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Mathematics LibreTexts

4.5.3: Inhomogeneous Wave Equations

Let \(\Omega\subset\mathbb{R}^n\) be a bounded and sufficiently regular domain. In this section we consider the initial-boundary value problem

\begin{eqnarray}
\label{waveinh1}\tag{4.5.3.1}
u_{tt}&=&Lu+f(x,t)\ \ \mbox{in}\ \Omega\times\mathbb{R}^1\\
\label{waveinh2} \tag{4.5.3.2}
u(x,0)&=&\phi(x)\ \ x\in\overline{\Omega}\\
\label{waveinh3} \tag{4.5.3.3}
u_t(x,0)&=&\psi(x)\ \ x\in\overline{\Omega}\\
\label{waveinh4} \tag{4.5.3.4}
u(x,t)&=&0\ \ \mbox{for} \ x\in\partial\Omega\ \mbox{and}\ t\in\mathbb{R}^1,
\end{eqnarray}

where \(u=u(x,t)\), \(x=(x_1,\ldots,x_n)\), \(f,\ \phi,\ \psi\) are given and \(L\) is an elliptic differential operator. Examples for \(L\) are:

  1. \(L=\partial^2/\partial x^2\), oscillating string.
  2.  \(L=\triangle_x\), oscillating membrane.
  3. $$Lu=\sum_{i,j=1}^n\frac{\partial}{\partial x_j}\left(a^{ij}(x)u_{x_i}\right),$$

where \(a^{ij}=a^{ji}\) are given sufficiently regular functions defined on \(\overline{\Omega}\). We assume \(L\) is uniformly elliptic, that is, there is a constant \(\nu>0\) such that

$$\sum_{i,j=1}^na^{ij}\zeta_i\zeta_j\ge\nu|\zeta|^2$$

for all \(x\in\Omega\) and \(\zeta\in\mathbb{R}^n\).

4. Let \(u=(u_1,\ldots,u_m)\) and

$$Lu=\sum_{i,j=1}^n\frac{\partial}{\partial x_j}\left(A^{ij}(x)u_{x_i}\right),$$

where \(A^{ij}=A^{ji}\) are given sufficiently regular \((m\times m)\)-matrices on \(\overline{\Omega}\). We assume that \(L\) defines an elliptic system. An example for this case is the linear elasticity.

Consider the eigenvalue problem

\begin{eqnarray}
\label{osceigen1} \tag{4.5.3.5}
-Lv&=&\lambda v\ \ \mbox{in}\ \Omega\\
\label{osceigen2} \tag{4.5.3.6}
v&=&0\ \ \mbox{on}\ \partial\Omega.
\end{eqnarray}

Assume there are infinitely many eigenvalues

$$0<\lambda_1\le\lambda_2\le\ldots\ \to\infty$$

and a system of associated eigenfunctions \(v_1,\ v_2,\ldots\) which is complete and orthonormal in \(L^2(\Omega)\). This assumption is satisfied if \(\Omega\) is bounded and if \(\partial\Omega\) is sufficiently regular.

For the solution of (\ref{waveinh1})-(\ref{waveinh4}) we make the ansatz

\begin{equation}
\label{oscinh1} \tag{4.5.3.7}
u(x,t)=\sum_{k=1}^\infty v_k(x)w_k(t),
\end{equation}

with functions \(w_k(t)\) which will be determined later. It is assumed that all series are convergent and that following calculations make sense.

Let

\begin{equation}
\label{oscinh2} \tag{4.5.3.8}
f(x,t)=\sum_{k=1}^\infty c_k(t)v_k(x)
\end{equation}

be Fourier's decomposition of \(f\) with respect to the eigenfunctions \(v_k\). We have

\begin{equation}
\label{oscinh3} \tag{4.5.3.9}
c_k(t)=\int_\Omega\ f(x,t)v_k(x)\ dx,
\end{equation}

which follows from (\ref{oscinh2}) after multiplying with \(v_l(x)\) and integrating over \(\Omega\).

Set

$$\langle\phi,v_k\rangle=\int_\Omega\ \phi(x)v_k(x)\ dx,$$

then

\begin{eqnarray*}
\phi(x)&=&\sum_{k=1}^\infty\langle\phi,v_k\rangle v_k(x)\\
\psi(x)&=&\sum_{k=1}^\infty\langle\psi,v_k\rangle v_k(x)
\end{eqnarray*}

are Fourier's decomposition of \(\phi\) and \(\psi\), respectively.

