$sin(kx-\omega t)$

\(f(x,t) = sin(kx-\omega t)\)

\(\frac{\partial f(x,t)}{\partial x} = kcos(kx-\omega t)\)

\(\frac{\partial^2 f(x,t)}{\partial x^2} = -k^2sin(kx-\omega t)\)

\(\frac{\partial f(x,t)}{\partial t} = -\omega cos(kx-\omega t)\)

\(\frac{\partial^2 f(x,t)}{\partial t^2} = -\omega^2 sin(kx-\omega t)\)

Phase speed \(v_{ph}\) and group speed \(v_{gr}\)

\(\frac{\omega^2}{k^2} \frac{\partial^2 f(x,t)}{\partial x^2} = \frac{\partial^2 f(x,t)}{\partial t^2}\)

$v_{ph} = \frac{\omega}{k}, v_{gr} = \frac{\partial \omega}{\partial k} $

Assumptions and Rayleigh formula

Dispersion Relationship: \(\omega(k)\)

let \(\omega = \frac{2\pi}{T}\), if \(k = \frac{2\pi}{x}\), then \(x = \lambda > 0\)

\(\frac{2\pi}{T} \frac{x} {2\pi} = \frac{x}{T} = \frac{\lambda}{T} = v_{ph}(\lambda)\)

\(v_{gr}(\lambda) = \frac{d \omega}{d k} = \frac{d \frac{2\pi}{T}}{d \frac{2\pi} {x}} = \frac{x^2}{T^2} \frac{d T}{d x} = \frac{\lambda^2}{T^2} \frac{d T}{d \lambda} = v_{ph} - \lambda \frac{dv_{ph}}{d\lambda}\)

No Units here now.