25Doppler's effect and Musical sound

📚
DOPPLER EFFECT AND MUSICAL SOUND
▢ Doppler Effect:
❖ Definition: Apparent change in frequency of sound heard by observer due to relative motion between source, observer and medium
❖ Shown By:
  • •
    Sound waves
  • •
    Light waves
  • •
    Electromagnetic waves
❖ Nature:
  • •
    Sound Doppler effect is not symmetrical
  • •
    Light Doppler effect is symmetrical
  • •
    Sound Doppler effect does not occur in transverse direction
  • •
    Light can show transverse Doppler effect
  • •
    Apparent frequency increases when source and observer approach
  • •
    Apparent frequency decreases when source and observer recede
❖ Key Concept:

Table 1: Effect of Motion

Motion
Change
Source motion
Changes apparent wavelength
Observer motion
Changes apparent frequency
Same speed same direction
No Doppler effect
Mutually perpendicular motion
No Doppler effect for sound
Only medium moves
No Doppler effect
Circular motion with constant distance
No Doppler effect
▢ General Doppler Formula for Sound:

Table 1: Symbols

Symbol
Meaning
n
Original frequency of source
n'
Apparent frequency
v
Velocity of sound in air
v_o
Velocity of observer
v_s
Velocity of source

Table 2: Formula

Case
Formula
General
\(n'=\frac{v\pm v_o}{v\mp v_s}\times n\)
Source and observer approaching
\(n'=\frac{v+v_o}{v-v_s}\times n\)
Source and observer receding
\(n'=\frac{v-v_o}{v+v_s}\times n\)
Observer towards source
\(+v_o\)
Observer away from source
\(-v_o\)
Source towards observer
\(-v_s\) in denominator
Source away from observer
\(+v_s\) in denominator
▢ Source Passing Listener:

Table 1: Frequency Jump

Case
Formula
Source passes stationary listener
\(\Delta n=\frac{2v_s}{v-v_s}n\)
If \(v_s\ll v\)
\(\Delta n=\frac{2v_s}{v}n\)
Listener passes stationary source
\(\Delta n=\frac{2v_o}{v}n\)
If \(v\gg v_o\)
\(\Delta n\approx\frac{2v_o}{v}n\)
▢ Wind Effect:

Table 1: Wind and Doppler Effect

Condition
Formula / Result
Wind in direction of sound
\(n'=\frac{v+v_w+v_o}{v+v_w-v_s}n\)
Wind opposite to sound
\(n'=\frac{v-v_w+v_o}{v-v_w-v_s}n\)
Wind at angle \(\theta\) with sound
\(v_w\cos\theta\) component used
Wind perpendicular to sound
No change in apparent frequency
▢ Reflection Doppler:

Table 1: Source / Observer Near Reflector

Case
Apparent Frequency
Car moving towards hill
\(n''=\frac{v+v_s}{v-v_s}n\)
Car moving away from hill
\(n''=\frac{v-v_s}{v+v_s}n\)
Astronaut / rocket signal reflected from reflector
Doppler shift occurs in two steps
Reflected light shift
\(\Delta n=\frac{2v}{c}n\)
Reflected wavelength shift
\(\Delta\lambda=\frac{2v}{c}\lambda\)
▢ Doppler Effect in Light:

Table 1: Light Doppler Effect

Point
Answer
Nature
Symmetrical
Depends on
Relative velocity only
Frequency shift
\(\Delta n=\frac{v}{c}n\)
Wavelength shift
\(\Delta\lambda=\frac{v}{c}\lambda\)
Star moving away
Red shift
Star approaching
Blue / violet shift
Red shift meaning
Apparent wavelength increases
Blue shift meaning
Apparent wavelength decreases and frequency increases
Expanding universe
Confirmed by red shift of distant galaxies
▢ Applications of Doppler Effect:

Table 1: Uses

Application
Purpose
RADAR
Locating distant objects using reflected radio/microwave waves
SONAR
Position of hidden submarines or icebergs
Stars and galaxies
Velocity measurement
Spectral lines
Determination of width
Sun
Speed of rotation
Tuning fork
Frequency determination
Twin / binary stars
Discovery and study
▢ Musical Sound and Noise:

