  The aim of the paper is to study the problem
  $u_{tt}-c^2\Delta u=0$ in $\mathbb{R}\times\Omega$,
  $\mu v_{tt}- \text{div}_\Gamma (\sigma \nabla_\Gamma v)+\delta v_t+\kappa
v+\rho u_t =0$ on $\mathbb{R}\times \Gamma_1$,
  $v_t =\partial_\nu u$ on $\mathbb{R}\times \Gamma_1$,$\partial_\nu u=0$ on
$\mathbb{R}\times \Gamma_0$,
  $u(0,x)=u_0(x)$ and $u_t(0,x)=u_1(x)$ in $\Omega$,
  $v(0,x)=v_0(x)$ and $v_t(0,x)=v_1(x)$ on $\Gamma_1$, where $\Omega$ is a open
domain of $\mathbb{R}^N$ with uniformly $C^r$ boundary ($N\ge 2$, $r\ge 1$),
$\Gamma=\partial\Omega$, $(\Gamma_0,\Gamma_1)$ is a relatively open partition
of $\Gamma$ with $\Gamma_0$ (but not $\Gamma_1$) possibly empty. Here
$\text{div}_\Gamma$ and $\nabla_\Gamma$ denote the Riemannian divergence and
gradient operators on $\Gamma$, $\nu$ is the outward normal to $\Omega$, the
coefficients $\mu,\sigma,\delta, \kappa, \rho$ are suitably regular functions
on $\Gamma_1$ with $\rho,\sigma$ and $\mu$ uniformly positive while $c$ is a
positive constant. This problem have been proposed long time ago by Beale and
Rosencrans, when $N=3$, $\sigma=0$, $r=\infty$, $\rho$ is constant,
$\kappa,\delta\ge 0$, to model acoustic wave propagation with locally reacting
boundary.
  In this paper we first study well-posedness in the natural energy space and
give regularity results. Hence we give precise qualitative results for
solutions when $\Omega$ is bounded and $r=2$, $\rho$ is constant,
$\kappa,\delta\ge 0$. These results motivate a detailed discussion of the
derivation of the problem in Theoretical Acoustics and the consequent proposal
of adding to the model the integral condition $\int_\Omega
u_t=c^2\int_{\Gamma_1}v$.
