2018-11-02

2018-12-18 本文已影响0人 carpediemmlf

Wave

the general n-dimensional wave equation

$\frac{\partial^2 \psi}{\partial t^2} = v^2 {\nabla^2 \psi}$
where in the special case of travelling in a tight string, an erroneous but good for practical purposes derivation shows

$v = \sqrt {\frac{T}{\rho}}$
this is the dispersion relation for a one dimensional non-dispersive wave equation
there exists two solutions propagating in opposite directions of the system namely

$\psi(\vec r, t) = f(\vec r \pm \vec v t)$

Generally we expect a harmonic solution in the form of

$\psi = \mathcal{Re} (Ae^{{i}{(\vec r \cdot \vec k - \omega t)}})$
where $\vec k = \frac{2\pi}{\lambda} \hat n$ is the wave vector and $\vec r \cdot \vec k - \omega t$ is the phase of the system
In 2D, e.g. a drumskin we get the normal form of a harmonic wave solution

Plane wave
Spherical waves

Cylindrical waves

$\frac{{\phi_y}^2}{{A_y}^2}+ \frac{{\phi_z}^2}{{A_z}^2} - \frac{{\phi_y}^2}{{A_y}^2}\frac{{\phi_z}^2}{{A_z}^2} \cos \phi = {\sin\phi}^2$

$y: \begin{pmatrix} A \\ 0 \end{pmatrix} e^{i \omega t}\ z: \begin{pmatrix} 0 \\ A \end{pmatrix} e^{i \omega t}$

We usually exclude the time dependent terms for simplicity in notation.
By convention, looking against the propagation direction of the wave, define the