1.3: Review of Classical Waves
- Page ID
- 49300
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A wave is a periodic oscillation. It is convenient to describe waves using complex numbers. For example consider the function
\[ \psi (x) = e^{ik_{0}x} \nonumber \]
where x is position, and \(k_{0}\) is a constant known as the wavenumber. This function is plotted in Figure 1.3.1 on the complex plane as a function of position, x. The phase of the function
\[ \phi = k_{0}x \nonumber \]
is the angle on the complex plane.
The wavelength is defined as the distance between spatial repetitions of the oscillation. This corresponds to a phase change of \(2\pi \). From Equations 1.3.1 and 1.3.2 we get
\[ k_{0} = \frac{2\pi}{\lambda} \nonumber \]
This wave is independent of time, and is known as a standing wave. But we could define a function whose phase varies with time:
\[ \psi(t) = e^{-i\omega_{0}t} \nonumber \]
Here t is time, and \(\omega\) is the angular frequency. We define the period, T, as the time between repetitions of the oscillation
\[ \omega_{0} = \frac{2\pi}{T} \nonumber \]