3.3.1.3 - Superposition and Stationary Waves

Superposition

The starting point is the principle of superposition. If two waves meet at the same place at the same time, the resultant displacement is found by adding the individual displacements.

If the two displacements are in the same direction, the waves reinforce each other and the resultant displacement is larger. This is constructive interference. If the displacements are in opposite directions, the waves partially or completely cancel. This is destructive interference.
In the figure, notice that the resultant pulse at the overlap is the sum of the individual displacements at that instant.

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Take upwards as positive. If one pulse gives $+3\,\mathrm{mm}$ and another $-5\,\mathrm{mm}$ at the same point, the resultant is $-2\,\mathrm{mm}$, not $8\,\mathrm{mm}$. Add corresponding positions, not the separate amplitudes regardless of sign. In a linear medium the original waves continue after overlap; destructive interference does not destroy them.

Formation of stationary waves

A stationary wave is formed when two progressive waves of the same frequency, wavelength, and amplitude travel in opposite directions through the same region. On a stretched string, one wave is usually the incident wave and the other is the reflected wave.

Follow corresponding positions in the three snapshots. At the first instant the two component shapes coincide, so the resultant displacement is twice either one. After a quarter period each component has advanced a quarter wavelength in its own direction: their displacements are now equal and opposite everywhere, and the string passes through its equilibrium line. After half a period the resultant shape is inverted.

Zero displacement at one instant does not make every point a node. Nodes are only the positions where the component displacements cancel at every instant. Other points pass through zero while moving.

The points of maximum oscillation are called antinodes.

Adjacent nodes are separated by $\lambda/2$, and the same is true for adjacent antinodes. Non-node particles between a given pair of adjacent nodes oscillate in phase, although their amplitudes differ. Particles in neighbouring loops are in antiphase. A node does not oscillate, so assigning it an oscillation phase is meaningless.
In the figure, follow how the two travelling waves combine at different instants to leave fixed nodes and antinodes in the resultant stationary pattern.

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The two progressive waves carry energy in opposite directions at equal average rates, so their energy transfers cancel and there is no net energy transfer along the stationary pattern. Energy is still present in the oscillating medium; a stationary wave is not a wave with no energy.

This is the ideal undamped pattern. In a driven laboratory string, energy supplied by the driver replaces losses, so the equal-amplitude model is an approximation.

Harmonics on strings

For a string fixed at both ends, there must always be a node at each end. That restriction means only certain wavelengths fit on the string, so stationary waves appear only at certain frequencies.

In the first harmonic, the string contains one loop, with an antinode in the middle and a node at each end. Since the string length $L$ is half a wavelength, $\lambda_1 = 2L$ and the frequency is $f_1 = c/(2L)$.

In the second harmonic, the string contains two loops, so $L = \lambda_2$. In the third harmonic, there are three loops, and so on. In this course, you should describe these patterns as first harmonic, second harmonic, third harmonic, and so forth.

Here, $n = 1, 2, 3, \dots$ is the harmonic number, $c$ is the wave speed on the string, and $L$ is the vibrating length of the string.
In the figure, notice how each higher harmonic fits an extra half-wavelength into the same string length while keeping nodes at both fixed ends.

[DIAGRAM: asset_name: 3.1.3 - Principle of Superposition of Waves and Formation of Stationary Waves - Diagram 3; asset_slug: 3.1.3 - Principle of Superposition of Waves and Formation of Stationary Waves - Diagram 3; recommended_method: retained_png; caption: The first three harmonics have one, two and three loops; description: Three strings of equal length fixed at both ends. First harmonic with one loop, second harmonic with two loops, third harmonic with three loops. Mark nodes and antinodes, and label the wavelength relationship for each pattern.]

As the harmonic number increases, more loops fit onto the string, the wavelength becomes shorter, and the frequency becomes higher.

For example, a $1.20\,\mathrm{m}$ string showing three loops contains three half-wavelengths: $\lambda=2(1.20)/3=0.80\,\mathrm{m}$. If the driving frequency is $150\,\mathrm{Hz}$, $c=f\lambda=120\,\mathrm{m\,s^{-1}}$. There are four nodes including the two ends, and three antinodes. The paired curves show opposite extreme positions, not two separate strings.

First harmonic formula

The wave speed on a stretched string depends on the tension $T$ and the mass per unit length $\mu$ of the string. A larger tension increases the speed, while a larger mass per unit length decreases it.

Substituting this into the first-harmonic expression $f_1 = c/(2L)$ gives the formula required by the specification.

In this equation, $f$ is the first harmonic frequency, $L$ is the vibrating length of the string, $T$ is the tension, and $\mu$ is the mass per unit length.

Here $T$ means tension in newtons, not the period used in $f=1/T$ elsewhere. The mass per unit length $\mu$ is the string's mass divided by its full measured length, in $\mathrm{kg\,m^{-1}}$. At fixed length and string type, multiplying tension by four doubles $f_1$. At fixed length and tension, doubling $\mu$ divides $f_1$ by $\sqrt2$. These follow from the square root; frequency is not directly proportional to tension.

Microwaves and sound

Stationary waves are not confined to strings. If microwaves are directed at a metal reflector, the reflected microwaves superpose with the incident microwaves and form a stationary pattern. A detector moved along the line of the beam finds equally spaced minima and maxima. The spacing between adjacent minima, or adjacent maxima, is $\lambda/2$.

For sound, a loudspeaker and a reflecting surface can produce a stationary pattern. In a suitable tube, fine powder gathers near displacement nodes, where air motion is smallest. A microphone instead responds to pressure variation: a displacement node is a pressure antinode, so a microphone signal maximum does not identify a displacement antinode. Adjacent pressure maxima are nevertheless also $\lambda/2$ apart. State what a detector measures before interpreting its readings.

These examples matter because they show that stationary waves are a general consequence of superposition, not a special trick that only works on strings.