becomes the sum of incoming and outgoing waves when $r\to \infty$ then the spectrum of the variable $k^2$ in the Helmholtz equation (\[2-3\]) will be positive and continuous.

Quasi-normal modes of a dielectric sphere
=========================================

The same situation, with regard to quasi-normal modes, takes place when we consider the oscillations of compound unbounded media. In this case in both the regions $D_{\text{in}}$ and $D_{\text{ex}}$ the wave equations are defined $$\begin{aligned}
\left ( \Delta -\frac{1}{c_{\text{in}}^2}\frac{\partial }{\partial
t^2} \right ) u_{\text{in}}(t,\mathbf{x}) &=& 0, \quad
\mathbf{x}\in D_{\text{in}}\,{,} \label{2-14}\\ \left ( \Delta
-\frac{1}{c_{\text{ex}}^2}\frac{\partial }{\partial t^2} \right )
u_{\text{ex} }(t,\mathbf{x}) &=& 0, \quad \mathbf{x}\in D_{\text{
ex}} \label{2-15}\end{aligned}$$ with the matching conditions at the interface $S$, for example, of the following kind $$\begin{aligned}
  u_{\text{ in}}(t,\mathbf{x})&=&u_{\text{ex}}(t,
\mathbf{x}){,} \label{2-16}
\\
\lambda_{\text{ in}}\frac{\partial u_{\text{in
}}(t,\mathbf{x})}{\partial n_{\text{in
}}(\mathbf{x})}&=&\lambda_{\text{ ex}}\frac{\partial u_{\rm
ex}(t,\mathbf{x})}{\partial n_{\text{ ex }}(\mathbf{x})}, \quad
\mathbf{x}\in S\,{,} \label{2-17}\end{aligned}$$ where $n_{\text{in}}(\textbf{x})$ and $n_{\text{ ex}}(\textbf{x})$ are the normals to the surface $S$ at the point $\textbf{x}$ for the regions $D_{\text{in}}$ and $D_{\text{ex}}$, respectively. The parameters $c_{\text{in}}$, $c_{\text{ex}}$, $\lambda
_{\text{in}}$, and $\lambda _{\text{ex}}$ specify the material characteristics of the media. At the spatial infinity the solution $u_{\text{ex}}(t, \mathbf{x})$ should satisfy the radiation conditions (\[2-5\]). For real $k$ we again have only zero solution 