Solutions To The Homogeneous Electromagnetic Wave Equation
The general solution to the electromagnetic wave equation is a linear superposition of waves of the form
for virtually any well-behaved function g of dimensionless argument φ, where ω is the angular frequency (in radians per second), and k = (kx, ky, kz) is the wave vector (in radians per meter).
Although the function g can be and often is a monochromatic sine wave, it does not have to be sinusoidal, or even periodic. In practice, g cannot have infinite periodicity because any real electromagnetic wave must always have a finite extent in time and space. As a result, and based on the theory of Fourier decomposition, a real wave must consist of the superposition of an infinite set of sinusoidal frequencies.
In addition, for a valid solution, the wave vector and the angular frequency are not independent; they must adhere to the dispersion relation:
where k is the wavenumber and λ is the wavelength.
Read more about this topic: Electromagnetic Wave Equation
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