Standing vs Traveling Waves: The Difference

The distinction, stated once

A standing wave is not a different kind of wave. It is what two traveling waves look like when they are trapped.

AP Physics 2 essential knowledge 14.6.B.1 says it in one sentence: standing waves can result from interference between two waves that are confined to a region and traveling in opposite directions. Every property in the rest of this page comes out of that sentence, and it has two conditions in it, not one.

Opposite directions is what freezes the pattern. Two waves running the same way keep their combined shape moving; two running against each other produce a shape whose peaks no longer travel. 14.6.B.1.i names the two extremes of that frozen shape: a node is a point on the standing wave where the amplitude is always zero, and an antinode is a point where the amplitude is always at maximum. The word doing the work in both is always. On a traveling wave every point is momentarily at zero displacement once per cycle, and no point is permanently at zero.

Confined to a region is what quantizes it. 14.6.B.1.ii says the possible wavelengths of a standing wave are determined by the size and boundary conditions of the region to which it is confined. A traveling wave has no such restriction: send any frequency down a long rope and it travels. Confine the rope at both ends and most frequencies will not produce a standing pattern at all.

That second consequence is the useful one. A traveling wave exists at every frequency you care to drive. A standing wave exists only at a discrete list of them, and finding the list is most of what Unit 14 asks you to do with Topic 14.6.

Standing against traveling, row by row

Read the last two rows together, because they are what the exam tests. Confining a wave changes nothing about how fast it travels; it only changes which wavelengths survive. So the speed comes from the medium, the wavelength comes from the geometry, and the frequency is whatever $\lambda = v/f$ then gives you. That order is the whole procedure.

The case that separates them: one string, two questions

Take a string of length $L = 1.20\ \text{m}$ fixed at both ends, carrying waves at $v = 240\ \text{m/s}$.

As a traveling wave. Flick one end and a pulse runs to the far end in $t = L/v = 1.20/240 = 5.0\ \text{ms}$. Drive the end at $150\ \text{Hz}$ instead and a wave of wavelength $\lambda = v/f = 240/150 = 1.60\ \text{m}$ travels along the string. Nothing about that calculation cares what $L$ is, and any frequency you pick gives an answer.

As a standing wave. Now the two ends are fixed, so both must be nodes. Sketch the shapes that satisfy that, which is what 14.6.B.3 asks for: one loop, two loops, three loops, and so on. One loop spans half a wavelength, so $L = n\lambda/2$ and the allowed wavelengths are

The first four are $2.40\ \text{m}$, $1.20\ \text{m}$, $0.80\ \text{m}$ and $0.60\ \text{m}$, and no wavelength between them is allowed. Convert each with $\lambda = v/f$ and the frequencies come out at $100\ \text{Hz}$, $200\ \text{Hz}$, $300\ \text{Hz}$ and $400\ \text{Hz}$.

Now look at the $150\ \text{Hz}$ wave again. Its wavelength is $1.60\ \text{m}$, and $1.60\ \text{m}$ is not on the list, because $L = 1.20\ \text{m}$ is not a whole number of half wavelengths of $1.60\ \text{m}$. The traveling wave at $150\ \text{Hz}$ is perfectly real. The standing wave at $150\ \text{Hz}$ does not exist.

Same string, same medium, same $\lambda = v/f$. The only thing that changed is whether the wave is confined, and that turned a continuous range of possible frequencies into a discrete list. This is the difference made visible, and it is the reason the CED files standing waves under interference rather than under wave properties.

Where the second wave comes from

14.6.B.1 needs two waves traveling in opposite directions, and in every real setup only one of them is the one you sent. The other is a reflection, which is why Topic 14.3 sits between the wave definitions and standing waves.

14.3.A.1 says a wave that travels from one medium to another can be transmitted or reflected, depending on the properties of the boundary separating the two media, and 14.3.A.1.i notes that a wave crossing between media results in both a reflected and a transmitted wave. Send a wave at a fixed end and enough of it comes back to interfere with what is still arriving.

Two details from 14.3 change what the reflected wave looks like:

• 14.3.A.1.ii: a reflected wave is inverted if the transmitted wave travels into a medium in which the speed of the wave decreases.
• 14.3.A.1.iii: a reflected wave is not inverted if the transmitted wave travels into a medium in which the speed of the wave increases.

A string clamped to a wall behaves like the first case, and that inversion is why a fixed end has to be a node: the arriving wave and its flipped reflection cancel there at every instant. A free or loose end behaves like the second, and it becomes an antinode instead. 14.6.B.1.iii lists the four cases the exam draws from, pipes with open or closed ends and strings with fixed or loose ends, and the end conditions are the entire difference between them.

One fact carries across the boundary untouched. 14.3.A.1.iv says the frequency of a wave does not change when it travels from one medium to another, so the reflected wave has the frequency of the incident wave and the two can interfere cleanly. If the frequency changed on reflection there would be beats instead of a standing pattern, which is what 14.6.A.6 describes for two waves of slightly different frequency.