In the following we will determine \(w_k(t)\), which occurs in ansatz (\ref{oscinh1}), from the requirement that \(u=v_k(x)w_k(t)\) is a solution of

$$u_{tt}=Lu+c_k(t)v_k(x)$$

and that the initial conditions

$$w_k(0)=\langle\phi,v_k\rangle,\ \ \ w_k'(0)=\langle\psi,v_k\rangle$$

are satisfied. From the above differential equation it follows

$$w_k''(t)=-\lambda_kw_k(t)+c_k(t).$$

Thus

\begin{eqnarray}
\label{oscinh4} \tag{4.5.3.10}
w_k(t)&=&a_k\cos(\sqrt{\lambda_k}t)+b_k\sin(\sqrt{\lambda_k}t)\\
&&+\frac{1}{\sqrt{\lambda_k}}\int_0^t\ c_k(\tau)\sin(\sqrt{\lambda_k}(t-\tau))\ d\tau,\nonumber
\end{eqnarray}

where

$$ a_k=\langle\phi,v_k\rangle,\ \ \ b_k=\frac{1}{\sqrt{\lambda_k}}\langle\psi,v_k\rangle.$$

Summarizing, we have

Proposition 4.2. The (formal) solution of the initial-boundary value problem (\ref{waveinh1})-(\ref{waveinh4}) is given by

\[u(x,t)=\sum_{k=1}^\infty v_k(x)w_k(t),\]

where \(v_k\) is a complete orthonormal system of eigenfunctions of (\ref{osceigen1}), (\ref{osceigen2}) and the functions \(w_k\) are defined by (\ref{oscinh4}).

The Resonance Phenomenon

Set in (\ref{waveinh1})-(\ref{waveinh4}) \(\phi=0\), \(\psi=0\) and assume that the external force \(f\) is periodic and is given by

$$f(x,t)=A\sin(\omega t)v_n(x),$$

where \(A,\ \omega\) are real constants and \(v_n\) is one of the eigenfunctions of (\ref{osceigen1}), (\ref{osceigen2}). It follows

$$c_k(t)=\int_\Omega\ f(x,t)v_k(x)\ dx=A\delta_{nk}\sin(\omega t).$$

Then the solution of the initial value problem (\ref{waveinh1})-(\ref{waveinh4}) is
\begin{eqnarray*}
u(x,t)&=&\frac{Av_n(x)}{\sqrt{\lambda_n}}\int_0^t\ \sin(\omega\tau)\sin(\sqrt{\lambda_n}(t-\tau))\ d\tau\\
&=&Av_n(x)\frac{1}{\omega^2-\lambda_n}\left(\frac{\omega}{\sqrt{\lambda_n}}\sin(\sqrt{\lambda_k}t)-\sin(\omega t)\right),
\end{eqnarray*}
provided \(\omega\not=\sqrt{\lambda_n}\). It follows

$$u(x,t)\to\frac{A}{2\sqrt{\lambda_n}}v_n(x)\left(\frac{\sin(\sqrt{\lambda_n}t)}{\sqrt{\lambda_n}}-t\cos(\sqrt{\lambda_n} t)\right)$$

if \(\omega\to\sqrt{\lambda_n}\). The right hand side is also the solution of the initial-boundary value problem if \(\omega=\sqrt{\lambda_n}\).

Consequently \(|u|\) can be arbitrarily large at some points \(x\) and at some times \(t\) if \(\omega=\sqrt{\lambda_n}\). The frequencies \(\sqrt{\lambda_n}\) are called critical frequencies at which resonance occurs.

A Uniqueness Result

The solution of the initial-boundary value problem (\ref{waveinh1})-(\ref{waveinh4}) is unique in the class \(C^2(\overline{\Omega}\times\mathbb{R}^1)\).

Proof. Let \(u_1\), \(u_2\) are two solutions, then \(u=u_2-u_1\) satisfies

\begin{eqnarray*}
u_{tt}&=&Lu\ \ \mbox{in}\ \Omega\times\mathbb{R}^1\\
u(x,0)&=&0\ \ x\in\overline{\Omega}\\
u_t(x,0)&=&0\ \ x\in\overline{\Omega}\\
u(x,t)&=&0\ \ \mbox{for} \ x\in\partial\Omega\ \mbox{and}\ t\in\mathbb{R}^n.
\end{eqnarray*}

As an example we consider Example 3 from above and set

$$E(t)=\int_\Omega\ (\sum_{i,j=1}^na^{ij}(x)u_{x_i}u_{x_j}+u_tu_t)\ dx.$$

Then
\begin{eqnarray*}
E'(t)&=&2\int_\Omega\ (\sum_{i,j=1}^na^{ij}(x)u_{x_i}u_{x_jt}+u_tu_{tt})\ dx\\
&=&2\int_{\partial\Omega}\ (\sum_{i,j=1}^na^{ij}(x)u_{x_i}u_tn_j)\ dS\\
&&+2\int_\Omega\ u_t(-Lu+u_tt)\ dx\\
&=&0.
\end{eqnarray*}

It follows \(E(t)=const.\) From \(u_t(x,0)=0\) and \(u(x,0)=0\) we get \(E(0)=0\). Consequently \(E(t)=0\) for all \(t\), which implies, since \(L\) is elliptic, that \(u(x,t)=const.\) on \(\overline{\Omega}\times\mathbb{R}^1\). Finally, the homogeneous initial and boundary value conditions lead to \(u(x,t)=0\) on \(\overline{\Omega}\times\mathbb{R}^1\).

\(\Box\)

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