 Envelope (waves)

A wave envelope is a slowly varying wave amplitude multiplying a rapidly oscillating carrier wave. For a complex wavefield $\psi =A(x,t)e^{i(k_{0}x-{\omega }_{c}t)}$, its nonnegative amplitude envelope is $|A|=|\psi |$; the complex $A$ can also encode a slowly varying wave phase. This choice distinguishes the amplitude envelope from a signed modulation factor, such as the cosine factor in a two-tone superposition. A slowly varying wave envelope is useful when the carrier's wavelength and period are short compared with its variation scales. Strong overlap between distinct packets or a broad frequency spectrum can spoil a unique carrier-envelope interpretation.

In the context of waves, an envelope refers to a smooth curve that outlines or envelops a set of oscillating waveforms. Specifically, it often describes the modulation shape of a wave packet or the amplitude variation of a group of waves. Here's a breakdown of the concept: 1. **Wave Packets**: When individual waves combine, they can interfere constructively or destructively. The resulting superposition can create a wave packet, which may have a varying amplitude over time.