Revision: Oscillations and Waves Physics (Theory) ISC (Science) ISC Class 11 CISCE

The number of oscillations produced by the particle per second is called frequency. It is denoted by f. SI unit for frequency is s⁻¹ or hertz (Hz).

Mathematically, frequency is related to the time period by f = `1/"T"`

Time period: The time period is defined as the time taken by a particle to complete one oscillation. It is usually denoted by T. For one complete revolution, the time taken is t = T, therefore,

`ω"T"` = 2π ⇒ T = `(2π)/ω`

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

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 pulses overlap, the resultant displacement is the algebraic sum of displacement due to each pulse.

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.