Physics 11. Unit 7 (Part 2) The Physics of Sound

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1 Physics 11 Unit 7 (Part 2) The Physics of Sound

$It possesses the properties and behaviors that are shared by other waves: reflection, refraction, diffraction, interference, etc.$

2 1. Sound waves As introduced in the previous section, sound is one of the many types of waves we encounter in our daily lives. It possesses the properties and behaviors that are shared by other waves: reflection, refraction, diffraction, interference, etc. When sound is transmitted from one location to the other, energy is carried. Unit 7 (Part 2) - The Physics of Sound 2

There are two types of waves categorized according to the mode of particle vibrations. Sound belongs to the longitudinal wave in which particle vibrate in the same direction as the wave propagation.

3 There are two types of waves categorized according to the mode of particle vibrations. Sound belongs to the longitudinal wave in which particle vibrate in the same direction as the wave propagation. When air particles vibrate back and forth, they form alternating regions of compression and rarefaction. These are the features unique to longitudinal waves. Unit 7 (Part 2) - The Physics of Sound 3

4 While the wavelength of a transverse wave can be determined by the distance between two successive crests or troughs, how is the wavelength of a longitudinal wave, which does not have crests or troughs, measured? We can measure the separation of two corresponding points in consecutive compressions or rarefactions. Unit 7 (Part 2) - The Physics of Sound 4

The measurement of the amplitude of a longitudinal wave is similar to that for a transverse wave, but instead of the particle movement perpendicular to the wave motion, the amplitude refers to the

5 The measurement of the amplitude of a longitudinal wave is similar to that for a transverse wave, but instead of the particle movement perpendicular to the wave motion, the amplitude refers to the maximum displacement of the particles from the equilibrium position. The amplitude can also be interpreted as the density of the particles at a compression relative to the density at undisturbed regions. Unit 7 (Part 2) - The Physics of Sound 5

When these compressions and rarefactions of air reach our ears, they induce the motion of our ear bones, creating a nerve pulse which is then transmitted

6 When these compressions and rarefactions of air reach our ears, they induce the motion of our ear bones, creating a nerve pulse which is then transmitted to our brains. This is why we can hear. There are three perspectives we perceive sound: loudness, pitch, quality. Unit 7 (Part 2) - The Physics of Sound 6

7 (1) Loudness It is also called sound intensity, which is a measure of how much power the sound has. This quantity is related to the amplitude of sound waves. Sound intensity is defined as I = P 4πr 2 where r is the distance between the source and the receiver. Unit 7 (Part 2) - The Physics of Sound 7

8 This value, however, is small compared to other forms of energy. Also, it varies greatly from one source to another. Hence, sound intensity is usually recorded in log scale called decibel (db): L = 10 log I I 0 I 0 is a reference value equal to W/m 2. db reading therefore depends on distance. For example, a reading of 100 db at 1 m becomes 60 db at 100 m. Note that human ears are sensitive to sound with intensities ranging from a whisper (20 db) to a jet plane taking off (140 db) a difference of a factor of about 10 13! Unit 7 (Part 2) - The Physics of Sound 8

9 The following chart lists out some common sources of sound and their approximate intensities. Unit 7 (Part 2) - The Physics of Sound 9

10 (2) Pitch It is determined by the frequency of the sound. Higher the frequency of the sound, higher the pitch we will hear. The frequency of the sound, and in turn its pitch, generated by an object depends heavily on its physical conditions. Everyone of us has a very unique voice with different pitch, tone and rate. Hence, our voices can serve as identifiers to tell who is who. Men usually have a larger vocal fold than women, resulting in a lower pitched voice for male (from 80 Hz to 700 Hz) than for female (from 170 Hz to 1100 Hz) Unit 7 (Part 2) - The Physics of Sound 10

