
#1
Nov2311, 07:37 PM

P: 69

1. The problem statement, all variables and given/known data
Boundaries at x=0,a y=0,b This is a waveguide. w = angular frequency and k is the wavenumber. Have the seperable solution to the wave equation. [itex]\psi[/itex] = X(x)Y(y)e[itex]^{i(kzwt)}[/itex] Where w=c[itex]\sqrt{k^{2}+\pi^{2}\left(\frac{n^{2}}{a^{2}}+\frac{m^{2}}{a^{2}}\right) }[/itex] I just need help with figuring out how to get this to an ODE and I should be able to figure out the boundary conditions. Thanks. 2. Relevant equations 3. The attempt at a solution [itex]\psi[/itex][itex]_{xx}[/itex]+[itex]\psi[/itex][itex]_{yy}[/itex][itex]\frac{1}{c^{2}}[/itex][itex]\psi[/itex][itex]_{tt}[/itex]=o Plugging in X''Ye[itex]^{i(kzwt)}[/itex] + Y''Xe[itex]^{i(kzwt)}[/itex]  k[itex]^{2}[/itex]XYe[itex]^{i(kzwt)}[/itex] + [itex]\frac{w^{2}}{c^{2}}[/itex]XYe[itex]^{i(kzwt)}[/itex] = 0 Now dividing by XY and factoring out e[itex]^{i(kzwt)}[/itex] e[itex]^{i(kzwt)}[/itex][itex][ \frac{X''}{X}[/itex] + [itex]\frac{Y''}{Y}[/itex]  k[itex]^{2}[/itex] + [itex]\frac{w^{2}}{c^{2}}][/itex] = 0 Does k[itex]^{2}[/itex] = [itex]\frac{w^{2}}{c^{2}}[/itex] necessarily? Do I set [itex]\frac{X''}{X} = \alpha[/itex] and [itex]\frac{Y''}{Y} = \beta[/itex] where [itex]\alpha , \beta[/itex] are constants and solve these ODE's. Doesn't seem to get me where I want to go though. Any advice would be very awesome. Thanks! 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 


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