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Mathematics LibreTexts

4.1: One-Dimensional Wave Equation

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

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