Revision: 12th Std >> Superposition of Waves MAH-MHT CET (PCM/PCB) Superposition of Waves

The phenomenon that occurs when a progressive wave travelling through a medium reaches a rigid boundary and gets reflected is called reflection of waves.

A wave in which the disturbance produced in the medium travels in a given direction continuously, without any damping and obstruction, from one particle to another, is called a progressive wave.

A wave in which vibrations of particles produce compressions and rarefactions along the direction of propagation of the wave is called a longitudinal wave.

A wave in which vibrations of particles are perpendicular to the direction of propagation of the wave and produce crests and troughs in their medium of travel is called a transverse wave.

When waves are incident on the boundary of two media, then a part of the incident waves is returned to the initial medium, which is known as reflection.

Rigid End:

• Wave reflects with phase change = 180° (π)
• Wave gets inverted (crest → trough)

Free End:

• No phase change
• Wave is not inverted (crest → crest)

The phenomenon in which, when two or more waves arrive at a point simultaneously, each wave produces its own displacement independent of the other, and the resultant displacement at a point is the vector sum of the displacements due to the individual waves, is called superposition of waves.

When two identical waves travelling along the same path in opposite directions interfere with each other, the resultant wave is called a stationary wave.

The vibration that occurs without any external periodic force after an initial disturbance, where the body vibrates at its own natural frequency, is called free vibration.

The vibration that occurs due to a continuous external force acting on the system, where the body vibrates at the frequency of the external periodic force, is called forced vibration.

Only those multiples of fundamental frequency which are actually present in a given sound are called overtones.

The musical instruments which are used for producing musical sounds by blowing air into them are called organ pipes.

All the frequencies that are integral multiples of the fundamental frequency are called harmonics.

The distance between the open end of the pipe and the position of antinode is called end correction.

Standing waves are formed on a string fixed at both ends when plucked at the centre, such that a node is formed at rigid ends and an antinode is formed in between them — this phenomenon is called vibrations produced in a string.

The minimum intensity point produced during the formation of beats is called waning.

The variation in intensity of sound with time at a particular position, due to the principle of superposition of two sound waves of slightly different frequencies, is called beats.

The periodic variation of intensity of sound between maximum and minimum due to superimposition of two sound waves of same amplitude and slightly different frequencies is called the phenomenon of beats.

The maximum intensity point produced during the formation of beats is called waxing.

The frequency of a particle executing S.H.M. is equal to the number of oscillations completed in one second. 

The musical instruments in which sound is produced by setting vibration in a stretched membrane are called percussion instruments. (Examples: Tabla, drum)

The musical instruments that consist of an air column in which sound is produced by setting vibrations of the air column are called wind instruments. (Examples: Mouth organ, flute)

The musical instruments that consist of a stretched string in which sound is produced by plucking of strings are called stringed instruments. (Examples: Tanpura, sitar, guitar)

y(x,t) = a sin(kx − ωt + ϕ₀​)

where:

• a = Amplitude
• k = Angular wave number
• ω = Angular frequency
• ϕ_0​ = Initial phase

If displacement is a linear combination of sine and cosine:

y(x,t) = A sin(kx − ωt) + B cos(kx − ωt)

y = 2A cos\[\left(\frac{2\pi x}{\lambda}\right)\sin(2\pi nt)\]

In terms of resultant amplitude R: y = R sin(2πnt)

where: R = 2A cos⁡\[\left(\frac{2\pi x}{\lambda}\right)\]

Combining all three laws:

n = \[\frac {1}{2l}\]\[\sqrt {\frac {T}{​m}}\]

where:

• n = fundamental frequency
• l = length of string
• T = tension in the string
• m = mass per unit length of the string

The number of beats formed per second is expressed as ∣v₁ − v₂∣, i.e., either (v₁ − v₂) or (v₂ − v₁), where v₁​ and v₂ are frequencies of the two sound waves.

