Chapter 21. Sound waves objectives - Weeblychunghuastpm.weebly.com/.../chapter21_sound_waves.pdf · 2018. 9. 7. · Sound waves Content 21.1 Propagation of sound waves 21.2 Sources

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Chapter 21. Sound waves

Content

21.1 Propagation of sound waves 21.2 Sources of sound 21.3 Intensity of sound 21.4 Beat 21.5 Doppler effect

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objectives

a) explain the propagation of sound waves in air in terms of pressure variation and displacement

b) interpret the equations for displacement, y = yo sin ( t kx), and pressure, p = po sin ( t kx + /2)

c) use the standing wave equation to determine the positions of nodes and antinodes of a standing wave along a stretched string

d) use the formula v = (T/ )1/2 to determine the frequencies of the sound produced by different modes of vibration of the standing waves along a stretched string

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objectives

describe, with appropriate diagrams, the different modes of vibration of standing waves in air columns, and calculate the frequencies of sound produced, including the determination of end correction define and calculate the intensity level of sound use the principle of superposition to explain the formation of beats use the formula for beat frequency, f = f1 f2 describe the Doppler effect for sound, and use the derived formulae (for source and/or observer moving along the same line)

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What are sound waves?

A mechanical wave that vibrates a medium (like air or water) with different frequencies. These frequencies are then picked up by our ears. They are created through a variety of interactions, but all are mechanical (Physical).

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When we use Sound Waves

Music ties into Sound waves and frequencies. Each note has a different frequency.

We talk through sound waves, and apply meaning to certain sounds. Dolphins and bats use sound wave (sonar ). Dolphins use it to communicate, like a language, and bats use them to fly due to poor eye sight.

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How they work

Sound waves travel in a longitudinal way (vertical fashion), as shown by the tuning fork in the picture. The sound vibrates the medium between the

whatever is straight in front of it.

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How they work

A sound wave is measured in hertz (Hz) => vibration/second These are High and Low frequency waves, they show the difference between the two.

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How they work The periods, T between the waves categorize their frequencies, f as low or high. f 1/T The higher frequency has a smaller amount of time between waves, while the lower frequency has a longer amount of time.

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Frequencies

Interval Frequency Ratio Examples Octave 2:1 512 Hz and 256 Hz Third 5:4 320 Hz and 256 Hz

Fourth 4:3 342 Hz and 256 Hz Fifth 3:2 384 Hz and 256 Hz

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This chart explains sound waves pertain to musicmake music. For instance, raising a note an octave would require multiplying the base note by 2 (take a low c, with frequency of 261.5, to raise it an octave: has frequency 523.)

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Frequency

This table shows the value in hertz of certain notes (rounding applies).

Note C C# D D# E F F# G G# A A# B C C# D

Octave 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2

Hz 262 278 294 311 330 349 370 392 415 440 466 494 523 554 587

Sound

A longitudinal traveling wave Produced by vibrations in a medium

The disturbance is the local change in pressure generated by the vibrating object It travels because of the molecular interactions.

The region of increased pressure (compared to the normal pressure) is called condensation The region of lower pressure is called rarefaction.

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Sound

The maximum increase in pressure (DPm) is the amplitude of the pressure wave. (measurable)

frequency: 20Hz to 20kHz. Pressure waves below 20 Hz are called infrasonic waves Pressure waves over 20kHz are called ultrasonic waves.

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21.1 Propagation of sound waves

1. The propagation of sound waves occurs due to the oscillations of individual particles with the medium producing traveling waves of pressure fluctuations

2. The general form of particle oscillation y(x, t) = yo cos(kx - t) or y = yo sin ( t kx)

where yo is the magnitude of the particle displacement

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21.1 Propagation of sound waves

3. The general equation for the pressure fluctuations:

P(x, t) = Po sin(kx - t) or P = Po sin ( t kx + /2)

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21.2 Sources of Sound

Musical instruments produce sounds in various ways vibrating strings, vibrating membranes, vibrating metal or wood shapes, vibrating air columns. The vibration may be started by plucking, striking, bowing, or blowing. The vibrations are transmitted to the air and then to our ears.

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21.2 Sources of Sound: Vibrating Strings

The strings on a guitar can be effectively shortened by fingering, raising the fundamental pitch. The pitch of a string of a given length can also be altered by using a string of different density.

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21.2 Sources of Sound: Vibrating Strings

A piano uses both methods to cover its more than seven-octave range: the lower strings (at bottom) are both much longer and much thicker than the higher ones.

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21.2 Sources of Sound: Vibrating Air Columns

Wind instruments create sound through standing waves in a column of air.

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21.2 Sources of Sound: Vibrating Strings and Air Columns

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A tube open at both ends (most wind instruments) has pressure nodes, and therefore

displacement antinodes, at the ends.

21.2 Sources of Sound: Vibrating Strings and Air Columns

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A tube closed at one end (some organ pipes) has a displacement node (and pressure antinode) at

the closed end.

