Skip to main content
Physics LibreTexts

6.1: The Wave Equation

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    A wave can be described by a function \(f(x,t)\), called a wavefunction, which specifies the value of a measurable physical quantity at each position \(x\) and time \(t\). For simplicity, we will assume that space is one-dimensional, so \(x\) is a single real number. We will also assume that \(f(x,t)\) is a number, rather than a more complicated object such as a vector. For instance, a sound wave can be described by a wavefunction \(f(x,t)\) representing the air pressure at each point of space and time.

    The evolution of the wavefunction is described by a partial differential equation (PDE) called the time-dependent wave equation: \[\frac{\partial^2 f}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}, \;\;\; v \in\mathbb{R}^+.\] The parameter \(v\), which we take to be a positive real constant, is called the wave speed, for reasons that will shortly become clear.

    Sometimes, we re-arrange the wave equation into the following form, consisting of a linear differential operator acting on \(f(x,t)\): \[\left(\frac{\partial^2}{\partial x^2} - \frac{1}{v^2} \frac{\partial^2}{\partial t^2}\right) \; f(x,t) = 0.\] This way of writing the wave equation emphasizes that it is a linear PDE, meaning that any linear superposition of solutions is likewise a solution.

    This page titled 6.1: The Wave Equation is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

    • Was this article helpful?