Source: PHYS 116 Final Exam Notes, University of Washington
Difficulty: Introductory–Intermediate | Prerequisites: Superposition and interference (Ch. 16.1), basic wave properties
Big Picture
Standing waves are what happen when superposition meets boundaries. A guitar string, a flute, a microwave oven: any time a wave bounces back and forth inside a confined space, you get a standing wave. This section connects the abstract idea of superposition to the physical reality of musical instruments and resonance. You need superposition (Ch. 16.1) down cold before this will make sense, and everything here feeds directly into standing sound waves (Ch. 16.4) and interference from two sources (Ch. 16.6).
A standing wave forms when two identical waves travel in opposite directions and superpose, typically because a wave reflects off a boundary and overlaps with the incoming wave. The result looks like it is oscillating in place rather than travelling. Only certain wavelengths and frequencies fit a given string or tube, and these are called the harmonics. The lowest is the fundamental frequency.
Standing wave
A wave pattern that appears to oscillate in place rather than travel. Individual points on the string move up and down, but the crests and troughs stay at fixed positions. In simple terms, it is the superposition of two identical travelling waves moving in opposite directions.
Node
A point on a standing wave where the displacement is always zero. Nodes are points of permanent destructive interference. The intensity at a node is zero.
Antinode
A point on a standing wave where the displacement reaches its maximum value. Antinodes are points of constructive interference. The intensity at an antinode is at its peak.
Mode number (m)
An integer (m = 1, 2, 3, ...) that labels each allowed standing wave pattern on a string or in a tube. The mode number equals the number of antinodes. A higher mode number means a shorter wavelength and a higher frequency.
Harmonic
Another name for an allowed standing wave frequency. The first harmonic (m = 1) is the fundamental frequency. The second harmonic (m = 2) has twice the fundamental frequency, and so on.
Fundamental frequency (f₁)
The lowest frequency at which a standing wave can exist on a given string or in a given tube. All higher harmonics are integer multiples of this frequency.
Boundary
The end of the medium (for example, the wall where a string is tied). A wave reaching a boundary reflects. At a fixed boundary the reflected pulse is inverted; all the wave's energy is reflected and the amplitude is unchanged.
Discontinuity
A point where the properties of the medium change (for example, where a light string meets a heavy rope). A wave hitting a discontinuity is partly transmitted and partly reflected, and both the transmitted and reflected pulses have smaller amplitude than the original.
Reflection
When a wave encounters a boundary or discontinuity, some or all of its energy bounces back. At a hard boundary the reflected pulse is inverted. At a discontinuity where wave speed increases, the reflected pulse is not inverted.
A standing wave is the superposition of two travelling waves with the same amplitude and wavelength, moving in opposite directions along the same medium.
The crests and troughs appear to "stand in place" rather than move. Individual points on the string oscillate up and down, but the wave pattern itself does not travel.
A standing wave is not a new kind of wave. It is purely the result of superposition.
Nodes are equally spaced points that never move. The displacement is always zero at a node. The distance between adjacent nodes is λ/2.
Antinodes sit midway between nodes. These are the points of maximum displacement. The distance between adjacent antinodes is also λ/2.
Nodes and antinodes alternate: node, antinode, node, antinode, and so on, each separated by λ/4.
At a node the intensity is zero. At an antinode the intensity is at its maximum.
When a wave pulse hits a hard, fixed boundary (such as a wall), it reflects with its amplitude unchanged but its shape inverted (a crest becomes a trough). All the wave's energy is reflected.
When a wave pulse hits a discontinuity (where the medium's properties change), part of the energy is transmitted and part is reflected. Both the transmitted and reflected pulses are smaller than the original.
If the wave speed increases at the discontinuity, the reflected pulse is not inverted. If the wave speed decreases, the reflected pulse is inverted (think of the slower side as behaving like a boundary).
Two waves of equal amplitude (A) and wavelength (λ) must travel in opposite directions along the string.
At both ends of a string fixed at x = 0 and x = L, the displacement must be zero at all times. Both ends are nodes.
The nodes must be equally spaced at intervals of λ/2.
Only specific wavelengths fit between the fixed ends of a string. These allowed wavelengths produce the standing wave modes.
The mode number m (m = 1, 2, 3, ...) equals the number of antinodes in the pattern.
m = 1 is the simplest pattern: two nodes (one at each end) and one antinode in the middle. This is the fundamental mode, or first harmonic.
m = 2 has three nodes and two antinodes. m = 3 has four nodes and three antinodes. Each successive mode adds one more node and one more antinode.
