Difference between revisions of "Template:Shallow depth one dimension"
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<center><math> | <center><math> | ||
| − | \partial_t^2 \zeta = \partial_x \left(h(x) \partial_x \zeta \right). | + | \rho(x)\partial_t^2 \zeta = \partial_x \left(h(x) \partial_x \zeta \right). |
</math></center> | </math></center> | ||
| − | subject to the | + | subject to the initial conditions |
<center><math> | <center><math> | ||
| − | \zeta_{t=0} = \zeta_0\,\,\,{\rm and}\,\,\, | + | \zeta_{t=0} = \zeta_0(x)\,\,\,{\rm and}\,\,\, |
| − | \partial_t\zeta_{t=0} = \partial_t\zeta_0 | + | \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
subject to the initial conditions
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).
