A plane wave in the time domain can be written (using notation for an electric field): bbE(r,t)=E_0e^(i(kr−omega t)) what is the corresponding expression for a plane wave, E(r,omega), in the frequency domain?

Leonel Schwartz

Leonel Schwartz

Answered question

2022-09-06

What is the explicit expression of a plane wave in the frequency domain?
A plane wave in the time domain can be written (using notation for an electric field):
E ( r , t ) = E 0 e i ( k r ω t )
what is the corresponding expression for a plane wave, E ( r , ω ), in the frequency domain?

Answer & Explanation

ralharn

ralharn

Beginner2022-09-07Added 15 answers

The key for all these formulas is
(1) R d e i x y d y ( 2 π ) d 2 = δ ( x ) ,
where δis the d-dimensional Dirac distribution. This question is probably concerned with d=1 only, but there is no extra difficulty in doing the general case.
To finally answer the question, we apply (1) to obtain
R E 0 e i k x ω t e i t s d s = ( 2 π ) 1 2 E 0 e i k x δ ( t + ω ) .
This is the Fourier transform of the given function, the "plane wave", in the time variable.

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