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Advanced Physics Topics like quantum mechanics and relativity have revolutionized our understanding of the universe.

## What is an Example of a Superposition of Waves?

### Principle of Superposition of Waves

The simultaneous progress of more than one waves through a region of space produces the phenomenon known as the superposition of waves. During superposition, each of the waves travels independently, i.e., a wave retains its individual properties though it overlaps on other waves. For example, when different musical instruments are played at a time, the noise from each instrument can be detected separately.

If more than one waves are incident simultaneously on a particle in a medium, the particle would have different displacements for each of the separate waves. But these displacements of the particle occur at the same time, i.e., a resultant displacement of the particle occurs for all the waves. Since displacement is a vector quantity, the resultant displacement is the vector sum of all the individual displacements. This is known as the principle of superposition of waves.

Superposition of two single waves or two wave pulses:

For example, let two wave pulses P and Q of equal displacement and of same sign (both upward) are travelling

with equal speed along the length of a string but in opposite direction [Fig.(a). Fig.(b) shows the two wave pulses at a later instant before superposition. Fig.(c) shows the result when the two pulse superpose each other at a time r. It is found from Fig.(c) that at time t, a wave pulse R of displacement equal to the sum of the displacements of wave pulses P and Q is formed. Then, after that, the wave pulses P and Q retaining their original shapes move along the string as shown in Fig.(d).

Now, let the two wave pulses P and Q having equal displacement hut of opposite sign are moving along the length of a string with equal speed in opposite direction [Fig.(a)]. Fig.(b) shows that after a small time, the two pulses get closer to each other. At the time r, the two wave pulses superpose and produce a resultant wave pulse R of zero displacement Fig.(c). After that the two wave pulses retain their original shapes and continue to move along the length of the string as shown in Fig.(d).

Statement: The resultant displacement of a particle in a medium due to more than one waves is equal to the vector sum of the different displacements produced by the individual waves separately.

Let n number of waves travelling in a medium superpose on each other. If \(\overrightarrow{y_1}\), \(\overrightarrow{y_2}\), y\(\overrightarrow{y_3}\), \(\overrightarrow{y_n}\) are the displacements at a point due to n waves, then the resultant displacement will be

\(\vec{y}\) = \(\overrightarrow{y_1}\) + \(\overrightarrow{y_2}\) + \(\overrightarrow{y_3}\) + ….. \(\overrightarrow{y_n}\)

If the displacements due to the two wave pulses are equal and in the same direction, i.e., |\(\overrightarrow{y_1}\)| = |\(\overrightarrow{y_2}\)| = A, then from the superposition principle, the magnitude of the resultant displacement will be, |\(\overrightarrow{y_1}\)| = A + A = 2A, as shown in Fig.(c). If the displacements are equal but in opposite direction, then the magnitude of the resultant displacement

will be

|\(\vec{y}\)| = A + (-A) = A – A = 0, as shown in Fig.(c).

If different displacements are collinear, it is sufficient to take their algebraic sum, i.e., to determine the resultant displacement1 two like vectors are added while two unlike vectors are subtracted. The principle of superposition is applicable to all types of waves, say, electromagnetic waves, sound waves, etc.

The best example of superposition of wave is the melody of musical instruments. Another classic example is the throwing of more than one stone in a lake. In fact most of the sounds we produce while speaking is a superposition. In case of light wave, this is also valid. Any light in nature we see, is in general a superposition. The superposition of similar waves gives rise to the following important phenomena:

Stationary waves: The superposition of two identical but oppositely directed progressive waves produces stationary waves.

Beats: Two progressive waves of the same amplitude and velocity, but of slightly different frequencies produce beats on superposition.

Interference: Two identical progressive waves on superposition with a constant phase difference, produce interference.

The first two of these three phenomena will be discussed in this chapter, with special emphasis on sound waves.

### Numerical Examples

**Example 1.**

The displacement of a periodically vibrating particle is y = 4cos^{2}(\(\frac{1}{2}\)t)sin(1000t). Calculate the number of harmonic waves that are superposed.

**Solution:**

y = 4 cos^{2}(\(\frac{1}{2}\)t)sin(1000t)

= 2 ᐧ 2 cos^{1}(\(\frac{1}{2}\)t) ᐧ sin(1000t)

= 2(1 + cost) + sin(1000t)

= 2sin(1000t) + 2sin(100t)cos t

= 2 sin(1000t) + sin(1000t + t) + sin(100t – t)

= 2 sin(1000t) + 1 sin(1001 t) + 1 sin(999t)

= y_{1} + y_{2} + y_{3}

Here each of y_{1}, y_{2} and y_{3} is in the form of A sinωt. Thus, each of them represents a harmonic wave.

Hence, the number of superposed harmonic waves = 3.

**Example 2.**

The displacements of a particle at the position x = 0 in a medium due to two different progressive waves are y_{1} = sin4πt and y_{2} = sin2πt, respectively. How many times would the particle come to rest in every second?

**Solution:**

According to the principle of superposition, the resultant displacement of the particle is

y = y_{1} + y_{2} = sin4πt + sin2πt

= 2sin\(\frac{4 \pi t+2 \pi t}{2}\)cos\(\frac{4 \pi t-2 \pi t}{2}\) = 2sin3πt ᐧ cosπt

The particle comes to rest (y = 0), when either sin3πt = 0 or cosπt = 0.

When sin3πt = 0, we have t = 0, \(\frac{1}{3}\)s, \(\frac{2}{3}\)s (t < 1s)

Again, when cos ir t = 0 , we have t = \(\frac{1}{2}\)s (t < 1 s)

∴ y = 0 , when r = 0, \(\frac{1}{3}\)s, \(\frac{1}{2}\)s, \(\frac{2}{3}\)s

In every second, the particle comes to rest 4 times.