1.4. Time-harmonic waves

Example 1.2

Consider the following simple one-dimensional wave example to find \(u:[0,T]\times[0,\infty)\to\mathbb R\) such that

(1.15)\[\begin{split}\begin{aligned} \partial_t^2 u-c^2\partial_x^2 u &= 0,&& \text{in }(0,T)\times (0,\infty),\\ u(\cdot,0) &= a,&&\text{in } (0,T],\\ u(0,\cdot ) &=0,&&\text{in } (0,\infty),\\ \partial_t u(0,\cdot ) &=0,&&\text{in } (0,\infty), \end{aligned}\end{split}\]

for a smooth function \(a:[0,\infty)\to\mathbb R\) with \(a(0)=a'(0)=a''(0)=0\). Then the (unique, strong) solution is given by

(1.16)\[\begin{split}u(t,x) = \begin{cases}a(t-x/c),&ct> x,\\ 0,&ct\leq x\end{cases}.\end{split}\]

More specifically consider \(a\) to be a “smoothened” cosine, such that \(a(t) = \cos(-\omega t)\) for \(t>\varepsilon\) and some \(\omega>0\) and \(a(0)=a'(0)=a''(0)=0\). Then for any bounded interval \([0,b]\) for some \(b>0\) we have that for \(t>b/c+\varepsilon\) the solution \(u\) can be written as

\[u(t,x) = \cos(-\omega(t-x/c))=\Re(\exp(-i\omega t)\exp(i\omega/c x))\]

In applications often scenarios are of interest where the solutions are (approximately) time-harmonic waves, i.e., are(approximately) of the form

(1.17)\[u(t,x) = \Re\left(\exp(-i\omega t) \hat u(x)\right),\]

where \(\omega >0\) is a given angular frequency and \(u(x)\) the amplitude as function in space.

Thus it is not necessary to simulate the full time-domain systems from Section 1.1 Section 1.2 Section 1.3 which in their second-order form can all be stated as

(1.18)\[\begin{aligned} \frac{1}{c^2}\partial_t^2 u-Du &= f \end{aligned}\]

where \(D\) is one of the second-order differential operators in space from the previous sections. For simplicity we assume homogeneous boundary conditions. Inserting the time harmonic wave (1.17) into (1.18) for a time harmonic right hand side

\[f(t,x)=\exp(-i\omega t)\hat f(x)\]

leads to the equation

(1.19)\[-\frac{\omega^2}{c^2} \hat u-D\hat u = \hat f,\]

Remark 1.3

again with homogeneous boundary conditions. Note that we neglected the initial conditions which are necessary to close the time-domain system (1.18). They have to be chosen fitting to a time-harmonic solution.

In the case of acoustic waves (i.e., \(D=\Delta\)) with a constant wave speed \(c\) this leads to the inhomogenous Helmholtz equation

(1.20)\[-k^2\hat u-\Delta\hat u = \hat f,\]

where the wave number \(k>\) is defined by \(k=\omega/c\). When exclusively the time-harmonic regime is of interest the \(\hat\cdot\) is usually ommitted.

Remark 1.4

A more general way to derive the time-harmonic counterparts to the time-domain equations is to introduce the Fourier transform of a function \(h\in L^1(\mathbb R)\) by

\[\hat h(\omega) = \int_{-\infty}^\infty h(t)\exp(-i\omega t) dt,\]

and apply it to the space-time solutions of the time-domain wave problems (with respect to the time variable).

Remark 1.5

Note that the sign convention to use \(\exp(-i\omega t)\) in the Fourier transform and the time harmonic ansatz (1.17) is arbitrary. Flipping the sign does not change the resulting time-harmonic equation. However, it corresponds to a time-reversal, thus whenever time-domain solutions, first-order time derivatives or (non-squared) factors \(\omega\) are used in any reasoning this has to be taken into account.

(Non)-locality

Example 1.3 (ctd.)

Considering again the one-dimensional example Example 1.2 we immediately obtain from (1.16) that for fixed \(t_0>0\) we have \(\mathrm{supp}(u(t_0,\cdot)) \subset [0,ct_0)\).

Looking e.g., at D’Alembert’s solution for the one-dimensional wave equation or the example above it is comprehensible that time-domain waves have a finite speed of propagation, e.g., for initial conditions with local support inside of a compact set \(\Omega\) in space and a finite time interval \([0,T]\) there exists another compact set \(\Omega'\) in space such that for every \(t\in [0,T]\) the support of the solution is contained in \(\Omega'\).

This can be made mathematically rigorous by looking at fundamental solutions of the respective time-domain wave equations. Similar properties hold for a compactly supported (in space) forcing term.

Thus, mathematically, the problem of solving the wave-equation in the free space for finite times can always be re-stated as solving the wave-equation on a large enough bounded domain with homogeneous boundary conditions. This heuristic argument already points out that for mathematical formulations (and numerical approximations) of time-domain waves one can always avoid the difficulties that unbounded domains cause (see example below).

Example 1.4 (ctd.)

