= Solution
Choose field $a$ to be $(\widehat{\mathbf u}^a)^*$ at <frequency> $-\omega_a$, and field $b$ to be $\widehat{\mathbf u}^b$ at $\omega_b$. Zero <body forces> and zero boundary <tractions> make all source and surface terms vanish. Therefore
$$
(\omega_b^2-\omega_a^2)\int_{\mathcal D}\rho(\widehat u_i^a)^*\widehat u_i^b\,dV=0.
$$
For distinct squared <frequencies> this proves \b[weighted <orthogonality>],
$$
\boxed{\langle\widehat{\mathbf u}^a,\widehat{\mathbf u}^b\rangle_\rho=\int_{\mathcal D}\rho(\widehat u_i^a)^*\widehat u_i^b\,dV=0.}
$$
Assume positive <mass density> bounded above and below and finite-energy fields in the ordinary bounded domain. The weighted space $H=L^2(\mathcal D,\rho\,dV)^3$ is a <separable Hilbert space>. Select one unit <eigenfunction> for each distinct squared <frequency>. They form an <orthonormal set>. Let $\{d_j\}$ be a countable dense subset of $H$; assign to each <eigenfunction> a point $d_j$ within distance $1/3$. Two <orthogonal> unit <eigenfunctions> have distance $\sqrt2$, so the same $d_j$ cannot be assigned to both. This gives an injection into a countable set.
Hence there are \b[at most countably many squared <eigenfrequencies>], and at most twice that many signed <frequencies>. Multiplicity at one <frequency> does not change the argument. A zero <frequency> associated with rigid translations or rotations is included. This proves the <countability of elastic eigenfrequencies> directly; proving discreteness or finite multiplicity would require additional spectral regularity arguments.
Back to article page
