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Can anyone clarify what does mean the angular frequency of a system in case of the vibrating membrane.

Angular frequency is measured in radians per second, what does this have with the vertical displacement in case of the wave equation.

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It is related to frequency by means of

$$ f = \frac{\omega}{2\pi} $$

and it is used in trig. functions like $\sin( \omega \cdot t)$ instead of $\sin( 2\pi f \cdot t)$ because it is simpler.

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Angular frequency applies to any sinusoidally varying with time frequency when the arguments of the trigonometric functions are in radians. For an "angle" or geometric interpretation: the trig function variation is the real part of the phasor quantity $\exp(i\,\omega\,t)$, which follows the unit circle at constant speed $\omega$ radians per second, and so this is the rate of increase of the angle this number's position vector makes with the $x$-axis, so we can be still talking about a true angle here.

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