11 Human ears are sensitive to sound with frequencies ranging from 20 Hz to 20,000 Hz in general. This is called the audible range. This range gets shortened when people get old; they become less sensitive to high-frequency sounds. Frequencies below 20 Hz are called infrasonic, while those above 20,000 Hz are called ultrasonic. Sound outside the audible range cannot be aware of by human. However, many animals can detect ultrasonic frequencies, such as dogs (~ 50,000 Hz) and bats (~ 100,000 Hz) Unit 7 (Part 2) - The Physics of Sound 11

12 (3) Quality When a piano and a flute play a note of the same loudness and pitch, such as middle C, we can clearly tell the difference. It is not about how good or bad the play is; it is about the type of the sound we perceive. This is what meant by quality (also called timbre or tone color) of a sound. Quality of the sound of a musical instrument depends on the mixing of the fundamental and overtones generated. If only the fundamental frequency and its overtones are present, the resulting sound will be more harmonious and is said to be in good quality. On the other hand, if many frequencies which are not related are mixed, the resulting sound will be in poor quality and no pitch will be discerned. It is therefore called noise. Unit 7 (Part 2) - The Physics of Sound 12

13 To analyze the quality of a sound, we can employ the skill of Fourier transform to resolve it into the frequency components (called sound spectrum). Unit 7 (Part 2) - The Physics of Sound 13

14 As mentioned earlier, sound is a mechanical wave whose propagation relies on vibration of air particles. Therefore, sound cannot transmit in space or within an evacuated container. In air (at 1 atm), the speed of sound can be approximated by the following formula: v T m/s It shows that higher the temperature, higher the speed of sound. Example: At 20 C, the speed of sound is v = = 343 m/s Unit 7 (Part 2) - The Physics of Sound 14

15 This formula does not apply to materials other than air, because the speed of a longitudinal wave depends on the density and stiffness of the substance in which it is moving. Materials Elastic modulus (GPa) Density (g/l) Speed (m/s) at 20 C Air Helium Hydrogen Water Sea water Hard wood Glass Iron Aluminum Unit 7 (Part 2) - The Physics of Sound 15

2. Sources of sound When an object vibrates, it

This is the general principle behind the music

Depending on the designs, various musical

the following means: striking, plucking, bowing and

16 2. Sources of sound When an object vibrates, it generates sound. This is the general principle behind the music created by musical instruments. Depending on the designs, various musical instruments can set the source into vibration by the following means: striking, plucking, bowing and blowing. Percussion instrument Stringed instrument Plucked instrument Wind instrument Unit 7 (Part 2) - The Physics of Sound 16

17 For all types of musical instruments, the pitch of a pure sound is determined by the frequency. The frequencies for different musical notes are systematized by the scientific pitch notation (SPN) which indicates the type of pitch and the octave. The current pitch standard uses A4 as exactly 440 Hz. Other pitches within the same octave can be determined by f = (n 9)/12 Unit 7 (Part 2) - The Physics of Sound 17

18 Using this formula, we can determine the frequencies for the octave starting from middle C. Note C C # D D # E F F # G G # A A # B n f The frequencies of the same tone in different octaves can be obtained by doubling the frequency of the preceding one. For example, for middle C: Octave f Unit 7 (Part 2) - The Physics of Sound 18

19 (1) Stringed and plucked instruments For both the plucked and stringed instruments, the sound comes from the vibration of the strings that forms a standing wave. Its pitch, or frequency, is found to be determined by four factors: length, diameter, tension and density. (i) Length (L): Frequency increases when length decreases (ii) Tension (T): Frequency increases when tension increases (iii) Diameter (d): Frequency increases when diameter decreases (iv) Density (ρ): Frequency increases when density decreases f 1 dl T ρ Unit 7 (Part 2) - The Physics of Sound 19

20 Example: The D string of a violin is 30.0 cm long and has a natural frequency of 288 Hz. Where must the violinist place his finger on the string to produce B (383 Hz)? Since frequency is inversely proportional to length, f 1 L So f 1 f 2 = L 2 L 1 And L 2 = L 1 f 1 f 2 = = 22.6 cm Unit 7 (Part 2) - The Physics of Sound 20

