= Solution
For a <material volume>, <mass conservation> is $d\int_{V(t)}\rho\,dV/dt=0$. The transport theorem and arbitrariness of that volume give $\boxed{\rho_t+\nabla\cdot(\rho u)=0}$. <Balance of linear momentum> includes the <pressure> <traction> and <body force>:
$$
\frac d{dt}\int_{V(t)}\rho u\,dV=-\int_{\partial V(t)}pn\,dS+\int_{V(t)}\rho g\,dV.
$$
Apply transport, the <divergence theorem> and the mass equation to obtain
$$
\boxed{u_t+(u\cdot\nabla)u=-\frac1\rho\nabla p+g.}
$$
For small <perturbations> of the uniform resting barotropic state, write $\rho=\rho_0+\rho'$, $p=p_0+c_0^2\rho'+\cdots$. Linearized mass and momentum are
$$
\rho'_t+\rho_0\nabla\cdot u=0,\qquad
u_t=-\frac{c_0^2}{\rho_0}\nabla\rho',\qquad\boxed{c_0^2=P'(\rho_0)>0.}
$$
Taking the <curl> shows that initially vanishing <vorticity> remains zero, so the acoustic <velocity> has a potential $u=\nabla\phi$. Integrating momentum, using the <additive freedom of a velocity potential> to remove the spatially constant integration term, gives $\rho'=-(\rho_0/c_0^2)\phi_t$. Continuity now yields $\boxed{\phi_{tt}=c_0^2\nabla^2\phi}$.
For <spherical symmetry>, set $\phi=e^{-i\omega t}F(r)$ and $k=\omega/c_0$. The radial equation is $(r^2F')'+k^2r^2F=0$, or $(rF)''+k^2(rF)=0$. Hence
$$
\boxed{\phi=\frac{e^{-i\omega t}}r[A\cos kr+B\sin kr].}
$$
Rigid boundaries require radial <velocity> $F'=0$ at both radii. A form automatically satisfying the condition at $a$ is
$$
F(r)=\frac C r\left[\cos k(r-a)+\frac1{ka}\sin k(r-a)\right].
$$
At $b$, the condition $b(rF)'-(rF)=0$ becomes
$$
k(b-a)\cos k(b-a)-(1+k^2ab)\sin k(b-a)=0.
$$
Therefore the nonzero oscillatory eigenfrequencies obey
$$
\boxed{(1+k^2ab)\tan k(b-a)=k(b-a),\qquad\omega=kc_0.}
$$
The undivided <sine>-<cosine> form avoids artificial issues from dividing by a <cosine>. The formal $k=0$ root is the spatially constant zero-frequency potential <additive freedom of a velocity potential>, not a propagating acoustic oscillation. This is the <radial acoustic spectrum between rigid concentric spheres>.
Back to article page
