Stationary Waves

Spec mapping (OCR H556 Module 4.5): Describe stationary waves as the superposition of two progressive waves of equal amplitude and frequency travelling in opposite directions; identify nodes and antinodes; derive harmonic frequencies on strings ( $f_n = nv/2L$ ) and in air columns (open–open: all harmonics; open–closed: odd harmonics only). PAG 3 anchor for measurement of the speed of sound or fundamental frequency on a stretched string using Melde's apparatus.

When two progressive waves of the same frequency and amplitude travel in opposite directions through the same region, their superposition produces a wholly different kind of pattern — a stationary wave (also called a standing wave). Rather than travelling through space, the wave appears to be "frozen" in place, with certain points (nodes) where the displacement is always zero and other points (antinodes) where the oscillation is maximum.

Stationary waves are the physics of musical instruments. Every string you have plucked, every pipe you have blown across, and every tuning fork you have struck produces sound by setting up stationary waves. This lesson introduces stationary waves in general, the distinction between nodes and antinodes, and the specific cases of stationary waves on strings (with both ends fixed) and in pipes (open and closed). The OCR A-Level specification expects quantitative treatment using:

$f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$

(fundamental frequency on a stretched string).

How Stationary Waves Form

Consider a progressive wave travelling to the right along a string, meeting a fixed end. The wave is reflected, with a $\pi$ phase shift (because a fixed end cannot move, the reflected wave must cancel the incoming displacement). The result is two waves of the same frequency and amplitude, travelling in opposite directions along the same string.

By the principle of superposition, the total displacement at any point is the sum of the two waves' displacements. Mathematically, the two opposing travelling waves combine to give:

$y(x, t) = 2A\sin(kx)\cos(\omega t)$

where $k = 2\pi/\lambda$ is the wavenumber and $\omega = 2\pi f$ the angular frequency of each constituent wave.

The spatial part $\sin(kx)$ is independent of time; the temporal part $\cos(\omega t)$ applies equally to every point. At positions where $\sin(kx) = 0$ — called nodes — the displacement is always zero. At positions where $\sin(kx) = \pm 1$ — called antinodes — the amplitude of oscillation is maximum ( $2A$ , twice that of either constituent wave).

This is the defining feature of a stationary wave: the pattern is stationary in space — nodes remain nodes and antinodes remain antinodes — while each point between them oscillates in simple harmonic motion at angular frequency $\omega$ .

Nodes and Antinodes

A node is a point on a stationary wave where the displacement is always zero. Nodes form at positions where the two travelling waves permanently cancel.

An antinode is a point where the amplitude of oscillation is maximum. Antinodes form at positions where the two travelling waves permanently reinforce.

Key facts about nodes and antinodes:

Adjacent nodes are $\lambda/2$ apart (where $\lambda$ is the wavelength of the underlying travelling wave).
Adjacent antinodes are $\lambda/2$ apart.
Adjacent node and antinode are $\lambda/4$ apart.
All points between two adjacent nodes oscillate in phase with each other.
Points on opposite sides of a node oscillate in antiphase ( $\pi$ rad apart).
Energy oscillates between kinetic (when displacement is zero everywhere — the wave is "flat") and potential (at maximum displacement — the wave is "tense"); no net energy is transported.

This list is worth memorising — it is the bedrock of every stationary-wave question.

Stationary Waves on a Stretched String (Both Ends Fixed)

A string of length $L$ , fixed at both ends (like a guitar or violin string) can support stationary waves only if the pattern has nodes at both ends, because the fixed ends cannot move.

The only stationary waves that fit are those with a whole number of half-wavelengths between the ends:

$L = \frac{n\lambda}{2}, \quad n = 1, 2, 3, \ldots$

So the allowed wavelengths are:

$\lambda_n = \frac{2L}{n}$

and the corresponding frequencies (using $v = f\lambda$ ) are:

$f_n = \frac{nv}{2L}, \quad n = 1, 2, 3, \ldots$

These are the harmonics of the string.

The Fundamental Frequency

The fundamental frequency (first harmonic) is the lowest allowed frequency:

$f_1 = \frac{v}{2L}$

It corresponds to $n = 1$ — a single "half-wavelength loop" with nodes at each end and one antinode in the middle.

Higher Harmonics

Second harmonic ( $n = 2$ ): $f_2 = 2f_1$ . Wavelength equals $L$ . Two loops, with one extra node in the middle.
Third harmonic ( $n = 3$ ): $f_3 = 3f_1$ . Three loops, four nodes total.
$n$ -th harmonic: $f_n = nf_1$ . Contains $n$ loops, $n+1$ nodes.

The overtones of a string are the harmonics above the fundamental. In music theory the second harmonic is the first overtone, the third the second overtone, etc.

flowchart LR
    Start[Stretched string of length L, both ends fixed]
    Start --> Cond[Node at each end]
    Cond --> Half[Length must contain integer number of half wavelengths]
    Half --> Form[Allowed wavelengths lambda equals 2L over n]
    Form --> Freq[Allowed frequencies f equals n v over two L]
    Freq --> F1[Fundamental f1 equals v over 2L]
    Freq --> Fn[All harmonics n equals 1 2 3 present]

Speed of a Wave on a Stretched String

The speed of a transverse wave on a stretched string is:

$v = \sqrt{\frac{T}{\mu}}$

where:

$T$ is the tension in the string (N);
$\mu$ is the mass per unit length of the string (kg m $^{-1}$ ).

Substituting into $f_1 = v/(2L)$ , the fundamental frequency of a stretched string becomes:

$f_1 = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$

This is the formula you must know. It tells you that the pitch of the string rises:

as $L$ decreases (shorter string — press your finger on a guitar fret);
as $T$ increases (tighten the string — tune up);
as $\mu$ decreases (thinner or lighter string).

