Revision: Oscillations and Waves JEE Main Oscillations and Waves

When a body oscillates under the influence of an external periodic force, not with its own natural frequency but with the frequency of the external periodic force, its oscillations are called forced oscillations.

The particular case of forced oscillations in which the frequency of the driving force equals the natural frequency of the oscillator, and the amplitude of oscillations is very large — such oscillations are called resonant oscillations and the phenomenon is called resonance.

The motion which repeats itself after equal intervals of time is called periodic motion.

The periodic (to and fro) and bounded motion of a body about a fixed point is called oscillatory motion.

When a body, capable of oscillation, is slightly displaced from its position of equilibrium and left to itself, it starts oscillating with a frequency of its own — such oscillations are called free oscillations.

The oscillations in which the amplitude decreases gradually with the passage of time are called damped oscillations.

When energy is continuously supplied from outside to an oscillating system at the same rate at which energy is lost, so that the amplitude remains constant, such oscillations are called maintained oscillations.

The constant φ in the equation of SHM x = Acos⁡(ωt + ϕ) is called phase constant or initial phase.

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 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"`

Any function that repeats itself at regular intervals of its argument is called a periodic function.

The physical quantity which represents the position and direction of motion at any instant of the particle executing SHM is called phase.

If the restoring force/torque acting on a body in oscillatory motion is always directly proportional to its displacement from the equilibrium position and directed towards it, then the motion is called simple harmonic motion (SHM).

The periodic functions that can be represented by a sine or cosine curve are called harmonic functions.

The periodic functions which cannot be represented by a single sine or cosine function are called non-harmonic functions.

The number of oscillations completed in unit time interval is called frequency of oscillation.

The distance of the particle from the mean position at any instant is called displacement.

The maximum value of displacement of the particle from its equilibrium position is called amplitude.

The rate of angular displacement per unit time is called angular frequency.

The smallest time interval after which the oscillatory motion gets repeated is called time period.

A wave in which particles of the medium vibrate in a direction perpendicular to the direction of propagation of the wave is called a transverse wave.

If, during propagation of a wave in a medium, the particles of the medium perform simple harmonic motion, then the wave is called a ‘simple harmonic progressive wave’.

• The equation of a progressive wave is \[y=A\sin(\omega t\pm kx)\]

Phase gives the state of the vibrating particle at any instant of time with regard to its position and direction of motion.

• Phase is the angular displacement from its mean position.
  \[\phi=(\omega t\pm kx)\]

The wave speed is the distance a wave travels per unit time.

When two waves of the same frequency, wavelength and velocity move in the same direction, on superposition, they result in interference.

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)

Node: Point where displacement is zero.

Antinode: Point where displacement is maximum.

Organ pipes are musical instruments that produce sound by blowing air into a pipe.

• Sound is due to longitudinal standing waves formed by the superposition of incident and reflected waves.

The fundamental mode, also known as the first harmonic, is the simplest form of vibration of a wave.

Harmonics are integral multiples of the fundamental frequency.

Two progressive waves having the same amplitude and time period/ frequency/ wavelength travelling with similar speed along the same straight line in opposite directions superimpose, forming another wave known as a stationary wave or standing wave.

A string is a stretched medium under tension in which transverse waves propagate, and standing waves are formed due to the superposition of incident and reflected waves.

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

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

The apparent change in the frequency of sound heard by a listener, due to relative motion between the source of sound and the listener is called Doppler effect in sound.

When the source and the observer are in relative motion with respect to each other and to the medium in which sound propagates, the frequency of the sound wave observed is different from the frequency of the source. This phenomenon is called Doppler Effect.

The apparent change in frequency of sound heard by a listener due to relative motion between the source and the listener is called the Doppler effect.

The phenomenon that occurs when two waves meet while travelling along the same medium is called wave interference.

The phenomenon of redistribution of energy on account of superposition of light waves from two coherent sources is called interference of light.

The points of maximum intensity in the regions of superposition of waves are said to be in constructive interference.

The points of minimum intensity in the regions of superposition of waves are said to be in destructive interference.

v = f λ

\[v=\sqrt{\frac{P}{\rho}}\]

\[v=\sqrt{\frac{\gamma P}{\rho}}\]

\[\gamma=\frac{C_p}{C_v}\]

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 waves of amplitudes a₁ and a_2​ interfere at a point where phase difference is ϕ, the resultant amplitude is:

\[A^2=a_1^2+a_2^2+2a_1a_2\cos\phi\]

I = I₁ ​+ I₂​ + 2\[\sqrt {I_1​I_2}\] ​​⋅ cos ϕ

When I₁ = I₂ = I₀:

I = \[2I_0(1+\cos\phi)=4I_0\cos^2\left(\frac{\phi}{2}\right)\]

\[\frac{I_{\max}}{I_{\min}}=\left(\frac{a_1+a_2}{a_1-a_2}\right)^2=\left(\frac{\sqrt{I_1}+\sqrt{I_2}}{\sqrt{I_1}-\sqrt{I_2}}\right)^2\]

When two or more pulses overlap, the resultant displacement is the algebraic sum of the displacements 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.

Thomas Young first demonstrated the phenomenon of interference of light with the help of a slit, using a monochromatic source and two slits S₁ and S₂​, producing alternating bright fringes (constructive interference) and dark fringes (destructive interference) on a screen.

Types of Waves

• Mechanical waves → Need medium (sound, water)
• Electromagnetic waves → No medium (light, X-rays)
• Matter waves → Associated with particles

Special Waves:

• Capillary waves → surface tension
• Gravity waves → gravity
• Progressive waves → energy transfer
• Stationary waves → no energy transfer

Important Terms

• Wavelength (λ) → Distance between two successive crests/troughs
• Frequency (f) → Number of waves per second
• Velocity (v) → Speed of wave
• Amplitude (A) → Maximum displacement
• Angular frequency (ω) → \[\omega=2\pi f=\frac{2\pi}{T}\]
• Wave number (k) → \[k=\frac{2\pi}{\lambda}\]
• Wave velocity relations → \[v=f\lambda=\frac{\omega}{k}\]

Types of Wave Motion

(a) Transverse Waves

• Particle motion ⟂ (perpendicular) to wave direction
• Examples: Light waves, waves on a string

(b) Longitudinal Waves

• Particle motion ∥ (parallel) to wave direction
• Examples: Sound waves

• 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.