Doppler's Effect and Musical Sound

 Doppler’s effect

When a source of sound or an observer or both are in motion relative to air, there is an apparent change in the frequency of sound as heard by the observer. This phenomenon is called doppler’s effect.

Doppler’s effect is shown by both light and sound wave.

Doppler’s effect holds true not only for sound waves but also for electromagnetic waves (microwave, radio waves) and visible light.

Motion of source changes the apparent wavelength whereas that of observer changes the apparent frequency (i.e., no. of waves received)

All cases of doppler effect can be written in single formula

i.e.,

$\textbf{sign convention}$

i. For same direction of velocities negative sign (-) is taken.

ii. For opposite direction of velocities positive sign (+) is taken.

-This formula is applied for either source or observer moving or stationary.

Various cases in Doppler's Effect

1. When source of sound moves towards the observer at rest
2. When source of sound moves away from the observer at rest
3. When observer moves towards the source in rest
4. When observer moves away from the source in rest.
5. When source and observer moves towards from each other.
6. When source and observer moves away from each other.
7. When source moves away from observer and observer moves towards source.
8. When observer moves away from source and source moes towards observer.

Doppler’s effect is not observed when

i. Relative velocity between source and observer is zero.

ii. Source and observer move in mutually perpendicular direction. (transverse direction)

iii. Only medium moves.

iv. Source and observer move in same direction with same velocity.

v. Distance between source and observer remains constant e.g., a source revolving in circular path around the observer at centre.

Doppler’s effect in light

Doppler’s effect for light is symmetrical, i.e., whether the source is moving towards a stationary observer or the observer is moving towards the stationary source, the doppler shift for the light is the same for a given relative velocity.

There occurs a transverse doppler shift for light,

$\textbf{Here for any case:}$

i. Change in frequency or doppler’s shift in frequency $∆f = \frac{v}{c} f$

Where v is the velocity in between source (star) and observer (earth) c is the velocity of light.

ii. Change in wavelength or doppler’s shift in wavelength

$∆λ = \frac{v}{c} λ$

Recessional red shift that a star is moving away earth. Hence apparent λ increases.

Violet or blue signifies that a star approaching earth. Hence, apparent λ decreases and frequency increases

When an astronaut in a rocket/ spaceship moving towards reflector receives its own signal reflected from reflector from reflector, doppler shift occurs in two steps.

• Due to motion of source and
• Due to motion of approaching observer

Then,

$∆f = \frac{2v}{c} f$

$∆λ = \frac{2v}{c} λ$

Musical sound

$Musical$ $sound$ $and$ $noise:$

A musical sound consists of a quick succession of regular and periodic rarefactions and compressions without any sudden change in amplitude.

A noise consists of slow succession of irregular and a periodic rarefactions compression accompanied by sudden change in amplitude.

Characteristics of musical sound

i. Pitch

-It depends on the frequency of incident sound waves. It helps to distinguish between a shrill or grave sound.

-Higher the frequency, higher is the pitch and shriller sound is heard. Similarly lower the frequency; lower the pitch and grave sound is produced is produced. So, pitch ∝ frequency.

-The humming of a mosquito or buzzing of a bee has high pitch but low loudness while due to pressure of more harmonics.

-In doppler effect, there is an actual change in pitch but an apparent change in frequencies only.

ii. Loudness

-Sensation received by our ears due to intensity of sound is known as loudness. -Weber – Fechner experimentally established that,

Or, L = K log I

i.e., higher is the amplitude of vibration, grater will be the intensity (I ∝ A2) and louder will be the sound as in a shout and lesser the intensity, feeble will be the sound as in whispering.

Loudness (L) = $log_{10} \frac{I}{I_o}$ (in bel)

= $10 log_{10} \frac{I}{I_o}$ (in decibel, dB)

Where, Io is threshold intensity which is equal to 10-1 W/ m2

-The unit of loudness intensity which is equal to intensity in dB of equally loud sound of 1 KHz (for which ear is most sensitive).

-Loudness depends on intensity and sensitiveness of ear and is subjective feeling.

iii. Quality or timbre

-Sensation received by our ears due to waveform is known as quality or timbre.

-It enables us to distinguish between 2 sounds having same intensity (or loudness) and frequency (or pitch).

-It depends on presence of overtones.

-One can recognize a person without seeing him and can distinguish the same note played on different instruments say guitar and veena due to different quality.

-Quality of two sounds is different because different overtones are present in them.