This is the full physics of how guitars, violins, pianos and harps are tuned and played.

Worked Example 1 — Fundamental of a Guitar String

Q. A guitar string has length $L = 0.65\text{ m}$ , mass per unit length $\mu = 1.0\times 10^{-3}\text{ kg m}^{-1}$ , and is under tension $T = 100\text{ N}$ . Calculate the fundamental frequency.

$v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{100}{1.0\times 10^{-3}}} = \sqrt{1.00\times 10^{5}} = 316\text{ m s}^{-1}$

$f_1 = \frac{v}{2L} = \frac{316}{2 \times 0.65} = 243\text{ Hz}$

Slightly lower than the $247\text{ Hz}$ of the B3 on a standard guitar — plausibly, a B string that needs a slight tune-up.

Worked Example 2 — Second Harmonic of the Same String

Q. For the same guitar string, calculate the frequency of the second harmonic.

A. $f_2 = 2f_1 = 2 \times 243 = 486\text{ Hz}$ .

Worked Example 3 — Tension for a Target Frequency

Q. A string of length $L = 0.50\text{ m}$ and $\mu = 2.0\times 10^{-3}\text{ kg m}^{-1}$ is to produce a fundamental of $300\text{ Hz}$ . Calculate the required tension.

A. From $f_1 = (1/2L)\sqrt{T/\mu}$ :

$300 = \frac{1}{2 \times 0.50}\sqrt{\frac{T}{2.0\times 10^{-3}}} = \sqrt{\frac{T}{2.0\times 10^{-3}}}$

$9.00\times 10^{4} = \frac{T}{2.0\times 10^{-3}} \Rightarrow T = 180\text{ N}$

Worked Example 4 — Diagnosing a Loose String

Q. A musician finds her guitar's open A-string sounds at $220\text{ Hz}$ when tuned correctly. After a temperature drop the same string sounds at $208\text{ Hz}$ . By what fraction has the tension changed?

A. $f_1 \propto \sqrt{T}$ , so

$\frac{T_{\text{new}}}{T_{\text{old}}} = \left(\frac{208}{220}\right)^2 = 0.894$

Tension has dropped by about $11\%$ — the typical effect of overnight cold contraction.

Energy Storage in a Stationary Wave

A stationary wave on a stretched string stores energy that oscillates between purely kinetic and purely potential forms — but the total energy stays constant. Consider the snapshots of the wave's shape across one period:

At $t = 0$ (extreme displacement): The string is bowed into its maximum shape. Every point is momentarily at rest at its extreme position. Kinetic energy is zero; all the energy is stored as elastic potential energy in the stretched string.
At $t = T/4$ (zero displacement): The string is momentarily flat (every point passes through equilibrium). Elastic potential energy is at its minimum; kinetic energy is at its maximum because every point is moving at peak speed.
At $t = T/2$ (opposite extreme): Same as $t = 0$ but inverted. Kinetic zero, potential maximum.

So the period of the energy oscillation is $T/2$ , half the period of the displacement oscillation. This is a generic feature of harmonic motion: kinetic and potential energies each oscillate at twice the displacement frequency, $\pi/2$ rad out of phase with each other, summing to a constant total.

For a string of length $L$ , linear mass density $\mu$ , vibrating in its $n$ -th harmonic with antinode amplitude $A$ , the total energy is

$E = \frac{1}{4}\,\mu \omega^2 A^2 L$

(where $\omega = 2\pi f_n$ ). You will not be asked to derive this at A-Level, but it tells you that doubling the amplitude quadruples the energy stored — important when discussing loudness of musical instruments or the energy dissipated when a vibrating string stops oscillating.

Resonance Curves and the Driven String

A real Melde's string driven by an oscillator does not vibrate at exactly one frequency — it responds with significant amplitude across a range of frequencies near each harmonic. This is a resonance curve: a peak in amplitude vs driving-frequency centred on each harmonic $f_n$ , with a finite width $\Delta f$ determined by damping.

The quality factor is defined as

$Q = \frac{f_n}{\Delta f}$

A high- $Q$ resonator (low damping) has a very sharp, tall peak — the string only responds significantly within a narrow frequency window of $f_n$ . A low- $Q$ resonator (heavily damped) has a broad, shallow peak.

For a Melde's-apparatus demonstration string, $Q \sim 10$ – $100$ is typical. Plucked piano strings reach $Q \sim 10^3$ ; high-quality crystal oscillators in electronics achieve $Q \sim 10^6$ or higher.

This explains a subtle observation from the lab: when you sweep the oscillator frequency through a harmonic, you don't see a sudden snap from "no pattern" to "perfect pattern" and back — the pattern builds smoothly, reaches a peak at $f_n$ , then fades. The width of that transition is $\Delta f$ . For sharp $Q$ resonance the transition is rapid; for weak resonance it is gradual.

Stationary Waves in Air Columns (Pipes)

The second classical case is stationary waves in a column of air — a pipe. Pipes come in two types, differing in whether the ends are open or closed.

Stationary Waves

Stationary Waves

How Stationary Waves Form

Nodes and Antinodes

Stationary Waves on a Stretched String (Both Ends Fixed)

The Fundamental Frequency

Higher Harmonics

Speed of a Wave on a Stretched String

Worked Example 1 — Fundamental of a Guitar String

Worked Example 2 — Second Harmonic of the Same String

Worked Example 3 — Tension for a Target Frequency

Worked Example 4 — Diagnosing a Loose String

Energy Storage in a Stationary Wave

Resonance Curves and the Driven String

Stationary Waves in Air Columns (Pipes)

Boundary Conditions for Sound in Pipes

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