21 Example: A piano string with a pitch of A (440 Hz) is under a tension of 140 N. What tension would be required to produce a high C (523 Hz)? Recall that f T. Hence Therefore f 1 f 2 = T 1 T 2 T 1 = T 2 f 1 f 2 2 = = 198 N Unit 7 (Part 2) - The Physics of Sound 21

22 (2) Wind instruments For the instruments whose sound is generated by means of blowing, their designs must share a common feature: an air column. Different notes are produced by creating standing waves with different wavelengths through resonance. Recall that every object has a natural frequency at which it will vibrate. When an object is free to vibrate due to the influence of a periodic force with the same frequency as the object s natural frequency, the object is said to be at resonance. During resonance, the amplitude of vibration of the object is greatly enhanced because of an efficient energy transfer from the source of the periodic force to the object. Unit 7 (Part 2) - The Physics of Sound 22

23 An air column can be divided into two types: Closed air column: closed at one end and open at the other Open air column: open at both ends (i) Closed air column For a closed air column such as a tube, a node is formed at the closed end because air cannot move freely at this point. At the open end, since air can move freely, an antinode is formed. Therefore, a loop is formed. For the fundamental mode, the length of the tube is equal to λ 4. Unit 7 (Part 2) - The Physics of Sound 23

24 Besides the fundamental, other modes of standing waves can also be formed in a closed air column. The available modes must have a node at the closed end and an antinode at the open end. With these constraints, we can sketch the possible modes as shown below. Unit 7 (Part 2) - The Physics of Sound 24

25 As shown, not all forms of standing wave are possible. They must fulfill the following relation for the wavelength: L = 2n + 1 λ n = 0, 1, 2, 4 This therefore yields the accessible modes of vibrations: f n = (2n + 1)v 4L = (2n + 1)f 0 n = 0, 1, 2, where f 0 = v/4l is the fundamental mode. From the above-mentioned equation, we can see that only the odd multiples of the fundamental are allowed for a closed air column. Unit 7 (Part 2) - The Physics of Sound 25

When a standing wave is formed, it must have antinodes at both ends.

26 (ii) Open air column For this type of air column, no node can be formed at either ends because they are all open. When a standing wave is formed, it must have antinodes at both ends. It suggests that there is at least one node within the column. For the fundamental mode, the length of the tube is equal to λ 2. Unit 7 (Part 2) - The Physics of Sound 26

27 Other modes possible to an open air column can be deduced in the same way as for the closed air column. From these figures, we see that the relationship between the length of the tube and the wavelength is given by L = n λ n = 0, 1, 2, Therefore the vibration frequencies allowed are: f n = n + 1 v 2L = (n + 1)f 0 n = 0, 1, 2, Unit 7 (Part 2) - The Physics of Sound 27

28 As seen, all multiples of the fundamental mode for open air column are possible, in contrast to the closed air column. In summary: Open air column Closed air column Overtone Harmonic Frequency Wavelength Harmonic Frequency Wavelength 0 1 v/2l 2L 1 v/4l 4L 1 2 2v/2L 2L/2 3 3v/4L 4L/ v/2L 2L/3 5 5v/4L 4L/ v/2L 2L/4 7 7v/4L 4L/ v/2L 2L/5 9 9v/4L 4L/9 Unit 7 (Part 2) - The Physics of Sound 28

29 Unit 7 (Part 2) - The Physics of Sound 29

30 Example: The first resonance of a closed air column occurs when the length is 18 cm. What is the wavelength of the sound? If the frequency of the sound is 512 Hz, what is the speed of sound? For the fundamental mode of a closed air column, λ = 4L Therefore, λ = 4 18 = 72 cm = 0.72 m If frequency is 512 Hz, the speed of sound is: v = f 0 λ = = 370 m/s Unit 7 (Part 2) - The Physics of Sound 30

