Wave Equation & Solutions Reference Cheat Sheet

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.