Components of E for a waveguide problem.

[atomicpedals]

Homework Statement

For a rectangular wave guide the components of E were found to be

E_x = A₁ cos( k₁ x) sin(k₂ y) e^{-i (\omegat-\gammaz)}

E_y = A₂ cos( k₁ x) sin(k₂ y) e^{-i (\omegat-\gammaz)}

E_z = 0

Show that in these equations A₁ = -A₂ k₂ / k₁

Homework Equations

See part (a)

The Attempt at a Solution

My first instinct is simply substitute this given A₁ into the x equation and show its equivalence to the second. But I'm not so sure that's the best way to go about it (or even the correct way to go about it). It would be tempting to solve this using matrices but that doesn't seem appropriate.

So, any suggestions as to if I'm on the right track or way off base are much appreciated!

 

[tiny-tim]

hi atomicpedals!

Show that in these equations A₁ = -A₂ k₂ / k₁

hmm … that's the same as k₁A₁ + k₂A₂ = 0 …

how could you manipulate the original equations to get that? ​

 

[atomicpedals]

Aww fer crying out loud, they're vector components (facepalm)... yeah, leave it to me to miss the bleedin' obvious!

Thanks!

 

[Phrak]

hi atomicpedals!


hmm … that's the same as k₁A₁ + k₂A₂ = 0 …

how could you manipulate the original equations to get that? ​

I don't get it.

 

[atomicpedals]

The fist thing I failed to see right off the bat was that each equation was a vector component. That is E = ( E_x , E_y , E_z )

So looking at it like that there are a number of ways to get it down to the simple algebraic form (divergence or laplacian ...I'll need to crunch the numbers).

 

[Phrak]

Ignoring Ez, in your problem statement you have two equations whos amplitudes are independent. There's a missing constraint.