Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

4.1: One-Dimensional Wave Equation

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

    The one-dimensional wave equation is given by

    where \(u=u(x,t)\) is a scalar function of two variables and \(c\) is a positive constant. According to previous considerations, all \(C^2\)-solutions of the wave equation are


    with arbitrary \(C^2\)-functions \(f\) and \(g\)

    The Cauchy initial value problem for the wave equation is to find a \(C^2\)-solution of


    where \(\alpha,\ \beta\in C^2(-\infty,\infty)\) are given.

    Theorem 4.1. There exists a unique \(C^2(\mathbb{R}^1\times\mathbb{R}^1)\)-solution of the Cauchy initial value problem, and this solution is given by d'Alembert's1 formula

    u(x,t)=\dfrac{\alpha(x+ct)+\alpha(x-ct)}{2}+\dfrac{1}{2c}\int_{x-ct}^{x+ct}\ \beta(s)\ ds.

    Proof. Assume there is a solution \(u(x,t)\) of the Cauchy initial value problem, then it follows from (\ref{wavegen}) that


    From (\ref{ini1}) we obtain


    which implies, together with (\ref{ini2}), that

    \label{12a} f'(x)&=&\dfrac{\alpha'(x)+\beta(x)/c}{2}\\


    f(x)&=&\dfrac{\alpha(x)}{2}+\dfrac{1}{2c}\int_0^x\ \beta(s)\ ds +C_1\\
    g(x)&=&\dfrac{\alpha(x)}{2}-\dfrac{1}{2c}\int_0^x\ \beta(s)\ ds +C_2.

    The constants \(C_1\), \(C_2\) satisfy


    see (\ref{ini1}). Thus each \(C^2\)-solution of the Cauchy initial value problem is given by d'Alembert's formula. On the other hand, the function \(u(x,t)\) defined by the right hand side of (\ref{waveform}) is a solution of the initial value problem.


    Corollaries. 1. The solution \(u(x,t)\) of the initial value problem depends on the values of \(\alpha\) at the endpoints of the interval \([x-ct,x+ct]\) and on the values of \(\beta\) on this interval only, see Figure 4.1.1. The interval \([x-ct,x+ct]\) is called {\it domain of dependence}.

     Interval of dependence

    Figure 4.1.1: Interval of dependence

    2. Let \(P\) be a point on the \(x\)-axis. Then we ask which points \((x,t)\) need values of \(\alpha\) or \(\beta\) at \(P\) in order to calculate \(u(x,t)\)? From the d'Alembert formula it follows that this domain is a cone, see Figure 4.2.1. This set is called domain of influence.

    Domain of influence

    Figure 4.2.1: Domain of influence

    1 d'Alembert, Jean Babtiste le Rond, 1717-1783