N = n₁ ​− n₂​

The beat period is the reciprocal of beat frequency:

T = \[\frac{1}{n_1-n_2}\] or T = \[\frac{1}{|v_1-v_2|}\]

When two or more waves arrive at a point simultaneously, each wave produces its own displacement independent of the other. The resultant displacement at a point is the vector sum (algebraic sum) of the displacements due to the individual waves.

The fundamental frequency of vibrations of a string is inversely proportional to the length of the vibrating string if tension and mass per unit length are constant. 

∴ n ∝ `1/l`    ...(if T and m are constant.)

The fundamental frequency of vibrations of a string is inversely proportional to the square root of mass per unit length (linear density) if the tension and vibrating length of the string are constant.

∴ `n ∝ 1/sqrtm`  ...(if T and l are constant.)

It states that the vibrations of a string is inversely proportional to the length of the vibrating string when tension (T) and mass per unit length (m) are constants.

It states that the fundamental frequency of vibration of a string is directly proportional to the square root of tension, when length (l) and mass per unit length (m) are constants.

It states that the fundamental frequency of vibrations of a stretched string is inversely proportional to the square root of mass per unit length, when tension (T) and length (l) are constants.

n ∝ \[\frac {1}{\sqrt m}\]​​(T and l constant)

Also expressed as: n ∝ \[\frac {1}{\sqrt ρ}\]​ ​n ∝ \[\frac {1}{r}\]

Consider two sound waves, having the same amplitude and slightly different frequencies n₁ and n₂. Let us assume that they arrive in phase at some point x of the medium. The displacement due to each wave at any instant of time at that point is given as

`y_1 = A sin {2pi (n_1t - x/lambda_1)}`

`y_2 = A sin {2pi (n_2t - x/lambda_2)}`

Let us assume for simplicity that the listener is at x = 0.

∴ y₁ = A sin (2πn₁t)     ...(i)

and y₂ = A sin (2πn₂t)     ...(ii)

According to the principle of superposition of waves,

y = y₁ + y₂

∴ y = A sin (2πn₁t) + A sin (2πn₂t)

By using formula,

sin C + sin D = 2 sin `((C + D)/2) cos ((C − D)/2)`

y = `A[2sin((2pin_1t + 2pi n_2t)/2 )] cos [((2pin_1t - 2pin_2t)/2)]`

y = `2A sin [2pi ((n_1 + n_2)/2)t] cos [2pi ((n_1 - n_2)/2)t]`

∴ y = `R sin [2pi ((n_1 + n_2)/2)t]`

y = R sin (2πnt)     ...(iii)

Where,

R = `2A cos[(2pi(n_1 - n_2))/(2)t]` and n = `(n_1 + n_2)/2`

Equation (iii) is the equation of a progressive wave having frequency `((n_1 + n_2)/2)` and resultant amplitude R.

For waxing,

A = ± 2a

`therefore 2A cos [2pi((n_1 - n_2)/2)t] = +- 2A`

`therefore cos [2pi ((n_1 - n_2)/2)]t = +-( 2A)/(2A)`

`therefore cos [2pi ((n_1 - n_2)/2)]t = +- 1`

This is possible if

`2pi ((n_1 - n_2)/2)t = 0, pi, 2pi, 3pi, ....`

i.e. t = 0, `1/(n_1 - n_2), 2/(n_1 - n_2), 3/(n_1 - n_2), ...`

∴ Period of beat T = `[1/(n_1 - n_2) - 0]`

T = `1/(n_1 - n_2)`

∴ Frequency of beats n = `1/T`

n = n₁ − n₂

Thus, the frequency of beats is equal to the difference between the frequencies of the two sound notes giving rise to beats.

• Beats are formed when two waves of same amplitude but slightly different frequencies superimpose.
• Waxing and waning are alternatively produced.
• The greater the difference in frequency between the two waves, the higher the beat frequency.