# Template:Shallow depth one dimension

$\displaystyle{ \rho(x)\partial_t^2 \zeta = \partial_x \left(h(x) \partial_x \zeta \right). }$
$\displaystyle{ \zeta_{t=0} = \zeta_0(x)\,\,\,{\rm and}\,\,\, \partial_t\zeta_{t=0} = \partial_t\zeta_0(x) }$
where $\displaystyle{ \zeta }$ is the displacement, $\displaystyle{ \rho }$ is the string density and $\displaystyle{ h(x) }$ is the variable depth (note that we are unifying the variable density string and the wave equation in variable depth because the mathematical treatment is identical).