Hi there, I'm kind of rusty on some stuff, so hope someone can help enlighten me.(adsbygoogle = window.adsbygoogle || []).push({});

I have an expression

[itex]E(r,w-w0)=F(x,y) A(z,w-w0) \exp[i\beta_0 z][/itex]

I need to substitute this into the Helmholtz equation and solve using separation of variables. However, I'm getting problems simplifying it to a form with can be separated... I reckon the problem lies with my understanding of the 2nd derivative, especially with more variables coming into play.

From the Helmholtz equation,

[itex]\nabla^2 E+\epsilon (w) k_0^{\phantom{0}2} E=0[/itex]

Working out

[itex]\nabla^2 E =\nabla(\nabla E)[/itex]

[itex]=\nabla(A\exp[i\beta_0 z] \frac{\partial F}{\partial x}+A\exp[i\beta_0 z] \frac{\partial F}{\partial y}+ FA (i \beta_0 \exp[i\beta_0 z])+F \exp[i\beta_0 z] \frac{\partial A}{\partial z})

[/itex]

[itex]

=A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial x^2}+i \beta_0 A \exp[i\beta_0 z] \frac{\partial F}{\partial x}+ \exp[i\beta_0 z] \frac{\partial A}{\partial z} \frac{\partial F}{\partial x} [/itex]

[itex]

+ A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial y^2}+i \beta_0 A \exp[i\beta_0 z] \frac{\partial F}{\partial y}+\exp[i\beta_0 z] \frac{\partial A}{\partial z} \frac{\partial F}{\partial y}

[/itex]

[itex]

+\frac{\partial F}{\partial x} \exp[i\beta_0 z] \frac{\partial A}{\partial z}+ \frac{\partial F}{\partial y} \exp[i\beta_0 z] \frac{\partial A}{\partial z}+ F A (i \beta_0)^2 \exp[i\beta_0 z]

[/itex]

[itex]

+i \beta_0 F \exp[i\beta_0 z] \frac{\partial A}{\partial z}+F \frac{\partial A}{\partial z} (i \beta_0) \exp[i\beta_0 z]+F \exp[i\beta_0 z] \frac{\partial^2 A}{\partial z^2}

[/itex]

Which gives

[itex]

=\exp[i\beta_0 z][A \frac{\partial^2 F}{\partial x^2}+2i \beta_0 A \frac{\partial F}{\partial x}+ 2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial x}+A \frac{\partial^2 F}{\partial y^2}+2i \beta_0 A \frac{\partial F}{\partial y}+ 2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial y}]

[/itex]

It seems like the terms [itex]2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial x}[/itex] and [itex]2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial y}[/itex] need to vanish...

------------------------------------------------------------------------------------------

Or have I done the derivative wrongly? Should it be the following instead?

[itex]

\nabla^2 E =\nabla_x ^{\phantom{0}2}E+\nabla_y ^{\phantom{0}2}E+\nabla_z ^{\phantom{0}2}E

[/itex]

where

[itex]

\nabla_x ^{\phantom{0}2}E = A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial x^2}

[/itex]

[itex]

\nabla_y ^{\phantom{0}2}E = A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial x^y}

[/itex]

[itex]

\nabla_z^{\phantom{0}2}E = \nabla_z [FA (i \beta_0) \exp[i\beta_0 z]+ F \exp[i\beta_0 z] \frac{\partial A}{\partial z}]

[/itex]

[itex]

=[F (i \beta_0) \exp[i\beta_0 z] \frac{\partial A}{\partial z}+FA (i \beta_0)^2 \exp[i\beta_0 z]+

F \exp[i\beta_0 z] \frac{\partial^2 A}{\partial z^2}]+ F \frac{\partial A}{\partial z} (i \beta_0) \exp[i\beta_0 z]

[/itex]

Thanks in advance!

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# 2nd order derivative (Nabla^2)

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