Table 1: Sound Types

Term
Meaning
Musical sound
Quick succession of regular and periodic rarefactions and compressions without sudden amplitude change
Noise
Irregular and aperiodic rarefactions and compressions with sudden amplitude change
Tone
Musical sound of single frequency
Note
Musical sound containing two or more tones
Fundamental tone
Lowest frequency tone
Overtones
Tones other than fundamental tone
❖ Noise Level:
  • •
    Sound intensity more than 100 dB is called noise
  • •
    130 dB sound is painful and may cause hearing impairment
  • •
    Ticking watch is noise due to low frequency
  • •
    Pure sine wave in sound is called tone
▢ Characteristics of Musical Sound:

Table 1: Pitch, Loudness and Quality

Characteristic
Depends on
Meaning
Pitch
Frequency
Distinguishes shrill and grave sound
Loudness
Intensity + sensitivity of ear
Sensation of sound strength
Quality / Timbre
Waveform + overtones
Distinguishes same pitch and loudness from different instruments
❖ Pitch:
  • •
    Higher frequency → higher pitch
  • •
    Lower frequency → lower pitch
  • •
    Mosquito / buzzing bee has high pitch but low loudness
  • •
    Pitch of female voice is higher than male voice due to presence of more harmonics
  • •
    Pitch depends on frequency
❖ Loudness:
  • •
    Loudness depends on amplitude
  • •
    Greater amplitude → greater intensity → louder sound
  • •
    Loudness is subjective
❖ Quality:
  • •
    Depends on overtones
  • •
    Different overtones make same note sound different on different instruments
  • •
    Sweetness depends on periodicity and regularity
▢ Loudness and Intensity Level:
❖ **table:
    ❖ Important Relations:
      ◈ _*type: bullet
    1. If loudness changes by 1 dB, intensity changes by about 26%
    2. If \(n_1\) and \(n_2\) are intensity levels: \(n_1-n_2=10\log*{10}\frac{I_1}{I_2}\)
    3. Intensity of harmonic wave depends on both frequency and amplitude
    ▢ Sound Intensity Levels:

    Table 1: Common Sound Levels

    Source
    Intensity level
    Threshold of hearing
    0 dB
    Churches / hospitals
    10–20 dB
    Normal conversation
    50–60 dB
    Normal city traffic
    60–70 dB
    Alarm clock
    70–80 dB
    Damage threshold under prolonged exposure
    80–85 dB
    Jetliner 150 m overhead
    100–115 dB
    Running motorcycle
    115–120 dB
    Threshold of pain
    120–140 dB
    Jet plane taking off
    140–150 dB
    Launching space rocket
    160–180 dB
    ❖ Maximum Tolerable Sound: 120 dB
    ▢ Musical Intervals:

    Table 1: Intervals

    Term
    Relation
    Musical interval
    \(\frac{p}{q}\), where p and q are frequencies of two tones
    Octave
    2 : 1
    Fifth
    3 : 2
    Concord
    Pleasant combination of two tones
    Discord
    Unpleasant combination of two tones
    ▢ Musical Instruments:

    Table 1: Important Points

    Point
    Answer
    Air box
    Increases intensity / loudness
    Stringed instrument frequency increase
    Shorten and tighten string
    Walls of music hall
    Absorb sound; should not amplify, reflect or transmit too much
    Good audibility
    Reverberation time nearly 1 sec
    Temple / college bell
    Large size helps produce loud sound
    ▢ Infrasonic and Ultrasonic Notes:

    Table 1: Special Sound Facts

    Fact
    Answer
    Animals sense hurricane
    Due to infrasonic waves
    Infrasonic source
    Hurricane, earthquake, ocean wave, volcano
    Frequency range of infrasonic
    < 20 Hz
    Ultrasonic
    > 20,000 Hz
    SONAR
    Uses ultrasonic sound
    ▢ Read and Digest:

