The phase of a wave is the value representing a fraction of a wave cycle. In a wave, a complete cycle, from crest to crest or trough to trough, is equal to 2π [rad]. Every fraction of that length, therefore, is less than 2π [rad]. Half a cycle is π [rad], while a quarter of a cycle is π/2 [rad]. The phase is measured in radians, which are non-dimensional units.
Fig. 1 - Wave cycles are divided into radians, with each cycle covering 2π [rad] of distance. Cycles repeat after 2π [rad] (red values). Every value larger than 2π [rad] is a repetition of the values between 0π [rad] and 2π [rad]
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:
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:
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.
\[\phi(x) = x \cdot \frac{180^{\circ}}{\pi}\]
Sometimes a wave can be represented by an expression such as \(y = A \cdot \sin(x - \phi)\). In these cases, the wave is out of phase by \(\phi\) radians.
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.
A wave with a maximum amplitude A of 2 metres is represented by a sine function. Calculate the wave phase when the wave has an amplitude of y = 1.
Using the \(y = A \cdot \sin (x)\) relationship and solving for x gives us the following equation:
\[x = \sin^{-1}\Big(\frac{y}{A}\Big) = \sin^{-1}\Big(\frac{1}{2}\Big)\]
This gives us:
\(x = 30^{\circ}\)
Converting the result to radians, we get:
\[\phi(30) = 30^{\circ} \cdot \frac{\pi}{180^\circ} = \frac{\pi}{6}\]
Now let’s say another wave with the same frequency and amplitude is out of phase with the first wave, with its phase at the same point x being equal to 15 degrees. What is the phase difference between the two?
First, we need to calculate the phase in radians for 15 degrees.
\[\phi(15) = 15^{\circ} \cdot \frac{\pi}{180^\circ} = \frac{\pi}{12}\]
Subtracting both phases, we obtain the phase difference:
\[\Delta \phi = \phi(15) - \phi(30) = \frac{\pi}{12}\]
In this case, we can see that the waves are out of phase by π / 12, which is 15 degrees.
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.
Seismic technology relies upon spring-mass systems to counteract the movement of seismic waves as, for instance, in the Taipei 101 tower. The pendulum is a sphere with a weight of 660 metric tons. When strong winds or seismic waves hit the building, the pendulum swings back and forth, swinging in the opposite direction to where the building moves.
Fig. 5 - The movement of the pendulum at the Taipei 101 tower is out of phase with the movement of the building by 180 degrees. Forces acting on the building (Fb) are counteracted by the pendulum force (Fp) (the pendulum is the sphere).
The pendulum reduces the oscillations of the building and also dissipates the energy, thus acting as a tuned mass damper. An example of the pendulum in action was observed in 2015 when a typhoon caused the pendulum ball to swing by more than a meter.
Phase Difference - Key takeaways
- The phase difference is the value representing a fraction of a wave cycle.
- In phase waves overlap and create a constructive interference, which increases their maximums and minimums.
- Out of phase waves create a destructive interference that creates irregular patterns. In extreme cases, when the waves are out phase by 180 degrees but have the same amplitude, they cancel each other out.
- Phase difference has been useful to create technologies in seismic mitigation and sound-cancelling technologies.
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