Template:Shallow depth one dimension

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[math] \rho(x)\partial_t^2 \zeta = \partial_x \left(h(x) \partial_x \zeta \right). [/math]

subject to the initial conditions

[math] \zeta_{t=0} = \zeta_0(x)\,\,\,{\rm and}\,\,\, \partial_t\zeta_{t=0} = \partial_t\zeta_0(x) [/math]

where [math]\zeta[/math] is the displacement, [math]\rho[/math] is the string density and [math]h(x)[/math] 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).