    Table 1: Important One-Liners

    Fact
    Answer
    Doppler effect in sound
    Not symmetrical
    Doppler effect in light
    Symmetrical
    Doppler effect frequency shift
    Independent of distance between source and observer
    Doppler effect not observed
    When relative velocity is zero
    Doppler effect not applicable
    Shock wave / supersonic wave
    Two photons approaching or receding
    Relative velocity = c
    Reflection of sound
    Causes reverberation
    Motion of source
    Changes apparent wavelength
    Motion of observer
    Changes apparent frequency
    Blue light from distant receding star
    Appears shifted towards red
    Ordinary conversation noise level
    About 65 dB
    Pitch
    Depends on frequency
    Loudness
    Depends on amplitude / intensity
    Quality
    Depends on overtones
    Sweetness
    Depends on periodicity and regularity
    Phon
    Unit of loudness
    Red shift
    Evidence of expanding universe
    ▢ High-Yield Recall:

    Table 1: Doppler and Musical Sound

    Fact
    Answer
    Doppler effect
    Apparent frequency change due to relative motion
    Approaching source-observer
    Frequency increases
    Receding source-observer
    Frequency decreases
    General sound formula
    \(n'=\frac{v\pm v_o}{v\mp v_s}n\)
    Light Doppler shift
    \(\Delta n=\frac{v}{c}n\)
    Wavelength shift
    \(\Delta\lambda=\frac{v}{c}\lambda\)
    Red shift
    Star/galaxy moving away
    Blue shift
    Star/galaxy approaching
    RADAR
    Radio Detection and Ranging
    SONAR
    Sound Navigation and Ranging
    Musical sound
    Regular periodic sound
    Noise
    Irregular aperiodic sound
    Tone
    Single frequency sound
    Note
    Two or more tones
    Fundamental tone
    Lowest frequency
    Overtones
    Remaining tones
    Pitch
    Frequency
    Loudness
    \(10\log*{10}(I/I_0)\)
    Threshold intensity
    \(10^{-12}Wm^{-2}\)
    Quality / timbre
    Overtones and waveform
    Octave
    2 : 1
    Fifth
    3 : 2
    Maximum tolerable level
    120 dB
    Q1.
    Observer moving towards stationary sound source hears higher frequency because
    📅CEE MBBS 2025
    Q2.
    A sound having intensity of 10⁻⁶ has loudness of:
    📅CEE Nursing 2024
    Q3.
    Two tuning fork A and B sounded together to produce 5 beats. The frequency of B is 512 Hz. When fork A is filed and sounded together, the beat frequency increases. The frequency of A is:
    📅CEE Nursing 2024
    Q4.
    A car is moving toward a cliff and blows a horn of frequency f, which is reflected back with frequency 2f. Then the velocity of car is:
    📅CEE MBBS 2024B
    Q5.
    The intensity levels due to two waves of the same frequency in a given medium are 1 bel and 5 bel. The ratio of their amplitudes is:
    📅CEE MBBS 2023
    Q6.
    A source of sound is moving away from an observer and the apparent frequency heard is half the original frequency. If the velocity of sound is \(V\), the velocity of the source is:
    📅CEE MBBS 2023
    Q7.
    If sound intensity is increased 20 times, the increase in sound level is approximately:
    📅CEE MBBS 2021
    Q8.
    If source and observer move towards each other each by half velocity of sound, then change in frequency is:
    📅CEE BDS 2023
    Q9.
    An observer is moving towards a stationary object with a velocity of half the speed of sound in air. The frequency of sound emitted by the object is 400 Hz. What will be the frequency heard by the observer? Assume speed of sound in air to be 340 m/s.
    📅CEE BDS 2023
    Q10.
    Loudness and pitch depend upon:
    📅CEE BDS 2021
