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For any two waves, the relation between the phase difference and the path difference can be stated as:

\[\Delta x = \frac{\lambda}{2 \pi} = \triangle \phi\]

The above is the phase difference and path difference relation.

Here,

\[\triangle x\] = path difference

\[\triangle \phi\] = phase difference

The path difference and the phase difference have no SI units that means their unit is one.

We define the phase difference between any two consecutive points in terms of radians, whereas the path difference is the integral number of wavelengths in a phase.

Now, let’s discuss the relationship between phase difference and path difference.

Displacement

The displacement is the distance a particle moved from its mean position. It is measured in meters.

Amplitude - Amplitude is the maximum displacement of a particle and it is measured in meters.

Wavelength - The wavelength of the particle is defined as the distance between two adjacent points on the full-wave. It is measured in meters.

Time Period - The time taken to complete a wave or a cycle is called the time period. The time period is measured in s.

Frequency - The number of full waves (or a complete cycle) in a second is called the frequency. It is measured in Hz.

Above these properties, waves have other two important properties viz: Phase difference and path difference.

The path difference of two varying waves is the difference in the distance they covered.

The path difference is the difference in the physical distance between the two sources to the observer, i.e., the difference in distance travelled from the source to the observer.

The path difference is physics for the waves having the same frequency and it is used to find the constructive and destructive interference in waves.

Phase Difference

Particles in waves oscillate. When they oscillate (move to-and-fro), the particles go through phases, from 0° to 360° or zero to 2π.

Where π is one period. The particles go through phases, from When the particle travels the distance of one wavelength (since a particle travels the distance of one wavelength in the time duration of one period).

Consider the displacement-time and displacement-phase graph of particles drawn below:

(Image will be uploaded soon)

In one period, the particle undergoes a phase change of 2π.

Now, take any two points in time where the particles’ motion and position are the same. The difference/variation in their phase is their phase difference.

In the example above, the period for both particles is 4s and the phase difference between the particles is π/2.

Similarly, we can draw a parallel between the distance between the source and the phase.

When particles make a displacement equal to their wavelength, they go through phases from 0 to 2π.

For better visualization, let’s consider Particle A and Particle B starting from s = 0 and s = - 2, respectively.

Over a certain period of time, both particles will go through a complete oscillation and go back to their respective starting positions. The time taken is known as the period, T (both their period is 4s.

Imagine that we are a big fan of these two particles. We take 360 photos of them within 4, while they are dancing. With this magical camera, we can capture both the position and the direction of velocity.

Now, what we have to do is, we have to take 360 photos of the same particle from time to time. Here, each photo represents each particle at different phases.

0°, 0°

1°, 1°

2°, 2°

3°, 3°, and so on.

Suppose that we notice that Particle B in Photo 90 looks exactly like Particle A in Photo 0°, i.e., their position and direction of velocity are the same. This means that Particle B “lags” behind Particle A by 90 photos. This means that they have a phase difference of 90° or π/2.

We define the phase difference of a sine wave as the time interval by which one wave leads or lags by another one. One must note that the phase difference is not a property of only one wave, it is the relative property to two or more waves.

We call the phase difference the “Phase offset” or the “Phase angle”. We represent the phase difference by the Greek letter Phi symbolized as to 𝜑.

The phase difference is represented by the following sine wave:

(Image will be uploaded soon)

The equation for the path difference and the phase difference relation is given by:

\[\text{Phase difference (in radians) = } \frac{2 \pi (\text{path difference in m})}{\lambda ({wavelength in meter})}\]

FAQ (Frequently Asked Questions)

Question 1: What is the Relationship Between Phase and Path Difference?

Answer: The relation between phase difference and path difference is very simple to understand. These two are directly proportional to each other.

For any two waves with the same frequency, Phase Difference and Path Difference are related as - Δx is the path difference between the two waves, while ΔΦ is the phase difference between two consecutive waves.

Question 2: What is the Phase Difference in Geometry?

Answer: If the frequencies are varying from each other, the phase difference ϕ increases linearly time t. The phenomenon of the change in period during periodic changes from reinforcement and opposition is called beating.

Question 3: What is the Formula for Path Difference?

Answer: If the distance travelled by the waves from two positions is the same, then the path difference is zero. Once you know the path difference, you can easily find the phase difference using the formula given below: Here, Δx is path difference, and ΔΦ is phase difference.

Question 4: What does the Phase Difference tell us?

Answer: We define the phase Difference (ϕ) between two particles or two waves tells us how much a wave or a particle is in front or behind another wave or a particle. The value of the phase difference ranges from 0 to 2π (in radians) The below diagram shows that P1 and P2 are in phase:

(Image will be uploaded soon)