Relation Between Phase Difference and Path Difference

An Introduction

Before we talk about the phase difference and the path difference of a wave and the relation between these two quantities, we should know what are the different types of waves and the properties of waves.

Types of Waves

There are two basic types of waves which are as follows:

1. Mechanical Waves: These are those types of waves that can be described as the oscillations of matter. These types of waves require a medium to transfer energy. Examples of mechanical waves can be sound waves, water waves, waves associated with a rope, etc. 

2. Electromagnetic Waves: These are those types of waves that do not require any medium to travel and transfer the energy possessed by them. This means that they can travel in a vacuum as well and these include x-rays, gamma rays, UV rays, etc.

Properties of Waves 

Wavelength: The distance between two consecutive crests or the distance between two consecutive troughs of a wave is known as its wavelength which is measured in meters as well.

Displacement: Displacement can be described as the total distance of a particle from its mean position which is always measured in meters.

Amplitude: Amplitude is the height of a crest or depth of a trough of a wavelength or it can be described as the maximum displacement of a particle. Amplitude like displacement is also measured in meters. 

Time Period: The time period of a wave is defined as the time taken by that wave to complete one full oscillation. The unit in terms of which the time period of a wave is measured is second. 

Time period = \[\frac{1}{frequency}\]

Frequency: The frequency of a wave can be described as the number of complete oscillations made by that wave in one second and the unit of frequency is Hertz (Hz). 

Frequency =\[\frac{1}{\text{time period}}\]

These were some common properties of waves, now we will talk about two additional properties of waves i.e., the phase difference and the path difference. These two properties of waves are discussed below. 

If we talk about the phase difference, it can be called the difference in the phase angle of the two waves while the difference in the path traversed by the two waves can be described as the path difference. The phase difference increases with an increase in the path difference and decreases with a decrease in the path difference which implies that both these quantities are directly proportional to each other. Here, in this article, we will talk about the phase difference, the path difference, and the relation between them.

Phase Difference and Path Difference

If we take two waves that have the same frequency, the relation between their path difference and phase difference is given as: 

\[\triangle\]x = (λ/2π).\[\triangle\]\[\phi\]

Where the path difference between the two waves of the same frequency is Δx and the phase difference between the two waves is\[\triangle\]\[\phi\]. 

Meter is the unit.

The above equation can be written in other ways as well. For example, the general equation that relates the path difference with the phase difference can be written as:

(\[\triangle\]x/λ) = (\[\triangle\]\[\phi\]/2π) 

If we check, we will find that this relation has no units.

From the above equation, we can derive the formula for the path difference which comes out to be:

\[\triangle\]\[\phi\] = (2π.Δx)/λ

After solving this equation, we find that the unit of path difference is Radian or degree. 

We can also derive the formula for the path difference from the relation between the path difference and phase difference which comes out to be:

\[\triangle\]x = (λ/2π).\[\triangle\]\[\phi\]

After solving the above equation, the unit of the path difference comes out to be 'meter'. 

So, we have discussed the relation between the phase difference and path difference relation. From that equation, we derived the equation of the path difference (\[\triangle\]x) in terms of the phase difference and found out that its unit is meter. 

Similarly, we derived the equation of phase difference (\[\triangle\]\[\phi\]) in terms of the path difference and also found that the unit of the phase difference is radian or degree. 

Let's now talk about some of the properties of waves. 

There's an important point to remember here which is that the phase difference can not be defined for a single wave. It is defined when two waves of the same frequency are taken into consideration.

Relation Between Phase Difference & Path Difference 

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

\[\triangle\]x=λ2π = \[\triangle\]\[\phi\]

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.

Path Difference Definition Physics

The path difference between the 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 traveled from the source to the observer.

Phase Difference

Articles 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:

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 periods are 4s.

Imagine that we are big fans 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.

Phase Difference Waves

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\[\phi\].

The phase difference is represented by the following sine wave:

Phase Difference and Path Difference Equation

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

Phase difference (in radians) = 2π(path difference in m)λ(wavelengths meter )

Difference Between Phase Difference and Path Difference

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