    Q11.
    A tuning fork (A) of unknown frequency gives 4 beats/sec when sounded with another fork B of frequency 256 Hz. The fork A is now loaded with a piece of wax and again 2 beats/sec are heard. Then the frequency of fork A is:
    📅CEE BDS 2021
    Q12.
    A car driver and an observer in another car move directly toward one another, each at \(45\,\mathrm{km/h}\). The driver sounds a horn at \(500\,\mathrm{Hz}\). With sound speed \(330\,\mathrm{m/s}\), what frequency does the observer hear?
    📅BP 2014
    Q13.
    Which can produce maximum pitch sound?
    📅BP 2013
    Q14.
    A sound has intensity level \(40\,\mathrm{dB}\). Find its intensity, taking \(I_0=10^{-12}\,\mathrm{W/m^2}\).
    📅BP 2013
    Q15.
    A window of area \(2\,\mathrm{m^2}\) faces noise of intensity level \(60\,\mathrm{dB}\). What sound power enters the window? Take \(I_0=10^{-12}\,\mathrm{W/m^2}\).
    📅BP 2009
    Q16.
    A source and a listener move in the same direction at the same speed \(v/2\) through still air, where \(v\) is the sound speed. What is the frequency change?
    📅IOM 2013
    Q17.
    A car approaches a stationary wall at \(20\,\mathrm{m/s}\) and emits a \(124\,\mathrm{Hz}\) horn. What frequency of the reflected wave reaches its driver if sound speed is \(330\,\mathrm{m/s}\)?
    📅MOE 2013
    Q18.
    A listener is moving away from stationary source of sound. The listener hears a sound of frequency:
    📅KU 2012
    Q19.
    A sound wave has intensity \(10^{-3}\,\mathrm{W/m^2}\). Find the intensity level in decibels, given \(I_0=10^{-12}\,\mathrm{W/m^2}\).
    📅KU 2012
    Q20.
    Which of these ordinary low-amplitude sounds propagates fastest through the same uniform air?
    📅BP 2013
    Q21.
    What is the frequency ratio of two musical notes separated by three octaves?
    📅BP 2014
    Q22.
    A star is moving away from the earth. The wavelength of the light notes by the observer on the earth will be:
    📅BP 2015
    Q23.
    Two waves of equal frequency travel through the same medium with intensity levels \(1\,\mathrm{bel}\) and \(5\,\mathrm{bel}\). Find their displacement-amplitude ratio.
    📅MOE 2011
    Q24.
    Two trains travel along a track in opposite directions at \(4\,\mathrm{m/s}\), one approaching and the other receding from a stationary listener. Both whistles emit \(240\,\mathrm{Hz}\). How many beats per second are heard if sound speed is \(320\,\mathrm{m/s}\)?
    📅IOM 2014
    Q25.
    A source emitting \(1200\,\mathrm{Hz}\) approaches a stationary listener at \(50\,\mathrm{m/s}\). Find observed frequency if sound speed is \(350\,\mathrm{m/s}\).
    📅MOE 2013
    Q26.
    A listener moving away from a stationary sound source hears a frequency 10% lower than the emitted frequency. Find listener speed if \(v=330\,\mathrm{m/s}\).
    📅KU 2010
    Q27.
    A sound source moving toward a stationary listener travels at the sound speed \(v\). In the ideal nondispersive model, what is the limiting behavior of wavefront arrival at the listener?
    📅BP 2012
    Q28.
    A stationary \(90\,\mathrm{Hz}\) tuning fork emits sound toward an observer approaching it at \(v/10\). Find observed frequency.
    📅MOE 2009
    Q29.
    A whistle moves on a circle of radius \(1\,\mathrm m\) at two revolutions per second. A distant stationary observer lies in the plane of rotation. With sound speed \(332\,\mathrm{m/s}\), approximately what is the ratio of maximum to minimum observed frequencies?
    📅IOM 2015
    Q30.
    A source moves past a stationary observer at \(50\,\mathrm{m/s}\). Before passing, observed frequency is \(1000\,\mathrm{Hz}\). Find observed frequency after passing if sound speed is \(350\,\mathrm{m/s}\).