A higher mode number means a shorter wavelength and a higher frequency.
The set of all allowed frequencies is called the harmonic series. Every harmonic frequency is an integer multiple of the fundamental: fₘ = m × f₁.
Wavelength of the mth mode (string fixed at both ends):
\lambda_m = \frac{2L}{m} \qquad m = 1, 2, 3, \ldotsL is the length of the string. m is the mode number.
Frequency of the mth mode:
f_m = m \left( \frac{v}{2L} \right) = m f_1v is the wave speed on the string.
Fundamental frequency:
f_1 = \frac{v}{2L} = \frac{1}{2L} \sqrt{\frac{T_s}{\mu}}Tₛ is the string tension. μ is the linear mass density (mass per unit length).
Rearranged form (useful for finding mode number):
m = \frac{f_m \cdot 2L}{v} \cdot 1 = f_m \cdot \frac{2L \sqrt{\mu}}{\sqrt{T_s}}Guitar strings produce sound through standing waves. Pressing a fret shortens the effective length L, raising the fundamental frequency and thus the pitch. Tightening the tuning peg increases tension Tₛ, which also raises the frequency. This is why both fret position and string tension matter for tuning.
Students often think a standing wave is a distinct type of wave. It is not a new kind of wave; it is simply the superposition of two ordinary travelling waves moving in opposite directions.
Some students confuse nodes with antinodes. Remember: nodes never move (zero displacement), antinodes move the most (maximum displacement).
A frequent error is forgetting that the mode number m equals the number of antinodes, not the number of nodes. For a string fixed at both ends, the number of nodes is m + 1 (counting the two endpoints).
Students sometimes assume a reflected pulse keeps its original orientation. At a fixed boundary, the reflected pulse is inverted. Whether it inverts at a discontinuity depends on whether the wave speed increases or decreases.
⚠️ Be prepared to identify the mode number from a diagram of a standing wave pattern (count the antinodes).
⚠️ Know how to calculate λₘ and fₘ from the string length and mode number. These formulas appear in nearly every standing wave problem.
⚠️ Understand what happens at a boundary vs. a discontinuity: full reflection with inversion at a fixed boundary, partial reflection at a discontinuity.
⚠️ The relationship fₘ = m × f₁ is heavily tested. If given two harmonic frequencies, you should be able to find the fundamental.
True or false: A standing wave is a fundamentally different type of wave from a travelling wave. (False, it is the superposition of two travelling waves.)
Fill in the blank: The distance between two adjacent nodes is ______. (λ/2)
True or false: The mode number equals the number of nodes. (False, it equals the number of antinodes.)
Fill in the blank: At a fixed boundary, a reflected pulse is ______. (inverted)
True or false: The third harmonic has a frequency of 3f₁. (True.)
Q: A string of length 1.2 m is fixed at both ends and vibrates in its third harmonic. What is the wavelength?
A: Using λₘ = 2L/m: λ₃ = 2(1.2)/3 = 0.80 m.
Q: A guitar string has a fundamental frequency of 220 Hz. What is the frequency of the fourth harmonic?
A: f₄ = 4 × f₁ = 4 × 220 = 880 Hz.
Q: A standing wave on a string has five antinodes. What is the mode number, and how many nodes are there (including the endpoints)?
A: The mode number m = 5 (one antinode per mode). The number of nodes = m + 1 = 6.
Q: A pulse on a string hits a wall (fixed boundary). Describe the reflected pulse.
A: The reflected pulse has the same amplitude as the incoming pulse but is inverted (a crest returns as a trough). All the wave's energy is reflected.
Q: What happens to a pulse when it reaches a point where the string becomes thicker (wave speed decreases)?
A: Part of the pulse is transmitted (continuing into the thicker string) and part is reflected. The reflected pulse is inverted. Both pulses have smaller amplitude than the original.
Standing waves on a string lead directly into standing sound waves in air columns (Ch. 16.4), which use the same formulas but with different boundary conditions (open vs. closed ends). The concept of harmonics connects to speech and hearing (Ch. 16.5), where the mix of harmonics determines timbre. Beats (Ch. 16.7) arise from a slightly different mechanism but rely on the same superposition principle.
Standing wave, stationary wave, nodes and antinodes, harmonic series, fundamental frequency, first harmonic, higher harmonics, mode number, string vibration, wave reflection, boundary reflection, inverted pulse, discontinuity, wavelength of standing wave, λ = 2L/m, f = mv/2L, PHYS 116, general physics waves, University of Washington