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## Homework Statement

Derive from Maxwell's equations these Hill equations for 's' and 'p' mode waves;

[itex]s\hspace{3mm} modes: E(r,t) = \Psi_{s}(z)e^{i(\beta x - \omega t)}y \\[/itex]

[itex]\hspace{10mm}Hill\, Equation for\, \Psi_{s}(z)\\[/itex]

[itex] \hspace{17mm} \dfrac{d^{2}\Psi_{s}(z)}{dz^{2}} + q^{2}(z)\Psi_{s}(z) = 0,\hspace{5mm}q(z+d) = q(z)\\[/itex]

[itex]p\hspace{3mm} modes: B(r,t) = \Psi_{p}(z)e^{i(\beta x - \omega t)}y \\[/itex]

[itex]\hspace{10mm}\mbox{General ODE with periodic coefficients for}\, \Psi_{p}(z)\\[/itex]

[itex] \hspace{17mm} \dfrac{d^{2}\Psi_{p}(z)}{dz^{2}} -\dfrac{2}{n(z)}\dfrac{dn(z)}{dz}\dfrac{d\Psi_{p}(z)}{dz}+q^{2}(z)\Psi_{p}= 0[/itex]

## Homework Equations

Maxwell's equations, combining Faraday's and Ampere's Laws and a vector identity.

## The Attempt at a Solution

I think I have the start down, I am just not sure on how to proceed from here;

Vector identity;

[itex] \nabla \times \nabla F = \nabla(\nabla \cdot F) - \nabla^{2}F[/itex]

applied to [itex] \nabla\cdot E = 0[/itex]

[itex]\nabla \times \nabla \times E = -\nabla^{2}E[/itex]

**Faraday's Law**then says that the

**curl of E**equals the negative partial derivative of

**B**with respect to

**t**, and

**B**equals

**μH**so therefore;

[itex] \nabla \times E = -\dfrac{\partial B}{\partial t} = -\mu \dfrac{\partial H}{\partial t}[/itex]

[itex]\nabla \times(\nabla \times E) = -\mu \dfrac{\partial}{\partial t}(\nabla \times H)[/itex]

and then as the curl of

**H**equals the partial derivative of

**D**with respect to

**t**plus the current density which in this case will equal zero and

**D**is equal to

**εE**I can say;

[itex] \nabla \times H = (\dfrac{\partial D}{\partial t} + J(equal0)) \hspace{10mm} D = \epsilon E[/itex]

[itex] \nabla \times \nabla \times E = -\mu \epsilon \dfrac{\partial}{\partial t}(\dfrac{\partial E}{\partial t}) [/itex]

[itex] \rightarrow \nabla^{2}E = \mu \epsilon \dfrac{\partial^{2}E}{\partial t^{2}}[/itex]

Then because the question gives an equation for

**E**I have tried to substitute this in and simplify but I can't make much sense of it.

Any help would be appreciated.

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