21.2 Sources of Sound: Vibrating Membrane

A piece of elastic membrane can vibrate in the modes as shown in the figure below:

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Vibrating Membrane

21.3 Intensity of sound

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Waves transport energy without transporting mass. The amount of energy transported per second is the power (P) of the wave (in W) Intensity is a measure of power transmitted by a wave per unit area:

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2 2medium wave m

Power PI = = = Area A


The energy transmission (power) is determined by the source. The power is distributed (spreads) in all directions. Far away from the source, the power is spread over a greater area. For a point source, intensity decreases inversely with the square of the distance from the source:

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2P PI(r) = = A 4 r

Loudness & Decibels

1. The human does not perceive sound intensity linearly but rather logarithmically

Perceived Loudness, Iperceived log (Iactual) 2. The average minimum perceivable sound

intensity: Io

-12 W/m2 3. The decibel scale was been developed to

ear perception (intensity level, ): = (10 dB). log(I/I0) = (10 dB). log(I + 12)

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21.4 Beats

1. When 2 sound waves the resultant wave pattern exhibits both constructive and destructive interference.

2. When the amplitudes of the 2 waves are similar but the frequencies are slightly different then: a. The frequency of the resultant wave is

roughly the average frequency of the 2 waves

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21.4 Beats

2. When the amplitudes of the 2 waves are similar but the frequencies are slightly different then:

a. The combined effect of interference produces periodic rises and drops in loudness called beats

b. The frequency of the beats (fbeat) is equal to the difference between the 2 sound frequencies: fbeat = f1 - f2

3. Musicians often tune their musical instruments by listening to beat frequency

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21.4 Beat The superposition of 2 sound waves:

fwave1=159.2 Hz fwave1=148.0 Hz

The resulting beat frequency: fbeat= fwave1 - fwave2 = 159.2 Hz - 148.0 Hz = 21.2 Hz

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21.4 Beat

When two sound waves of different but nearly equal frequency (f1 and f2) superimpose, we an intensity variation at the difference frequency The intensity variation is called beats The beat frequency is equal to the difference frequency | f1 - f2|

1 beat

Used to tune musical instruments to same pitch

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CP 535

21.4 Beat

Superimpose oscillations of equal amplitude, but different frequencies Modulation of amplitude frequency of pulses is | f1-f2 |

Oscillation at the average frequency

Not examinable 1 2

1 2 1 2

1 2 1 2

sin(2 ) sin(2 )( ) ( )2 sin(2 )cos(2 )

2 2( ) ( )2 cos(2 ) sin(2 )

2 2

A f t A f tf f f fA t t

f f f fA t t

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21.4 Beat interference in time Consider two sound sources producing audible sinusoidal waves at slightly different frequencies f1 and f2. What will a person hear? How can a piano tuner use beats in tuning a piano? If the two waves at first are in phase they will interfere constructively and a large amplitude resultant wave occurs which will give a loud sound. As time passes, the two waves become progressively out of phase until they interfere destructively and it will be very quite. The waves then gradually become in phase again and the pattern repeats itself. The resultant waveform shows rapid fluctuations but with an envelope that various slowly.

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21.4 Beat interference in time

The frequency of the rapid fluctuations is the average frequencies = The frequency of the slowly varying envelope =

1 22

f f

1 2

2f f

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beat 1 2f f f

CP 535

21.4 Beat interference in time

Since the envelope has two extreme values in a cycle, we hear a loud sound twice in one cycle since the ear is sensitive to the square of the wave amplitude. The beat frequency is

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0

10

20

30

40

50

60

0 0.05 0.1 0.15 0.2 0.25time

CP 535

f1 = 100 Hz f2 = 110 Hz frapid = 105 Hz Trapid = 9.5 ms fbeat = 10 Hz Tbeat = 0.1 s (loud pulsation every 0.1 s)

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0

10

20

30

40

50

60

0 0.05 0.1 0.15 0.2 0.25time

f =100f = 120beats

CP 535


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0

10

20

30

40

50

60

0 0.05 0.1 0.15 0.2 0.25time

f =100f = 104beats

CP 535


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One might wonder why the siren on a moving ambulance seems to produce sound with a higher pitch when it passes an observer and decreases when it recede the observer. Is this simply because of the relative distance between the observer and the ambulance (sound)? Or is it because of the loudness of the sound produced by the siren?

21.5 Doppler effect

21.5 Doppler effect

Christian Johann Doppler (1803-1853) Studied motion related frequency changes (1842)

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oo s

s

v vf fv v

Source (s) Observer (o)

21.5 Doppler effect

Doppler effect is the change in frequency of a wave (or other periodic event) for an observer moving relative to its source.

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oo s

s

v vf fv v

Source (s) Observer (o)

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21.5 Doppler effect

of waves and the observer are approaching each other, the sound heard by the observer becomes higher in pitch, whereas if the source and observer are moving apart the pitch becomes lower. For the sound waves to propagate it requires a medium such as air, where it serves as a frame of reference with respect to which motion of source and observer are measured.

