Complex Waves

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Real-Valued, Harmonic Traveling Waves

Before we get to complex waves, we'll first review real-valued traveling waves. A real-valued, harmonic traveling wave is a function of the form:

$$\psi(x,t) = A \sin ( kx - \omega t).$$
where:

Note: the wave travels in the positive $x$ direction if $k$ and $\omega$ are both positive or are both negative.

The wave number determines the number of wave crests per unit length, i.e. it determines how "wiggly" the wave is in space. The angular frequency determines the number of wave crests per unit time, i.e. it determines how "wiggly" the wave is in time.

The speed $v$ of the wave can be found from the wave parameters using the following relations:

$$v=f\lambda,$$ $$v=\frac{\omega}{k}.$$

We can create a wave that travels to the left (i.e. in the negative $x$ direction) by changing the sign in front of the angular frequency (where $k$ and $\omega$ assumed to have the same sign):

$$\psi(x,t) = A \sin ( kx + \omega t).$$

Complex Traveling Waves

In many areas of physics and engineering, the traveling wave $\psi(x,t) = A \sin ( kx - \omega t)$ is often written as a complex exponential:

$$\psi(x,t) = A e^{i(kx-\omega t)},$$

where the amplitude $A$ can, in general, be complex-valued.

What does it mean for a wave to be complex-valued and why would we want to define such a thing? The main reason why we use complex exponentials is that they are often easier to work with than sine or cosine functions. Think, for example, how easy it is to differentiate or integrate an exponential function. We can always get a real-valued sine or cosine function back by taking the real or imaginary part of the complex exponential.

\begin{align*} \text{Im}\left[ A e^{i(kx-\omega t)} \right] &= A \sin ( kx - \omega t) \\ \text{Re}\left[ A e^{i(kx-\omega t)} \right] &= A \cos ( kx - \omega t) . \end{align*}

Example: Show that complex exponentials $\psi(x,t) = A e^{i(kx-\omega t)}$ are solutions to the wave equation
$$\frac{\partial^2 \psi}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2},$$
where $v$ is the speed of the wave.

Solution: We use direct substitution. We start by finding the first and second partial derivatives of $\psi(x,t)$ with respect to $x$ and $t$:

$$\frac{\partial \psi}{\partial x} = \frac{\partial }{\partial x} \left[A e^{i(kx-\omega t)}\right] = i k A e^{i(kx-\omega t)}$$ $$\frac{\partial^2 \psi}{\partial x^2} = \frac{\partial }{\partial x} \left[i k A e^{i(kx-\omega t)}\right] = -k^2 A e^{i(kx-\omega t)}$$ $$\frac{\partial \psi}{\partial t} = \frac{\partial }{\partial t} \left[A e^{i(kx-\omega t)}\right] = -i\omega A e^{i(kx-\omega t)}$$ $$\frac{\partial^2 \psi}{\partial t^2} = \frac{\partial }{\partial t} \left[-i\omega A e^{i(kx-\omega t)}\right] = -\omega^2 A e^{i(kx-\omega t)}.$$
We then substitute these expressions into the wave equation and see if we get the same result on both sides:
$$\frac{\partial^2 \psi}{\partial x^2} \stackrel{?}{=} \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2}$$ $$ -k^2 A e^{i(kx-\omega t)} \stackrel{?}{=} \frac{1}{v^2} \left[-\omega^2 A e^{i(kx-\omega t)}\right]$$
Next, we write the wave number in terms of the angular frequency and speed of the wave $k = \omega / v$:
$$-\frac{\omega^2}{v^2} A e^{i(kx-\omega t)} = -\frac{\omega^2}{v^2} A e^{i(kx-\omega t)} \;\;\;\;\;\;\;\;\; ✅$$
Since both sides of the equation are equal, we have shown that the complex exponential $\psi(x,t) = A e^{i(kx-\omega t)}$ is a solution to the wave equation.

The other advantage of using complex exponentials is economy. If you wanted to write an equation describing how the phase and amplitude of a wave evolves in time, you would need two equations (one for each quantity). However, because a complex-valued function contains two pieces of information (the real and imaginary parts of the complex number), you can use a single complex-valued function to describe how the wave evolves. A more common way of saying this is that complex numbers contain phase information.

Usually the physical wave "height" is taken to be the real part of the complex wave.

Extracting the Amplitude and Phase from a Complex Wave

How do we extract the amplitude and phase from the complex wave $\psi(x,t)$? We simply find the modulus of the wave:

$$A = |\psi| =\sqrt{\psi^* \psi }= \sqrt{ |\psi|^2}$$

The phase may be extracted a number of ways, one of which is

$$\phi = \arccos \left( \frac{\text{Re}[\psi]}{|\psi|} \right)$$

Complex Wave Visualization

This animation shows the real and imaginary parts of the complex wave $\psi(x,t) = A e^{i(kx-\omega t)}$ as functions of position $x$ at different times. The wave propagates to the right or left depending on the parameter values.

Legend: Blue curve = Real part $\text{Re}[\psi] = A\cos(kx - \omega t)$, Red curve = Imaginary part $\text{Im}[\psi] = A\sin(kx - \omega t)$

Wave speed: 1.0 m/s | Wavelength: 6.28 m | Period: 6.28 s