Phase Difference

The wave phase formula

To calculate the wave phase in an arbitrary position, you need to identify how far this position is from the beginning of your wave cycle. In the simplest case, if your wave can be approximated by a sine or cosine function, your wave equation can be simplified as:

\[y = A \cdot \sin(x)\]

Here, A is the maximum amplitude of the wave, x is the value on the horizontal axis, which repeats from 0 to 2π for sine/cosine functions, and y is the wave height at x. The phase of any point x can be determined using the equation below:

\[x = \sin^{-1}(y)\]

The equation gives you the value of x in radians, which you need to convert to degrees to obtain the phase. This is done by multiplying x by 180 degrees and then dividing by π.

\[\phi(x) = x \cdot \frac{180^{\circ}}{\pi}\]

The phase difference in waves

The phase difference of waves occurs when two waves move and their cycles do not coincide. The phase difference is known as the cycle difference between two waves at the same point.

Overlapping waves that have the same cycle are known as waves in phase, while waves with phase differences that do not overlap are known as out-of-phase waves. Waves that are out of phase can cancel each other out, while waves in phase can amplify each other.

The phase difference formula

If two waves have the same frequency/period, we can calculate their phase difference. We will need to calculate the difference in radians between the two crests that are next to each other, as in the following figure.

Fig. 2 - The difference in phases between two waves i(t) and u(t) that vary regarding time (t) causes a space difference in their propagation

This difference is the phase difference:

\[\Delta \phi = \phi_1 - \phi_2\]

Here is an example of how to calculate the wave phase and the wave phase difference.

In phase waves

When waves are in phase, their crests and troughs coincide with each other, as shown in figure 3. Waves in phase experience constructive interference. If they vary in time (i(t) and u(t)), they combine their intensity (right: purple).

Fig. 3 - Constructive interference

Out-of-phase waves

Waves that are out of phase produce an irregular pattern of oscillation, as the crests and troughs don’t overlap. In extreme cases, when the phases are shifted by π [rad] or 180 degrees, the waves cancel each other out if they have the same amplitude (see the figure below). If that is the case, the waves are said to be in anti-phase, and the effect of that is known as destructive interference.

Fig. 4 - Out of phase waves experience destructive interference. In this case, waves \(i(t)\) and \(u(t)\) have a \(180\) degrees phase difference, causing them to cancel each other out

The phase difference in different wave phenomena

The phase difference produces different effects, depending on the wave phenomena, which can be used for many practical applications.

• Seismic waves: systems of springs, masses, and resonators use cyclical movement to counteract vibrations produced by seismic waves. Systems installed in many buildings reduce the amplitude of the oscillations, thus reducing structural stress.
• Noise-cancelling technologies: many noise-cancelling technologies use a system of sensors to measure the incoming frequencies and produce a sound signal that cancels those incoming sound waves out. The incoming sound waves thus see their amplitude reduced, which in sound is directly related to the noise intensity.
• Power systems: where an alternating current is being used, voltage and currents can have a phase difference. This is used to identify the circuit as its value will be negative in capacitive circuits and positive in inductive circuits.