stylistic:tone, structural:scholar_edition
Amplitude and modulation

The amplitude of a wave may be constant (in which case the wave is a c.w. or continuous wave), or may be modulated so as to vary with time and/or position. The outline of the variation in amplitude is called the envelope of the wave. Mathematically, the modulated wave can be written in the form: u (x, t) = A (x, t) sin (k x − ω t + ϕ), {\displaystyle u(x,t)=A(x,t)\sin \left(kx-\omega t+\phi \right),} where A (x, t) {\displaystyle A(x,\ t)} is the amplitude envelope of the wave, k {\displaystyle k} is the wavenumber and ϕ {\displaystyle \phi } is the phase. If the group velocity v g {\displaystyle v_{g}} (see below) is wavelength-independent, this equation can be simplified as: u (x, t) = A (x − v g t) sin (k x − ω t + ϕ), {\displaystyle u(x,t)=A(x-v_{g}t)\sin \left(kx-\omega t+\phi \right),} showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation.