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Mathematics LibreTexts

4.5.3: Inhomogeneous Wave Equations

  • Page ID
    2181
  • [ "article:topic", "showtoc:no" ]

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    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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