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 2022-09-06 Answered
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?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

ralharn
Answered 2022-09-07 Author has 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.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-02-11
Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in M2×4 with the property that FA = 0 (the zero matrix in M3×4). Determine if H is a subspace of M2×4
asked 2021-05-17
Find the scalar and vector projections of b onto a.
a=(4,7,4),b=(3,1,1)
asked 2021-05-29
Which of the following expressions are meaningful? Which are meaningless? Explain.
a) (ab)c
(ab)c has ? because it is the dot product of ?.
b) (ab)c
(ab)c has ? because it is a scalar multiple of ?.
c) |a|(bc)
|a|(bc) has ? because it is the product of ?.
d) a(b+c)
a(b+c) has ? because it is the dot product of ?.
e) ab+c
ab+c has ? because it is the sum of ?.
f) |a|(b+c)
|a|(b+c) has ? because it is the dot product of ?.
asked 2021-05-29

Find the vector and parametric equations for the line segment connecting P to Q.
P(0, - 1, 1), Q(1/2, 1/3, 1/4)

asked 2022-08-11
Imagine I have an arbitrary vector v R n and say it can be represented as
v = ( v 1 , . . . , v n ) T
where it's understood that the indexes are in increasing order. Imagine now the vector u R n 1 obtained by simply removing entry with index i of v. What's the best (and most "economic") way to represent such vector? My idea was to simply interpret the vector as a sequence and write
u = { v j } j i T
but I'm not sure whether this is correct and/or there is a better way of writing it.
asked 2021-11-19
Tell the opposite to a cross product.
For example, a×b=c. How to find c, if you know only a and b?
asked 2022-08-19
In Δ A B C, A P A Q = B P B Q = C P C Q = 2 find circumcentre given coordinates of P,Q
Let A , B , C be the vertices of Δ A B C. Let P ( 1 , 1 , 2 ) and Q ( 4 , 7 , 8 ) be two points such that A P A Q = B P B Q = C P C Q = 2. Find the coordinates of circumcentre of Δ A B C

New questions