When it costs a mark

• Calling a standing wave a wave that does not move. The medium moves, and at an antinode it moves more than a traveling wave of the same amplitude would. What does not move is the pattern, and specifically the nodes.
• Measuring a wavelength as the distance between adjacent nodes. It is half a wavelength. One loop of a standing wave is a half wavelength, so a five-loop pattern on a string holds two and a half wavelengths.
• Assuming every harmonic exists. 14.6.B.2 gives the naming and then adds the exception in the same statement: a standing wave with the longest possible wavelength is called the fundamental or first harmonic, the second-longest wavelength is typically called the second harmonic, the third-longest is called the third harmonic, and so on, however, for a standing wave with a node at one end and an antinode at the other end, only odd harmonics can be established. Quoting the first half of that sentence and stopping is how a pipe closed at one end gets a second harmonic it cannot have.
• Thinking confinement changes the wave speed. It does not. 14.1.A.3 sets the speed from the type of wave and the properties of the medium, and clamping the ends is neither. Tighten the string and the speed changes, by 14.1.A.3.ii; hold the same string at a different length and it does not.
• Using a memorised harmonic formula on the wrong end condition. 14.6.B.3 asks you to get the relationship from a visual representation, which is safer than recalling a formula, because the formula changes between a string fixed at both ends, a pipe open at both ends, and a pipe closed at one end, while the drawing does not.
• Reading a standing-wave diagram of a pipe as a shape. The loops are the displacement amplitude of air moving back and forth along the pipe, not a picture of bent air. Transverse vs longitudinal waves works that reading problem directly.

What the equation sheet prints, and what it leaves to you

The AP Physics 2 equation sheet's Waves, Sound, and Optics panel prints fifteen equations. Going through them one at a time, the ones that touch this comparison are $\lambda = v/f$, $T = 1/f$, $v_{\text{string}} = \sqrt{F_T/(m/\ell)}$, $x(t) = A\cos(\omega t) = A\cos(2\pi f t)$, $y(x) = A\cos(2\pi x/\lambda)$ and $\lvert f_{\text{beat}} \rvert = \lvert f_1 - f_2 \rvert$. The other nine are the optics lines: $n = c/v$, Snell's law, the thin lens and mirror equation, magnification, and five path-length and fringe relations.

Nothing in that list connects a harmonic number to the length of a region, and no other panel on the sheet does either. There is no printed $\lambda_n = 2L/n$, no printed $f_n = nv/2L$, and no printed standing-wave function. That is deliberate: 14.6.B.3 makes the visual representation the tool, so you are expected to draw the pattern, read the wavelength off the geometry, and then use the printed $\lambda = v/f$.

Two of the printed lines are worth separating, because they are the closest the sheet comes to describing a wave and neither is a standing wave.

• $x(t) = A\cos(\omega t)$ is displacement at one location as a function of time.
• $y(x) = A\cos(2\pi x/\lambda)$ is displacement at one instant as a function of position.

14.2.A.2 introduces both, and calls them example equations rather than relevant equations. Neither has position and time in it together, so neither describes a wave moving, and neither describes a node either. The AP Physics 2 formula sheet page lists all of them in printed order.

One more line is easy to misfile. $v_{\text{string}} = \sqrt{F_T/(m/\ell)}$ comes from 14.1.A.3.ii and gives the speed of a wave or pulse on a string from the tension and the mass per length. It is a traveling-wave fact that you use inside a standing-wave problem, which is exactly the pattern of most Unit 14 questions.

When they coincide, and why that lulls you

Almost every number you compute is computed the same way for both, which is why the distinction can sit unnoticed until a question turns on it.

Both have a wavelength, a frequency and a period. Both obey the printed $\lambda = v/f$. Both take their speed from 14.1.A.3, the type of wave and the properties of the medium. Both can be transverse or longitudinal. Both are built out of 14.6.A.3, superposition, which says the resulting displacement of overlapping waves is found by adding the individual displacements: a standing wave is that addition performed on two counter-propagating waves, and a traveling wave is what you get when there is only one term in the sum.

So a student can find the wavelength of the third harmonic and then the frequency without ever deciding which object is in front of them, and get full credit. The distinction only bites in three places.

1. Which frequencies are allowed. Continuous for a traveling wave, discrete for a standing one.
2. Whether a given point ever moves. Every point on a traveling wave moves; a node never does.
3. How far apart the marked features are. Crest to crest is one wavelength on a traveling wave, and node to node is half a wavelength on a standing one.

There is a quieter coincidence too. A standing wave is genuinely two traveling waves at all times, so anything true of traveling waves in general is still true of the two components: they still travel at $v$, they still carry the same wavelength, and they still keep their frequency at a boundary. The standing pattern is a description of their sum, not a replacement for them. Period vs frequency covers the other conversion these problems always need.