Principle of superposition of waves, different cases, practice problems, FAQs

If you and your friend are standing at some distance and holding the two ends of the string. When both of you start oscillating corners of string up and down, you will notice that the oscillation of one of you will reach the other end by crossing the oscillation of the other.This is called the principle of superposition of wave.

Table of content

• Superposition of waves
• Practice problem
• FAQs

Superposition of waves

If any external agent produces different disturbances simultaneously causing different vertical disturbances (or amplitude) in the same string, then the resultant vertical displacement in the string will be the vector sum of the individual displacements. 

Mathematically, if , , ,............. , are the individual displacements given in the string, then the resultant vertical displacement will be,

• Case 1 (Two wave pulses have positive amplitude and move along the same plane): 

Suppose two wave pulses propagate in the opposite direction of one another with positive vertical displacements and , respectively, then at the moment of superposition the resultant displacement becomes, 

As shown in the following figure:

                         (Recreated all)

• Case 2 (Both the pulses move along the same plane, where one has a negative amplitude and the other has a positive amplitude): 

In this case, at the moment of superposition the resultant displacement becomes,

 As shown in the following figure. Here, if the vertical displacement of the two wave pulses become the same in magnitude, then the net displacement becomes zero. This means that the pulses cancel each other at the moment of superposition. The given figures justify this scenario.

• Case 3 (Two pulses move along two planes perpendicular to each other with two different amplitudes): 

In this case, at the moment of superposition the resultant amplitude becomes, 

and the resultant pulse makes an angle of 45o with the planes on which the individual pulse was moving. 

Now Consider two harmonic traveling waves on a stretched string, both with the same (angular frequency) and (wave number), and, therefore, the identical wavelength. Their wave speed will be equal. consider that their amplitudes are the same and both are traveling in the positive x direction.Describe by the function

Net displacement given by principle of superposition

Using trigonometric formula

The resultant is also a harmonic traveling wave in the positive direction of the x-axis, with the same frequency and wavelength. And its initial phase angle is . The amplitude is a function of the phase difference between the two waves

When , i.e. the waves are in phase and the resultant wave has amplitude 2a, the largest possible value for amplitude. This is the case of constructive interference.

For , the waves are completely, out of phase and the resultant wave has zero displacement everywhere at all times. This is the case of destructive interference.

Practice problem

Q 1.Two waves passing through a region are represented as follows:

y1 = (5 mm) sin [(2π cm-1 )x − (50π s-1)t] and

y2 = (10 mm) sin [(π cm-1 )x − (100π s-1)t]

Find the displacement of the particle at x = 1 cm at time t = 5.0 ms.

A. Given,

y1 = (5 mm) sin [(2π cm-1 )x − (50π s-1)t] ……………………(i)

y2 = (10 mm) sin [(π cm-1 )x − (100π s-1)t] ………………….(ii)

By putting x = 1 cm and t = 5.0 = 5.0 × 10-3 s in equations (i) and (ii), we get,

y1 = (5 mm) sin [(2π − (50π s-1)(5.0 × 10-3 s)] 

Now,

y2 = (10 mm) sin [(π cm-1 )(1 cm) − (100π s-1)(5.0 × 10-3 s)]

Therefore, the resultant displacement of the particle at x = 1 cm at time t = 5.0 ms is given by,

Q 2.Consider the two identical pulses in a stretched string are initially at the distance of 8 cm. They are moving towards each other, as shown in the figure. The speed of each pulse is 2 cm s-1. After 2 s, what will be the total energy of the pulses?

a. Zero
b. Purely kinetic
c. Purely potential
d. Partial kinetic and partial potential

A.The distancetraveled by the first wave in 2 s is given as follows:

 x1= v × t

⇒ x1= 2 × 2 = 4 cm

The distancetraveled by the second wave in 2 s is given as follows: 

x2= v × t

⇒ x2= 2 × 2 = 4 cm

After 2 s, both the waves will be at the same position. 

When they interfere, the net displacement will be zero. Since there is no stretching, there is no potential energy at this point.

Thus, PE = 0

However, the velocity of the wave pulses is not zero at that point. Therefore, the KE is greater than zero.

I.e. KE ≠ 0

Hence,

Total energy = Kinetic energy + Potential energy

⇒ TE = KE + PE

⇒ TE = KE

At this point, the total energy is purely kinetic.

Thus, option (B) is the correct answer.

Q 3. Two wave represented by , are superimposed, find the resultant amplitude. 

A.Given

And

The resultant amplitude of superimposed wave is given by 

, and putting all values

Q 4. If is expressed as then find the value of ?

A.

Given:  

Phase difference between Two part waves is

We know that for superposition of two wave 

FAQs

Q 1. What are coherent waves?
A. Two waves are said to be coherent if they have the same frequency and constant phase difference.

Q 2. What does it mean that Two waves are in phase?
A. If two coherent waves align in such a way that their phase difference is zero then they are called in phase.

Q 3. The superposition of two waves of slightly different frequency is called.
A. When two waves of nearly equal frequency superimposed on each other then beats are produced.

Q 4. Echo is an example of 

a. Reflection
b. Refraction
c. Transmision
d. None of these

A. Echo is an example of reflection by a rigid boundary.

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