Waves as Vectors

Summary: Waves can be represented by vectors. The length of the vector represents the wave's amplitude and the direction the vector points represents the wave's phase. The wave's oscillation can be represented by rotating the vector.

Waves as vectors

A wave can be described by its amplitude, frequency, and phase. Phase describes how a wave evolves through time. As a quantity, it generally only makes sense as a phase shift relative to some other wave. For instance, these two waves are shifted by a quarter of their wavelength.

We would describe this as a phase shift of 90° or, more typically, π/2 in radians. At first, the notion of describing a phase shift (which in this graph looks like a movement of the wave left or right) as a rotation may be confusing. It can be helpful to imagine these waves as three-dimensional helices viewed from the side, rather than 2D waves:

Here, we see that a sine wave can be modeled as the rotation of a vector (blue arrow, right of the animation) with a length equal to the wave’s amplitude at a rotational velocity of $\omega = 2\pi{}f$, where $f$ is the wave’s frequency. Put another way, the speed of rotation represents frequency — a vector which spins faster traces out a wave which oscillates more frequently.

Now, consider what happens when we rotate the vector by π/2:

The pink wave appears to be shifted forward in time compared to the blue wave by a quarter of the wavelength.

Adding waves

Summary: Adding two waves together can change their amplitude, phase, or both.

The vector representation is especially useful when we begin to consider sums of waves rather than individual sine waves. For instance, it is an intuitive result that when we add two sine waves with the same frequency and phase together we get another wave of the same frequency and phase, but with a greater amplitude:

Using the vector notation for this simple example, we observe the same behavior:

Adding the two vectors together produces a vector pointing in the same direction, rotating at the same speed, but with a longer total length. In this case, the utility of thinking about adding waves this way may be unclear, but consider the following surprising result:

In this example, the red wave has the same frequency as the blue wave, but a much smaller amplitude and a π/2 phase shift. When we add together these two waves, we get the purple wave as a result — it looks like a phase-shifted version of the blue wave! This result is less surprising if we represent the waves as vectors instead:

Because the red vector is always pointing perpendicular to the blue vector, adding the two has the effect of rotating the blue vector with only a very modest effect on the final magnitude. Recalling that a rotation is equivalent to a phase shift, we have arrived at the same result as directly adding each point of the wave.

The animation of vector rotation is helpful for developing a sense of what these vectors represent, but makes the figures cumbersome. For the rest of this guide, we will only draw the waves at some static position — implicitly, the vectors rotate as time passes or, equivalently, as we move through space. For instance, the above animation would be drawn like so:

Using this method, it is clear that the resulting wave has approximately the same amplitude as wave 1 (precisely $\sqrt{|Wave\ 1|^2 + |Wave\ 2|^2}$, where $|Wave\ 1|$ is the magnitude of Wave 1’s vector), but has a phase shift of $\arctan{\frac{|Wave\ 2|}{|Wave\ 1|}}$. If $|Wave\ 2| \ll |Wave\ 1|$, we can approximate the Result wave by shifting the phase of Wave 1 by the magnitude of Wave 2. This approximation is closely related to the Weak Phase Object approximation, covered in Contrast in Cryo-EM.