Revision: Class 11 >> Wave NEET (UG) 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)

Harmonics are integral multiples of the fundamental frequency.

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

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.

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.

Node: Point where displacement is zero.

Antinode: Point where displacement is maximum.

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 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 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.

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 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.

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

Transverse Waves:

Longitudinal 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.