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21.5 Doppler effect

Applications: police microwave speed units speed of a tennis ball speed of blood flowing through an artery heart beat of a developing fetous burglar alarms sonar ships & submarines to detect submerged objects detecting distance planets observing the motion of oscillating stars.

21.5 Doppler effect

Consider source of sound at frequency fs, moving speed vs, observer at rest (vo = 0) Speed of sound v What is frequency fo heard by observer?

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21.5 Doppler effect

On right - source approaching source catching up on waves wavelength reduced frequency increased On left - source receding source moving away from waves wavelength increased frequency reduced

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SITUATION 1 Stationary Source and Observers (NO DOPPLER EFFECT)

A stationary sound source S emits a spherical wavefronts of

v relative to the medium air.

In time t, the wavefronts move a distance vt toward the observers, O1 & O2.

The number of wavelengths detected by the observer infront and behind the source are the same and equal to vt

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SITUATION 1 Stationary Source and Observers (NO DOPPLER EFFECT)

Thus, the frequency f heard by both stationary observers is given by,

f - frequency of sound source v - speed of sound waves t - time - wavelength

vt

vtf /

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21.5 Doppler effect

What if both of the observers in figure 1 are moving, is there any change in the frequency and wavelength of the source?

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SITUATION 2 Stationary Source; Moving Observers

Observer 1 moves a distance vot toward the source at speed vo

We had known earlier that wavefronts also move at speed v towards O1 in time t at distance vt. The distance traveled by the wavefronts with respect to O1 becomes vt + vOt.

The number of wavelengths intercepted by O1 at this distance is (vt + v0t

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SITUATION 2 Stationary Source; Moving Observers This shows that there is an increase in the

frequency heard by O1 as it goes nearer to the sound source as given by,

(2)

Since = v/f, then (3)

00 /)(' vvttvvtf

vvvff 0'

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If observer 2 moves away from the sound source, the distance traveled by the wavefronts with respect to O2 in time t, is vt vot. Consequently, there would be a decrease in the frequency heard by O2 as given by,

(4) vvvff 0'

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(5)


In these situations only the frequency heard by the observers changes due to there motion relative to the source. However the wavelength of sound remains constant.

vvvff 0'

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SITUATION 3 Moving Source; Stationary Observers

As the source moves a distance vST (T=1/f period of wave) toward O1 there is a decrease in the wavelength of sound by a quantity of vsT. The shortened

vsT


The frequency of sound wave heard by O1 increases as given by,

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(6)

Tvvvfs'

'fvfv

v

s/

svvvff '


With respect to observer 2, the wavelength of sound increases, where

vsT. The frequency of sound wave heard by O2 decreases as given by,

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svvvff '


Combining Equations (6) and (7), we have

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(8)

(7)

(6)

svvvff '

svvvff '

svvvff '

SITUATION 4 Moving Source and Observer

From the equations (5) and (8), we can now derive the equation of general Doppler Effect by replacing f in equation (5) with of equation (8). This result to,

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(Moving source and observer)

(9) svvvvff 0'


The ± signs correspond to the direction of the source or observer when they are moving relative to the other. These would determine whether there is an increase or decrease on the frequency heard by the observer during the motion.

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(9)svvvvff 0'


Approaching observer, receding source If vo> vs , observed frequency increases If vo< vs , observed frequency decreases 58

svvvvff 0'

svvvvff 0'

Receding observer, receding source Decrease in observed frequency


Approaching observer, approaching source

Observed frequency increases

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Receding observer, approaching source If vo> vs , observed frequency decreases If vo< vs , observed frequency increases

svvvvff 0'

svvvvff 0'

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Problem

A train has a whistle, which emits a 400 Hz sound. You are stationary and you hear the whistle, but the pitch is 440 Hz. How fast is train moving towards or away from you?

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Solution:

The pitch is higher, so the train is moving towards you. Its speed relative to you is found from

f = f0v/(v-vs). We have (v-vs) = f0v/f = (400/s)(330 m/s)/(440/s)

= 300m/s. Therefore vs = 330m/s 300m/s

= 30m/s

Summary: Sound waves

Propagation

Sources of sound

Intensity level

Beat

Doppler effect

y = yo sin ( t kx) p = po sin ( t kx + /2)

Open ends tube: Ln= n /2, fn = nv/2L

One Closed End tube: Ln = n /4, fn = nv/4L

I = P/4 r2

= (10 dB). log(I/I0)

fbeat =| f1 - f2 |

f f[(v v0)/(v vs)]

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Documents

Chapter 21. Sound waves objectives - Weeblychunghuastpm.weebly.com/.../chapter21_sound_waves.pdf · 2018. 9. 7. · Sound waves Content 21.1 Propagation of sound waves 21.2 Sources