    📅BPKIHS 2011
    Q31.
    A \(450\,\mathrm{Hz}\) whistle approaches a stationary listener at \(33\,\mathrm{m/s}\). Take sound speed as \(330\,\mathrm{m/s}\). Find heard frequency.
    📅MOE 2010
    Q32.
    Find the musical frequency interval (higher frequency divided by lower) between \(320\,\mathrm{Hz}\) and \(240\,\mathrm{Hz}\).
    📅BP 2013
    Q33.
    Light from a receding star has observed wavelength 0.4% greater than emitted wavelength. Estimate recession speed for \(v\ll c\).
    📅IOM 2016
    Q34.
    Waves of the same frequency in the same medium have intensity levels \(20\,\mathrm{dB}\) and \(60\,\mathrm{dB}\). Find their displacement-amplitude ratio.
    📅KU 2012
    Q35.
    A sound source moves directly away from a stationary observer at sound speed \(v\). What frequency is observed if the source frequency is \(n\)?
    📅KU 2010
    Q36.
    A \(450\,\mathrm{Hz}\) whistle approaches a stationary listener at \(33\,\mathrm{m/s}\). Find observed frequency if \(v=330\,\mathrm{m/s}\).
    📅MOE 2010
    Q37.
    A radar transmits at \(7.8\times10^9\,\mathrm{Hz}\) and detects a reflected frequency shift of magnitude \(2.7\times10^3\,\mathrm{Hz}\) from a slowly moving aeroplane. Estimate its radial speed.
    📅IOM 2007
    Q38.
    Increasing the volume of air enclosed by an otherwise identical resonating musical instrument generally increases which property if it provides better acoustic coupling?
    📅IOM 1997
    Q39.
    Radio waves can often reach a receiver around obstacles that block visible light. Which wave phenomenon best explains this?
    📅MOE 2063/KU 08/09
    Q40.
    Quality of two sounds is different because:
    📅MOE 2062
    Q41.
    A slab transmits 90% of incident sound intensity. What fraction emerges after two identical slabs?
    📅MOE 2009
    Q42.
    Sound levels are \(100\,\mathrm{dB}\) and \(20\,\mathrm{dB}\). Find \(I_1/I_2\).
    📅IE 2005
    Q43.
    A driver approaches a stationary wall at \(2\,\mathrm{m/s}\) while sounding a \(700\,\mathrm{Hz}\) horn. Find the beat frequency between the horn and its reflected sound heard by the driver if \(v=350\,\mathrm{m/s}\).
    📅BPKIHS 2015
    Q44.
    A stationary policeman blows a \(300\,\mathrm{Hz}\) whistle toward a driver approaching him at \(5\,\mathrm{m/s}\). Find frequency heard if \(v=320\,\mathrm{m/s}\).
    📅ITE 2010
    Q45.
    A person approaches a stationary \(510\,\mathrm{Hz}\) siren at \(40\,\mathrm{m/s}\). Find frequency heard, with \(v=300\,\mathrm{m/s}\).
    📅BPKIHS 2007
    Q46.
    A bus approaches a stationary wall at \(5\,\mathrm{m/s}\). A passenger hears the \(200\,\mathrm{Hz}\) horn directly and after reflection. Calculate the beat frequency if \(v=350\,\mathrm{m/s}\).
    📅BPKIHS 1996
    Q47.
    Each sound-absorbing slab reduces intensity by 10%. What percentage reduction results after two such slabs?
    📅MOE 2008
    Q48.
    A car moving toward a hill at \(30\,\mathrm{m/s}\) emits a \(600\,\mathrm{Hz}\) horn. Find the echo frequency heard by its driver if \(v=330\,\mathrm{m/s}\).
    📅BPKIHS 2014
    Q49.
    Two trains approach and recede from a stationary observer at \(4\,\mathrm{m/s}\). Each emits \(240\,\mathrm{Hz}\). Find the beat frequency for \(v=320\,\mathrm{m/s}\).
    📅MOE 2012
    Q50.
    A \(400\,\mathrm{Hz}\) source passes a stationary observer at \(200\,\mathrm{m/s}\). Find the difference between frequencies just before and just after passing if \(v=300\,\mathrm{m/s}\).
    📅IOM 2015
    Q51.
    Sound intensity is \(10^6\) times the threshold intensity. Find intensity level in decibels.
    📅BP 2011
    Q52.
    An engine travels in a circle of radius \(100\,\mathrm m\) at constant speed \(20\,\mathrm{m/s}\) while emitting \(500\,\mathrm{Hz}\). What frequency does a stationary listener at the centre hear (ignoring propagation effects other than Doppler)?
    📅IOM 2016
    Q53.
    A sound source and observer approach each other, each at speed \(v/2\) relative to still air. Find percentage increase of received frequency.
    📅MOE 2014
    Q54.
    A source emits sound of wavelength \(40\,\mathrm{cm}\) when at rest. It travels east at \(v/4\). What wavelength is measured in the air behind it by a stationary observer to its west?
    📅BP 2010
    Q55.
    How fast must a source move away from a stationary observer for the observed sound frequency to be half its emitted frequency? (Sound speed is \(v\).)
    📅KU 2011
    Q56.
    A galaxy approaches Earth at \(10^5\,\mathrm{m/s}\). Estimate the magnitude of blueshift of a \(5700\,\mathrm{\mathring A}\) spectral line.
    📅IOM 2017
    Q57.
    Whistle A is stationary at \(600\,\mathrm{Hz}\). Whistle B emits \(596\,\mathrm{Hz}\) and moves together with a listener at \(30\,\mathrm{m/s}\) directly away from A (in still air). Find the beat frequency heard if sound speed is \(330\,\mathrm{m/s}\).
    📅BP 2016
    Q58.
    A \(500\,\mathrm{Hz}\) whistle rotates at radius \(1.2\,\mathrm m\) and \(400\,\mathrm{rev/min}\). Find its approximate observed frequency range for a distant stationary listener in its plane if \(v=340\,\mathrm{m/s}\).
    📅MOE 2015
    Q59.
    A celestial line of rest wavelength \(4700\,\mathrm{\mathring A}\) is observed at \(4747\,\mathrm{\mathring A}\). Estimate radial speed and whether the source is approaching or receding.
    📅IOM 2018
    Q60.
    A stellar spectral line is observed at wavelength 0.05% shorter than its laboratory value. Estimate radial speed and direction.
    📅BPKIHS 2017
    Q61.
    An astronaut in a rocket approaching the Moon transmits \(5\times10^9\,\mathrm{Hz}\) radio waves toward it and receives their reflection at a frequency \(9\times10^6\,\mathrm{Hz}\) higher. Find relative radial speed, using the small-speed double-Doppler approximation.
    📅IOM 2019
    Q62.
    Two cars approach a crossing on perpendicular roads at \(72\,\mathrm{km/h}\) and \(36\,\mathrm{km/h}\). The first car emits \(280\,\mathrm{Hz}\). When the line joining the cars makes \(45^\circ\) with both roads, what frequency is heard by the second driver? Take sound speed \(330\,\mathrm{m/s}\).
    📅MOE 2016
    Q63.
    A point source radiates sound power \(200\,\mu\mathrm W\) uniformly. Find the intensity level at radius \(10\,\mathrm m\) with \(I_0=10^{-12}\,\mathrm{W/m^2}\).
    📅BP 2017
    Q64.
    A stationary observer hears a train whistle at \(220\,\mathrm{Hz}\) as it approaches and \(184\,\mathrm{Hz}\) as it recedes at constant speed. Find its emitted frequency.
    📅BPKIHS 2015
    Q65.
    A car approaches a rock at \(30\,\mathrm{m/s}\) while sounding its \(600\,\mathrm{Hz}\) horn. Find frequency of the reflected sound heard by the driver if \(v=330\,\mathrm{m/s}\).
    📅IOM 2016
    Q66.
    A stationary observer hears twice the source frequency as a source approaches. Find the source speed if \(v=332\,\mathrm{m/s}\).
    📅IOM 2016
    Q67.
    Find the intensity level of a sound having intensity \(10^{-4}\,\mathrm{W/m^2}\) relative to \(I_0=10^{-12}\,\mathrm{W/m^2}\).
    📅KU 2017
    Q68.
    A source and observer both move in the same direction at \(v/2\) relative to still air. Find the percentage increase in the received frequency.
    📅IOM 2017