Dispersive Wave: Difference between revisions
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Since the [https://belize.math.uwaterloo.ca/mediawiki/index.php/Glossary#Phase_Velocity phase velocity] is given by <math>c_p=\frac{\omega}{k}</math> and the group velocity is given by <math>c_g=\partial_k\omega</math>, if <math>\omega(k) = c k</math> for some scalar <math>c</math>, then <math>c_p=c_g=c</math> independent of <math>k</math>, so that all waves, no matter their wavelength, travel at the same speed. If the dispersion relation is some other function of <math>k</math>, waves of different wavelengths travel at different speeds, leading to dispersion. | Since the [https://belize.math.uwaterloo.ca/mediawiki/index.php/Glossary#Phase_Velocity phase velocity] is given by <math>c_p=\frac{\omega}{k}</math> and the group velocity is given by <math>c_g=\partial_k\omega</math>, if <math>\omega(k) = c k</math> for some scalar <math>c</math>, then <math>c_p=c_g=c</math> independent of <math>k</math>, so that all waves, no matter their wavelength, travel at the same speed. If the dispersion relation is some other function of <math>k</math>, waves of different wavelengths travel at different speeds, leading to dispersion. | ||
All of this was in the case of one dimension. In multiple dimensions we must turn to the definitions <math> \vec{c_p}</math> = <math> \frac{\omega}{|\vec{k}|}\hat{k}</math> and <math> \vec{c_g}</math> = <math> \nabla_{\vec{k}} \omega</math>. We then define a dispersive wave as a wave for which <math> \vec{c_g} | All of this was in the case of one dimension. In multiple dimensions we must turn to the definitions <math> \vec{c_p}</math> = <math> \frac{\omega}{|\vec{k}|}\hat{k}</math> and <math> \vec{c_g}</math> = <math> \nabla_{\vec{k}} \omega</math>. We then define a dispersive wave as a wave for which <math> \vec{c_g}\neq \vec{c_p}</math>. | ||
Revision as of 17:53, 25 May 2015
Since the phase velocity is given by and the group velocity is given by , if for some scalar , then independent of , so that all waves, no matter their wavelength, travel at the same speed. If the dispersion relation is some other function of , waves of different wavelengths travel at different speeds, leading to dispersion.
All of this was in the case of one dimension. In multiple dimensions we must turn to the definitions = and = . We then define a dispersive wave as a wave for which .
