Phase
Phase
The angle (expressed in radians) in the argument of the sine in the wave function is called the phase:
\[\phi(x, t) := k x - \omega t= 2 \pi \left(\frac{x}{\lambda} - \frac{t}{T} \right).\]
The phase determines the state in which the wave can be found at any time and at any position. Crests (or compressions for longitudinal waves) correspond to phase values of
\[\phi_{\mathrm{crest}} = \frac{\pi}{2} + i \cdot 2 \pi,\]
(where \(i\) is an arbitrary integer, reflecting the periodicity of the sine function), whilst troughs (or rarefactions for longitudinal waves) can be found where the phase assumes
\[\phi_{\mathrm{trough}} = \frac{3\pi}{2} + i \cdot 2 \pi.\]
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In the figure, we can see how the phase can merge the position dependence at a given time and the time dependence at a given location into a single angle which describes the state of the wave within a cycle. We can also observe how to express the delay of one wave (the maroon one) as compared to another (the green one) as a phase difference \(\Delta\phi.\) |
Initial phase
The form of the wave function
\[y(x, t) = A \cdot \sin\left(k x - \omega t\right)\]
would always suggest an initial value of zero at the origin:
\[y(0, 0) = 0,\]
which cannot account for cases where the wave function assumed a non-zero value at the origin at the beginning. To preserve generality, we need to introduce the initial phase \(\phi_0\), which is the phase value whose sine yields the initial wave function at the origin as a fraction of the amplitude:
\[y(0, 0) = A \cdot \sin\left(\phi_0\right).\]
With the initial phase, the general form of the wave function becomes
\[y(x, t) = A \cdot \sin\left(k x - \omega t + \phi_0\right).\]