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 $\overline{\Omega}\times\mathbb{R}^1$.

$\Box$

Contributors and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.