Difference between revisions of "Template:Shallow depth one dimension"

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\partial_t\zeta_{t=0} = \partial_t\zeta_0(x)
 
\partial_t\zeta_{t=0} = \partial_t\zeta_0(x)
 
</math></center>
 
</math></center>
 +
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).

Latest revision as of 23:45, 22 March 2009

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

subject to the initial conditions

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

where [math]\displaystyle{ \zeta }[/math] is the displacement, [math]\displaystyle{ \rho }[/math] is the string density and [math]\displaystyle{ 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).