5 Harmonic waves
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5.1 Harmonic waves
Harmonic waves are waves that are described by sines and cosines. A travelling harmonic wave can be written as \[ y(x,t)=a \cos\left(kx - \omega t + \phi\right) \tag{5.1}\] or \[ y(x,t)=a \cos\left(2\pi(\hat{k}x - \nu t) + \phi\right) \tag{5.2}\] where \(a\) is the amplitude, \(\hat{k}\) the wave number, \(\nu\) the frequency, \(\phi\) the phase, and where \[ k= 2\pi\hat{k} \quad \text{and} \quad \omega = 2\pi \nu \tag{5.3}\] are the angular wave number and the angular frequency, respectively. Note that \(k\) and \(\omega\) are used more often by mathematicians than \(\hat{k}\) and \(\nu\) and that the prefix ‘angular’ is often discarded.
- Frequency \(=\) number of cycles (oscillations) per unit time.
- Wave number \(=\) number of cycles (oscillations) per unit length.
- Period \(=\) time \(P\) needed to complete one cycle (oscillation): \[ P=\frac{1}{\nu}=\frac{2\pi}{\omega}. \tag{5.4}\]
- Wave length \(=\) distance between two consecutive wave crests (peaks): \[ \lambda = \frac{1}{\hat{k}}=\frac{2\pi}{k}. \tag{5.5}\]
- Wave speed \(=\) speed at which the wave is travelling: \[ c = \frac{\lambda}{P}=\frac{\nu}{\hat{k}}=\frac{\omega}{k}. \tag{5.6}\] (Sometimes, the wave speed is also called the phase speed.)
The harmonic wave Eq. 5.1 can be written as \[\begin{split} y(x,t)&=a \cos\left(kx - \omega t + \phi\right) \\ &= \operatorname{Re}\left( a e^{i\phi} e^{i(kx-\omega t)}\right)\\ &=\operatorname{Re}\left( A e^{i(kx-\omega t)}\right), \end{split} \tag{5.7}\] where we have included the phase factor \(e^{i\phi}\) into the complex amplitude: \(A= a e^{i\phi}\).
Consider now the complex function \[ y(x,t)= A \, e^{i(kx-\omega(k) t)} \tag{5.8}\] for any \(A\in\mathbb{C}\), any \(k\in\mathbb{R}\) and some \(\omega(k)\). Substituting this into the wave equation gives us an equation for \(\omega(k)\): \[ \partial_t^2 y =c^2 \partial_x^2 y \quad \Rightarrow \quad -\omega^2 A =-c^2 k^2 A. \tag{5.9}\] Therefore, if \[ \omega(k) = \pm ck \tag{5.10}\] then the complex function in Eq. 5.8 is a solution of the wave equation. We will refer to these complex solutions as complex harmonic waves. They are often more convenient to work with than their real counterpart in Eq. 5.1.
Eq. 5.10 is an example of a dispersion relation. It states that for the wave equation, \(\omega\) is proportional to \(k\). But complex harmonic waves can also solve other PDEs, as we will see in the next subsection, and that will lead to more complicated dispersion relations.
5.2 Solving PDEs with harmonic waves
Any linear homogeneous PDE (in variables \(x\) and \(t\)) with constant coefficients has complex harmonic wave solutions Eq. 5.8 for some \(\omega(k)\).
Example 5.1 Consider the damped string (with friction force proportional to velocity): \[ \partial_t^2 \, y= c^2 \, \partial_x^2 \, y - p\, \partial_t \, y\quad \text{where }\ p>0. \tag{5.11}\] Substituting the complex harmonic wave from Eq. 5.8 into this equation, we obtain the dispersion relation \[ -\omega^2 y =- c^2 k^2 y +ip\,\omega y . \tag{5.12}\] Cancelling the \(y\) we obtain a quadratic equation for \(\omega^2\) which has the complex solution \[ \omega = -\frac{ip}{2}\pm \sqrt{c^2 k^2 - \frac{p^2}{4}}. \tag{5.13}\] Thus we have the following solution for the damped string: \[\begin{split} y(x,t)&= A \, e^{i\left(kx+\frac{ip}{2}t\pm \sqrt{c^2 k^2 - \frac{p^2}{4}} t\right)} \\ &=A \, e^{-\frac{pt}{2}}e^{ik\left(x\pm \sqrt{c^2 - \frac{p^2}{4k^2}} t\right)} \end{split} \tag{5.14}\] The factor \(e^{-pt/2}\) shows that we have a wave with exponentially decreasing amplitude. This is a consequence of the damping. The wave speed is now dependent on the wave number \(k\): \[ c(k)=\sqrt{c^2 - p^2/(4k^2)}. \tag{5.15}\] If we want to, we can get a real solution by taking the real part of the complex solution: \[ \operatorname{Re}(y(x,t))=a\, e^{-pt/2} \cos\left[k\left(x \pm t \, \sqrt{c^2 - p^2/(4k^2)}\right)+\phi\right] \tag{5.16}\] where \(a=\vert A\vert\) and \(\phi=\arg(A)\).
Note that the imaginary part of \(\omega\) produces the damping exponential and the real part of \(\omega\) determines the wave speed.