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Why is V=fl a dispersion relationship ?

  1. Jul 11, 2013 #1
    Why is V=fl a "dispersion relationship"?

    I've heard of V=fλ being referred to as "the simplest dispersion relationship", however it seems to be used to describe non dispersive travelling and standing waves, where the wave velocity is determined by the medium (fixed) and either f is a dependent variable (and λ independent), or vice versa. Is it correct to call this a dispersion relationship in this context?

    For example, in sound, V is constrained by the properties of the gas. The frequency of excitation is set by the source (this could be considered the dependent variable), and the wavelength (indep var) is determined by the wave velocity and frequency.

    Standing waves seem to have a constrained wave velocity and fixed wavelength, which selects the frequency (independent variable) when excited (by some broadband excitation like a string being plucked or a closed pipe being tapped).

    Is there a situation where a wave is governed by the V=fλ relationship, and the velocity is the independent variable. For example, a wave that has both its frequency and wavelength determined by the physical excitation mechanism that then seeks an appropriate wave velocity?
     
  2. jcsd
  3. Jul 11, 2013 #2
    It is a dispersion equation in the same sense as ##y=x## is a quadratic equation, since ##y=0x^2+x## is indeed a quadratic.
     
  4. Jul 12, 2013 #3
    A dispersion relation is an expression that relates the frequency of a wave to its wave length. We often deal with angular frequency [itex]\omega = 2\pi f[/itex] and wave number [itex]k = 2\pi / \lambda[/itex].

    Mathematically a dispersion relation is a equation of the form
    [itex]\omega = g(k) [/itex]

    The phase velocity of the wave is

    [itex]V_p=\frac {\omega} {k} = \frac {g(k)}{k} [/itex]

    and the group velocity of the wave is
    [itex]V_g=\frac{\partial \omega} {\partial k} = \frac{\partial g(k)} {\partial k} [/itex]

    If the phase velocity depends on the wave number, then the wave is dispersive. But all waves have a dispersion relation.
     
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