Slowly varying envelope approximation

In physics, the slowly varying envelope approximation (SVEA, sometimes also called slowly varying amplitude approximation or SVAA) is the assumption that the envelope of a forward-travelling wave pulse varies slowly in time and space compared to a period or wavelength. This requires the spectrum of the signal to be narrow-banded—hence it also referred to as the narrow-band approximation. In physics, the slowly varying envelope approximation (SVEA, sometimes also called slowly varying amplitude approximation or SVAA) is the assumption that the envelope of a forward-travelling wave pulse varies slowly in time and space compared to a period or wavelength. This requires the spectrum of the signal to be narrow-banded—hence it also referred to as the narrow-band approximation. The slowly varying envelope approximation is often used because the resulting equations are in many cases easier to solve than the original equations, reducing the order of—all or some of—the highest-order partial derivatives. But the validity of the assumptions which are made need to be justified. For example, consider the electromagnetic wave equation: If k0 and ω0 are the wave number and angular frequency of the (characteristic) carrier wave for the signal E(r,t), the following representation is useful: where ℜ { ⋅ } {displaystyle scriptstyle Re {cdot }} denotes the real part of the quantity between brackets. In the slowly varying envelope approximation (SVEA) it is assumed that the complex amplitude E0(r, t) only varies slowly with r and t. This inherently implies that E0(r, t) represents waves propagating forward, predominantly in the k0 direction. As a result of the slow variation of E0(r, t), when taking derivatives, the highest-order derivatives may be neglected: