# Wave Envelope

The image on the left provides a simple demonstration of the distinction between a wave envelope and individual wave crests. The red curves depict the wave envelope, while the blue curve indicates the individual wave crests (in the context of a surface wave, the blue curve can be viewed as the water surface). The blue curve is not propagating while the red curve is steadily propagating to the right, highlighting the distinction between the propagation of an individual wave crest and the propagation of the overlaying envelope.

A brief discussion of envelope propagation is presented from the perspective of the KdV equation, and table is included that demonstrates three different cases: envelope propagation faster than crest propagation, envelope propagation equal to crest propagation, and envelope propagation slower than crest propagation. Both a continuous wave train and a single wave packet are shown.

## Contents

### Derivation using KdV

The linear KdV equation is $\frac{\partial}{\partial t}\eta+c\frac{\partial}{\partial x}\eta + \beta\frac{\partial^3}{\partial x^3}\eta = 0$. Using the method of multiple scales, suppose that the wave envelope evolves on slower temporal and spatial scales than the wave crests. Denote the slower scales by $(\tau,\chi)=(\epsilon t,\epsilon x)$, for $0<\epsilon\ll1$, and separate using $\eta=A(\tau,\chi)\phi(x,t)$. After substituting into the KdV equation, the equation becomes: $A\phi_t + \epsilon A_{\tau}\phi + cA\phi_x + \epsilon cA_{\chi}\phi + \beta\left(A\phi_{xxx} + 3\epsilon A_{\chi}\phi_{xx} + 3\epsilon^2 A_{\chi\chi}\phi_{x} + \epsilon^3 A_{\chi\chi\chi}\phi \right) = 0$,

where subscripts denote partial derivatives.

##### $\mathcal{O}(1)$ Problem:

Assuming that $A\not\equiv0$, the first order problem is found to be $\left[\frac{\partial}{\partial t}+ c\frac{\partial}{\partial x}+ \beta\frac{\partial^3}{\partial x^3}\right]\phi=0$. If a plane-wave solution of $\phi=e^{i\left(kx-\omega t \right)}$ is taken, then the problem reduces to $\omega = ck-\beta k^3$. This gives the dispersion relation for the linear problem.

##### $\mathcal{O}(\epsilon)$ Problem:

Again assuming that $\phi=e^{i\left(kx-\omega t \right)}$, the first order problem is found to be $\left[\frac{\partial}{\partial \tau}+ \left(c-3\beta k^2\right)\frac{\partial}{\partial \chi}\right] A=0$. Writing $c_g = c-3\beta k^2$, the problem reduces to $\left[\frac{\partial}{\partial \tau}+ c_g\frac{\partial}{\partial \chi}\right] A=0$, and so $A(\tau,\chi)=A(\chi-c_g\tau)$, showing that the propagation speed of the wave envelope is given by $c_g$. In particular, $c_g=\frac{\partial}{\partial k}\omega$.

## More Examples

In each animation, the red curves denote the envelope, the blue curves denote the individual waves, and the green dots track a single wave crest for illustrative purposes.

Various Demonstrations of Wave Envelope vs. Wave Crest Propagation
Case Continuous Wave Train Single Wave Packet $c_g>c_p$: Envelope propagation faster than crest propagation. $c_g=c_p$: Envelope propagation equal to crest propagation. $c_g: Envelope propagation slower than crest propagation.