse

Figure 9 is a three-dimensional plot, but each of the three dimensions represents something different. The spatial x-coordinate increasas from the left to the lower right. Time increases from the front to the right rear. The ordinate of the wave function is more-or-less vertical up the page.

If you look along the axis where it says “snapshot”, that corresponds to taking a snapshot of what the wave looks like, across all positions, along a contour of constant time=0. The wave is moving left-to-right, although you cannot tell that from a single snapshot. At time=0, the main peak is centered at position=0, and there is a subsidiary peak at position=7.

If you look along the other axis, where it says “oscilloscope trace”, that corresponds to sitting at a fixed location, namely position=20, and using an oscilloscope to observe what the wavefunction does at that one location, as a function of time. The subsidiary peak arrives at time=13. Then the main peak arrives at time=20.

### 2.3  Standing Waves

A wave does not need to propagate. There are such things as standing waves. For a linear medium, we can describe the situation as follows: The standing wave has a definite shape as function of position (independent of time). It just sits there and gets bigger and smaller in proportion to a function of time (independent of position). So there is a separation of variables. Also, you can think of a standing wave as the superposition of a running wave with an equal-and-opposite running wave.

Examples include:

1. Standing waves on a violin string. (S) (AH)
2. Standing waves in a cylindrical bore, such as a flute, clarinet, et cetera. (S)
3. Acoustical or electromagnetic standing waves in a rectangular cavity. (S)
4. Acoustical or electromagnetic standing waves in a non-rectangular cavity.
5. Standing waves on the surface of a non-rectangular swimming pool.
6. Atomic wavefunctions such as |1s⟩ and |2pz⟩, as discussed in reference 2. (AH!)
7. Standing waves on an ordinary (non-rectangular) drumhead.
8. Standing waves in a conical bore such as an oboe, saxophone, et cetera.
9. Standing waves on a chain with non-uniform tension. (This can easily happen if the chain’s own weight makes a nontrivial contribution to the tension.) (AH)

The items marked (S) exhibit a more-or-less sinusoidal pattern as a function of position, whereas all the others don’t. The items marked (AH) are perceptibly anharmonic (i.e. nonlinear) at readily-achievable amplitudes.

In all cases, the criterion for a wave to be a standing wave is that it does not transport energy from place to place. For example, consider a nice simple sinusoidal standing wave on a string. The ordinate of the wavefunction is the height of the wave, denoted ϕ, which is implicitly a function of x and t. At each point x, the ordinate is ϕ=sin(ωt). The corresponding velocity is ϕ‌·=ωcos(ωt). The potential energy is proportional to ϕ2 and the kinetic energy is proportional to ϕ‌·2. The constants of proportionality are such that the total energy is proportional to sin2 + cos2 ... which is a constant. There is no energy flowing from place to place.

## 3  What is a Wave Equation?

In section 2 we defined a wave as something that is a solution to a wave equation. That leaves us with the obvious question, what is a wave equation?

There are lots of different wave equations, some of which are more complicated than others. In particular, the wave equation for an electromagnetic wave in a waveguide is more complicated than for an electromagnetic wave in free space. In quantum mechanics, the wave equation – i.e. the Schrödinger equation – has complexities all its own.

Here are the key things that all these have in common: The equation involves at least two variables: one timelike variable and one spacelike variable (which may belong to a very strange abstract space). It is possible, at least in principle, to have a solution that moves from an initial location at one time to a nearby location at a later time, maintaining the same shape, or at least almost the same shape. This is called a running-wave solution.

To repeat: To qualify as a wave equation, it must have some running-wave solutions.

On the other hand, as mentioned in section 2.3, we do not require that all solutions be running waves. Very commonly it is possible to combine running waves so as to make a standing wave, i.e. a solution that does not propagate from place to place. This does not disqualify the equation from being a wave equation. Similarly, the standing wave counts as a wave – because it is a solution to a wave equation – even though you couldn’t necessarily ascertain this just by looking at a snapshot of the solution. On the other hand, if an equation has only non-propagating solutions, then it probably shouldn’t be considered a wave equation.

Tangential remark: It is fairly common for the solutions of an equation to have a different symmetry from the equation itself. For example, a roulette wheel has N=37 or N=38 slots, each of which is supposed to have the same size, same depth, and same probability. This gives us an equation with N-fold symmetry. However, on any particular play of the game, the marble winds up in only one of the slots. The ensemble of all possible outcomes is symmetrical, but any given outcome is not symmetrical. I mention this to support the idea that it is OK for the wave equation to have standing-wave solutions as well as running-wave solutions.

## 4  Reflection from an Impedance Mismatch

See reference 3. This is the classic video from 55 years ago. Note that even today, the torsion-wave demonstration apparatus is sometimes called a Shive machine.

For reasons discussed in reference 4, I disagree with the way he uses the words “cause” and “effect”, but that is unimportant and tangential to the points he is making.

A shorter video showing just the reflection situation is reference 5.

In both of these demos, they missed a trick insofar as they kept the torsional spring constant the same and varied only the moment of inertia of the ribs. Therefore the wave speed changes along with the impedance. Students can’t tell from the demo whether the important thing is the impedance or the speed. Those who know about Snell’s law will be fixated on the speed.

Suggestion: The smart thing would be to change the torsional stiffness in proportion to the moment of inertia, so that the wave speed stays the same even as the impedance changes. I have not yet found video of that.

On the other side of the same coin, figure 10 shows what happens when a wave propagates from one medium to another. On the left side of the diagram, the medium is polyester rope. On the right side of the diagram, the medium is brass chain. However, I arranged things so that the impedance is the same, so there is no reflection.

The problem with using a rope (or chain) instead of a torsion apparatus is that the wave speed is inconveniently high. A the acceleration of gravity being what it is, the rope will always be under a fair bit of tension, just to support its own weight. This gives rise to a minimum wave speed. You can increase the speed by increasing the tension. There are ways of reducing the tension, e.g. by supporting some of the weight of the rope in the middle, but this is not super-convenient. Unless the rope is very long, the high speed makes hard to visualize what is going on. You can alleviate the problem somewhat by viewing the rope almost end-on, so that the distance along the rope is foreshortened. I used a fair amount of foreshortening in figure 10.

Figure 10: Wave on a Rope + Chain

## 5  Dispersion, Dissipation, Nonlinearity, and Shocks

In the simplest case, all waves in a given medium travel at the same speed. However, this is definitely not always the case.

You can easily have a situation where a large-amplitude wave travels at a different speed from a small-amplitude wave. This is called nonlinearity. This is quite common for water waves in a shallow channel. In some cases this can lead to a tidal bore, which is a kind of shock (aka shock wave). (Experts tend to call such things shocks, rather than shock waves.)

It is also possible to have a situation where a long-wavelength wave travels at a different speed from a short-wavelength wave. This is very common for waves on the surface of deep water. There’s actually a minimum in the wave speed; shorter waves are faster (due to surface tension) and longer waves are also faster (due to gravity). You may be aware that a tsunami is a long-wavelength wave, moving at tremendous speed, commonly 800 to 1000 km per hour, almost the speed of sound in air. It is well known that the ordinary waves you see at the beach (wavelength on the order of meters) do not travel nearly so fast.

Last but not least, it must be pointed out that the wave equation in polar coordinates is dispersive, even in situations where plane waves in the same medium would be non-dispersive. This has tremendous practical consequences. For one thing, you may have notices that an explosion up close makes a short “snap” noise, whereas an explosion farther away makes a “boom” noise.

• Part of the difference is explained by dispersion; different frequency components travel at different speeds.
• Part of the difference is explained by absorption; shorter wavelengths die out more quickly.

It is also worth noting that in the real world, a wave loses energy as it moves along. For example, a sound wave in air can lose energy do to viscosity, and also due to thermal conductivity. At the peaks of the sound pressure field, the air is always slightly hotter than in the troughs, so there can be losses due to thermal conductivity, although this effect is small when the wavelength is long. Similarly, compressing and decompressing the air dissipates energy via bulk viscosity. There can be vastly greater dissipation at boundaries, via shear viscosity. (This is how “acoustical tile” works.)

In the typical introductory course, nobody worries about dispersion, dissipation, nonlinearity, or shocks. However, such things are worth mentioning at least once, in hopes of cutting down on the number of misconceptions. It would be a misconception to think that all waves are linear, non-dispersive, and/or non-dissipative.

### 6.1  UNSW

The Music Science site at UNSW is awesome. It contains an enormous amount of material, most of it very readable, with very high levels of technical accuracy. See reference 6.

### 6.2  Indiana University Southeast

Reference 7 has a collection of high-quality animations. I particularly like the longitudinal wave animation, because it seems physically correct for a lattice of regularly-space particles. It doesn’t pretend to be something it isn’t.

### 6.3  Penn State

Reference 8 is well worth a bookmark. There are dozens of pages, each with several animations. The first few are quite basic.

• On the first page, there is an adorable animation of stick figures “doing the wave”.

However, I don’t like the "dictionary definition". It rules out standing waves. I know a lot of chemists who would get cranky if you told them that s-waves and p-waves are not waves.

I also don’t like the diagrams that look like molecular dynamics simulations of particles in a gas (on the first page and elsewhere). They are fake data pretending to be something more. (I usually don’t object to data that is obviously fake.) You can’t have a sound wave in a gas if the gas particles themselves aren’t moving. These animations are just begging to be misinterpreted.

• On the second page, there is an animation to help explain the relationship between a snapshot and an oscilloscope trace.
• On later pages, there are the usual animations of reflection from an impedance mismatch ... although I prefer video of real experiments, whenever possible, rather than simulations.
• After the first few pages, things rapidly get more sophisticated.

### 6.4  Mullin et al.: Fundamentals of Sound

Reference 9 has some nice animations.

## 7  References

1.
John Denker,
“Words Acquire Meaning
from How They Are Used”
www.av8n.com/physics/meaning.htm

2.
John Denker,
“Models and Pictures of Atomic Wavefunctions”
www.av8n.com/physics/wavefunctions.htm

3.
John N. Shive,

4.
John Denker,
“Cause and Effect”
www.av8n.com/physics/causation.htm

5.
“Bell Labs Wave Machine: Mismatched Impedance”
http://video.mit.edu/watch/bell-labs-wave-machine-mismatched-impedance-7049/

6.
Joe Wolfe et al.,
“Music Science”
http://newt.phys.unsw.edu.au/music/

7.
Kyle Forinash,
“Waves: An Interactive Tutorial”
http://homepages.ius.edu/kforinas/WJS/WavesJS.html

8.
Dan Russell,
“Acoustics and Vibration Animations”
http://www.acs.psu.edu/drussell/demos.html

9.
W. J. Mullin, W. J. Gerace, J. P. Mestre, and S. L. Velleman,
FUNDAMENTALS OF SOUND With Applications to Speech and Hearing
http://www.ablongman.com/mullin/
http://www.ablongman.com/mullin/AnimationHome.html
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