Wave Equation & Solutions Reference Cheat Sheet

This cheat sheet covers the main equations and solution forms used to describe mechanical and electromagnetic waves. Students need it to connect wave motion, graphs, and physical quantities such as speed, frequency, wavelength, amplitude, and phase. It is especially useful for solving problems involving traveling waves, standing waves, and wave behavior on strings or in other media.

The most important ideas are that wave speed depends on the medium, frequency is set by the source, and wavelength changes when speed changes. A traveling sinusoidal wave is often written as $y(x,t)=A\sin(kx-\omega t+\phi)$ or $y(x,t)=A\sin(kx+\omega t+\phi)$ . The core relationships are $v=f\lambda$ , $k=\frac{2\pi}{\lambda}$ , and $\omega=2\pi f$ .

Standing waves form when waves traveling in opposite directions interfere, creating nodes, antinodes, and allowed frequencies.

Key Facts

The basic wave speed equation is $v=f\lambda$ , where $v$ is speed, $f$ is frequency, and $\lambda$ is wavelength.
The period and frequency are related by $T=\frac{1}{f}$ and $f=\frac{1}{T}$ .
Angular frequency is $\omega=2\pi f=\frac{2\pi}{T}$ , measured in radians per second.
Wave number is $k=\frac{2\pi}{\lambda}$ , measured in radians per meter.
A wave traveling in the positive $x$ direction can be written as $y(x,t)=A\sin(kx-\omega t+\phi)$ .
A wave traveling in the negative $x$ direction can be written as $y(x,t)=A\sin(kx+\omega t+\phi)$ .
The one-dimensional wave equation is $\frac{\partial^2 y}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2 y}{\partial t^2}$ .
For a string fixed at both ends, the allowed wavelengths are $\lambda_n=\frac{2L}{n}$ and the allowed frequencies are $f_n=\frac{nv}{2L}$ for $n=1,2,3,\ldots$ .

Vocabulary

Amplitude: Amplitude is the maximum displacement of a wave from its equilibrium position.
Wavelength: Wavelength is the distance between matching points on a repeating wave, such as crest to crest, and is represented by $\lambda$ .
Frequency: Frequency is the number of complete wave cycles passing a point each second, measured in hertz.
Phase: Phase describes the position of a point in a wave cycle and appears in expressions such as $kx-\omega t+\phi$ .
Node: A node is a point in a standing wave that always has zero displacement.
Antinode: An antinode is a point in a standing wave where the displacement reaches a maximum amplitude.

Common Mistakes to Avoid

Confusing wave speed with particle speed is wrong because $v=f\lambda$ describes how fast the wave pattern moves, not how fast a point in the medium oscillates.
Using $y(x,t)=A\sin(kx+\omega t)$ for a wave moving in the positive $x$ direction is wrong because the plus sign indicates motion in the negative $x$ direction.
Forgetting to convert frequency and period is wrong because $T=\frac{1}{f}$ , so a frequency of $50\,\text{Hz}$ means a period of $0.020\,\text{s}$ , not $50\,\text{s}$ .
Mixing angular frequency and frequency is wrong because $\omega=2\pi f$ , so $\omega$ and $f$ are not the same numerical value unless units and factors of $2\pi$ are handled.
Using any wavelength for a standing wave on a fixed string is wrong because boundary conditions require $\lambda_n=\frac{2L}{n}$ .

Practice Questions

1 A wave has frequency $f=12\,\text{Hz}$ and wavelength $\lambda=0.80\,\text{m}$ . Find its speed using $v=f\lambda$ .
2 A sinusoidal wave is described by $y(x,t)=0.040\sin(6.0x-18t)$ . Find the amplitude $A$ , wave number $k$ , angular frequency $\omega$ , and wave speed $v=\frac{\omega}{k}$ .
3 A string fixed at both ends has length $L=1.20\,\text{m}$ and wave speed $v=48\,\text{m/s}$ . Find the first three allowed frequencies using $f_n=\frac{nv}{2L}$ .
4 Two waves on the same string have equal amplitude and frequency but travel in opposite directions. Explain why a standing wave can form and describe what happens at nodes and antinodes.

Understanding Wave Equation & Solutions Reference

The wave equation expresses a physical balance. On a stretched string, a small curved piece is pulled sideways by tension from its neighbours. Greater curvature produces a greater restoring effect.

The mass of that small piece resists acceleration. These two effects determine how rapidly a disturbance moves along the string. A tight, light string carries a pulse faster than a loose, heavy one.

This is why the material and conditions matter. In air, sound speed depends mainly on temperature.

In a solid, stiffness and density are important. The equation works for many wave types because each has a restoring influence and inertia, though the physical details differ.

A sinusoidal expression contains more information than a graph at one instant. Amplitude tells the largest displacement from equilibrium. Phase tells where a point is in its repeating motion.

Points with the same phase move together, while points separated by half a wavelength move in opposite directions. To identify travel direction, hold the phase constant and follow a crest over time. A form containing wave number times position minus angular frequency times time moves toward increasing position.

Reversing that sign changes the direction. Students often confuse the vertical motion of a bit of string with the horizontal motion of the wave. The string section moves up and down, while the pattern carries energy along the string.

Superposition means that overlapping disturbances add their displacements at every location. Two small pulses can make a larger pulse during overlap, then continue onward with their original shapes in an ideal medium. This does not mean that matter travels from one pulse to the other.

It means the medium responds to the combined displacement. Noise cancelling headphones use this idea by producing sound with a carefully chosen phase. Water ripples, musical sounds, and radio signals can interfere too.

Constructive interference occurs when displacements reinforce. Destructive interference occurs when they oppose. Complete cancellation is possible only when the amplitudes match at that place and time.

Standing waves need special attention because their pattern does not seem to travel. They result from repeated interference between waves moving in opposite directions, usually after reflection at boundaries. Nodes remain at zero displacement.

Antinodes have the greatest motion. Energy still moves back and forth locally, even though there is no steady energy flow along the whole pattern. A guitar string supports only patterns that fit its fixed length exactly.

The lowest pattern is the fundamental. Higher patterns are harmonics and produce higher pitches. When solving these problems, sketch the endpoints first, count the loops or segments, then connect that shape to the wavelength.

Check units throughout. Frequency is measured in hertz, wavelength in metres, speed in metres per second, and phase quantities are measured in radians.

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Wave Equation & Solutions Reference Cheat Sheet

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