2 cos u cos v = cos (u + v) + cos (u - v)

to the amplitude modulation:

A = a cos (ω t - k x) + \frac{b}{2} cos ((ω + ω_{m}) t - k x) + \frac{b}{2} cos ((ω - ω_{m}) t - k x) .

(1)

The amplitude modulation sidebands have angular frequencies

ω + ω_{m}

and

ω - ω_{m}

, each with amplitude

b /2

in this real cosine convention. The carrier wave has angular frequency

ω

and amplitude

a

This identity is exact for the supplied space-time expression. A physical qualification is needed if every term is meant to be a freely propagating electromagnetic wave with speed of light

c

: the dispersion relation requires

k_{\pm} = (ω \pm ω_{m}) / c

, whereas the prescribed modulation at every position leaves all three wavenumbers equal to

k = ω / c

. A freely propagating modulated wave instead has envelope

a + b cos (ω_{m} (t - x / c))

; its analogous expansion has

k_{\pm}

. For example, at

x = t = 0

the prescribed expression has

A_{tt} - c^{2} A_{xx} = - b ω_{m}^{2}

, which is nonzero for nontrivial modulation. The given formula can describe modulation imposed throughout space or the temporal signal at one position.