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2018-11-02

2018-11-02

作者: carpediemmlf | 来源:发表于2018-12-18 23:55 被阅读0次

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)

Solutions for systems of different geometries

  • 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

Elliptical polarization

  • For a wave travelling in the positive x direction

\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

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