The time-harmonic counterpart to (1.15) can be stated as

\[\begin{split}-\omega^2/c^2 u-u'' &= 0,&&\text{in }(0,\infty),\\ u(0)&=1,\end{split}\]

However, although this equation is solved by \(u(x)=\exp(i\omega/c x)\) the solution is not unique since also \(u_-(x)=\exp(-i\omega/c x)\) (and every linear combination of \(u,u_-\)) is also a solution. Thus the time-harmonic equation needs to be supplied with another (boundary) condition to obtain well-posedness.

Transferring the solution \(u_-\) back to time-domain, i.e.,

\[\Re(\exp(-i\omega t)u_-(x))=\Re(\exp(-i\omega(t+x/c)))=\cos(\omega(t+x/c))\]

shows that \(u_-\) corresponds to a time-harmonic wave travelling from right to left.

Sommerfeld radiation condition

To obtain unique solvability of the time-harmonic equations stated above we need to pose an additional condition. As motivated in Example 1.2 and Example 1.4 this condition should separate incoming from outgoing waves.

The sought condition is the so-called Sommerfeld radiation condition named after Arnold Sommerfeld which is given by

(1.21)\[\lim_{|x|\to\infty} |x|^{(d-1)/2}\left(\frac{\partial u}{\partial|x|}-iku\right) = 0\]

where \(d\) is the spacial dimension.

In two or three dimensions it can be motivated similar to the one-dimensional case by looking at an explicit representation of the solution.

Scattering by an obstacle

A typical application of (1.20) on an unbounded domain is the so-called scattering problem. Thereby an incident field \(u^i\) on \(\mathbb R^d\) which can be e.g. a plane wave

\[u^i(t,x) = \exp(ik e\cdot x-\omega t)\]

for a given vector \(e\in\mathbb R^d\) with \(|e|=1\) and an angular frequency \(\omega>0\) and a compactly supported obstacle \(\Omega\subset\mathbb R^d\) is given. The total field \(u:\mathbb R^d\setminus \Omega\) can be decomposed into the incident field \(u^i\) and a scattered field \(u^s\) and should satisfy

\[\begin{split}\begin{aligned} -k^2 u-\Delta u&=0,&\text{on }&\mathbb R^d\setminus\bar\Omega,\\ u &= u^i+u^s,&\text{on }&\mathbb R^d\setminus\bar\Omega,\\ u&=0,&\text{on }&\partial\Omega,\\ \lim_{|x|\to\infty} |x|^{(d-1)/2}\left(\frac{\partial u^s}{\partial|x|}-iku^s\right) &= 0. \end{aligned}\end{split}\]

By linearity it is clear that the scattered field also satisfies the Helmholtz equation with inhomogeneous Dirichlet boundary conditions on \(\partial\Omega\) and the Sommerfeld radiation condition for \(|x|\to\infty\), i.e.,

\[\begin{split}\begin{aligned} -k^2 u^s-\Delta u^s&=0,&&\text{on }&\mathbb R^d\setminus\bar\Omega,\\ u^s&=-u^i,&&\text{on }&\partial\Omega,\\ \lim_{|x|\to\infty} |x|^{(d-1)/2}\left(\frac{\partial u^s}{\partial|x|}-iku^s\right) &= 0. \end{aligned}\end{split}\]

The resonance problem

Resonance problems in general are problems where not only the function \(u\) but also the frequency \(\omega\) is unknown. This usually lead to eigenvalue problems.

Resonances on bounded domains

Let \(\Omega\subset \mathbb R^d\) be a bounded convex Lipschitz domain. Then it is well-known that the spectrum of the negative Laplacian (as operator on \(L^2\)) consists of a countable set of positive real numbers \(\lambda_j\) and the according eigenvectors \(u_j\) form a complete orthonormal system of \(L^2(\Omega)\) (we assume that \(u_j\) are normalized). Thus we may expand the solution \(\hat u\) of the Helmholtz problem (1.20) and the right-hand side \(\hat f\) with respect to \(u_j\) to obtain

(1.22)\[-k^2\sum_{j=0}^\infty c_j u_j+\lambda_j u_j = \sum_{j=0}^\infty d_j u_j\]

where the coefficients are given by

\[c_j = (u,u_j)_{L^2(\Omega)},\quad d_j = (f, u_j)_{L^2(\Omega)}.\]

Testing (1.22) with \(u_l\) yields due to the orthonormality that

\[c_l = \frac{d_l }{\lambda_l-k^2} = \frac{(f,u_l)_{L^2(\Omega)}}{(\lambda_l-k^2}\]

and thus

\[\hat u = \sum_{j=0}^\infty \frac{(f,u_j)_{L^2(\Omega)}}{(\lambda_j-k^2}u_j.\]

If we now choose \(k^2\approx \lambda_j\) for some \(j\) we obtain that \(\hat u\) is dominated by the eigenfunction \(u_j\), i.e.,

(1.23)\[\hat u \approx \frac{(f,u_j)_{L^2(\Omega)}}{(\lambda_j-k^2}u_j.\]

Resonances on unbounded domains

If we modify the Helmholtz resonance problem by posing it on an unbounded domain, the negative Laplacian is in general no self-adjoint, compact operator any more, hence we cannot use the same reasoning as above. However, if there exists an eigenvalue close to the given wavenumber \(k^2\) the approximation (1.23) still remains valid. Thus, it is an interesting problem to look for acoustic resonances on unbounded domains. The most interesting ones are usually the ones that are close to or lie on the real axis.