In this post I would like to summarize what I understand about the wave equation.
The wave equation look like this $ u(x, t) = A \sin ( k x - \omega t + \phi) $.
The number $ k $ is the wave vector. This confused me because it looks like a scalar to me.
Suppose time freezes, if we advance $ x $ by $ \frac{2\pi}{k} $, we get $ u(x + \frac{2\pi}{k}, t) = A \sin ( k (x + \frac{2\pi}{k}) - \omega t + \phi) = A \sin ( k x + 2\pi - \omega t + \phi) = A \sin ( k x - \omega t + \phi) $. Therefore, $ \frac{2\pi}{k} $ is the wave length.
Switching to the time scale, if position freeze and we advance time by $ \frac{2\pi}{\omega} $. It is apparent that using the same trick we get the same value. This means $ \frac{2\pi}{\omega} $ is the period, $ \frac{\omega}{2\pi} $ is the frequency, and $ \omega $ is the angular frequency, nothing special about this.
To advance one wavelength, one need one period of time. Therefore the speed of the wave is $ v = \frac{\lambda}{T} = f\lambda = \frac{\omega}{2\pi}\frac{2\pi}{k} = \frac{\omega}{k} $.
The wave equation look like this $ u(x, t) = A \sin ( k x - \omega t + \phi) $.
The number $ k $ is the wave vector. This confused me because it looks like a scalar to me.
Suppose time freezes, if we advance $ x $ by $ \frac{2\pi}{k} $, we get $ u(x + \frac{2\pi}{k}, t) = A \sin ( k (x + \frac{2\pi}{k}) - \omega t + \phi) = A \sin ( k x + 2\pi - \omega t + \phi) = A \sin ( k x - \omega t + \phi) $. Therefore, $ \frac{2\pi}{k} $ is the wave length.
Switching to the time scale, if position freeze and we advance time by $ \frac{2\pi}{\omega} $. It is apparent that using the same trick we get the same value. This means $ \frac{2\pi}{\omega} $ is the period, $ \frac{\omega}{2\pi} $ is the frequency, and $ \omega $ is the angular frequency, nothing special about this.
To advance one wavelength, one need one period of time. Therefore the speed of the wave is $ v = \frac{\lambda}{T} = f\lambda = \frac{\omega}{2\pi}\frac{2\pi}{k} = \frac{\omega}{k} $.
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