31 Example: What will be the fundamental frequency and first three overtones for a 26-cm organ pipe at 20 C if it is open? At 20 C, the speed of sound is v = = 343 m/s For an open pipe, the fundamental frequency is f 0 = v 2L = 343 = 660 Hz 2(0.26) Since all harmonics are possible for open pipes, the first three overtones are respectively: f 1 = = 1320 Hz f 2 = 1980 Hz f 3 = 2640 Hz Unit 7 (Part 2) - The Physics of Sound 31

32 3. Wave phenomena of sound As introduced at the beginning of this section, sound is a type of wave in which particles move longitudinally. Because of this, sound inherits all the typical properties that waves demonstrate. (1) Echo and reverberation Sound waves radiating out from a source are reflected when they strike a rigid obstacle. The reflected angle will be equal to the angle of incidence. If the sound is reflected by a wall or a cliff, echoes are produced. Unit 7 (Part 2) - The Physics of Sound 32

33 Human ears are able to distinguish two sources of sound if there is a time difference of 0.1 second or longer. Hence, echoes can be heard if the reflected sound returns to the speaker at least 0.1 second after. In practice, it corresponds to about 17 m of distance between the observer and the reflecting surface. If the reflecting surface is less than 17 m away, the echo will follow very closely to the original sound, and they will be mixed together, making the original sound to be apparently longer. This phenomenon is called reverberation. Suitable amount of reverberation may raise up the quality of sound in a concert, but excess reverberation is undesirable. This is the most important concern of the architect when designing a concert hall. Unit 7 (Part 2) - The Physics of Sound 33

taken for a sound wave to hit the bottom of the sea and return to the receiver.

34 Reflection of sound and echoes have been applied in many aspects. For example, the echo-sounder is a device that measures the depth of the sea by measuring the time taken for a sound wave to hit the bottom of the sea and return to the receiver. A similar device called sonar (sound navigation and ranging) is employed in fish industry to locate schools of fish in the ocean. Unit 7 (Part 2) - The Physics of Sound 34

$(2) Diffraction Sound, like water waves, can diffract around a corner.$ $It has been shown that waves with large wavelength diffract more than those$

35 (2) Diffraction Sound, like water waves, can diffract around a corner. That s why we can still hear a loudspeaker located behind a wall. It has been shown that waves with large wavelength diffract more than those with shorter wavelengths, resulting in a smaller shadow region. Unit 7 (Part 2) - The Physics of Sound 35

$http://hyperphysics.phy-astr.gsu.edu/hbase/sound/diffrac.$

36 Consider a situation when you are about to meet a marching band at a cross intersection. Which sound will you hear first, piccolo or drum? Unit 7 (Part 2) - The Physics of Sound 36

$(3) Refraction of sound Sound transmits at different speeds in air when temperature is different. Higher the temperature, higher the speed of sound.$

37 (3) Refraction of sound Sound transmits at different speeds in air when temperature is different. Higher the temperature, higher the speed of sound. When sound moves from air at one temperature to air at another temperature at a certain angle, it is refracted in the same way as other waves. Unit 7 (Part 2) - The Physics of Sound 37

$Consider an aeroplane flying over the downtown area. During night time, air near the ground is colder. It therefore refracts the sound from the plane towards the surface.$

38 Consider an aeroplane flying over the downtown area. During night time, air near the ground is colder. It therefore refracts the sound from the plane towards the surface. The sound can be heard more easily. During day time, since air near the surface is warmer, the sound is refracted upward, getting farther away from the surface. Hence, the sound is not heard much. Unit 7 (Part 2) - The Physics of Sound 38

$The concept of sound refraction can also be used to explain why Jill can hear Jack but not Dana.$

39 The concept of sound refraction can also be used to explain why Jill can hear Jack but not Dana. Since Jack is talking tail wind, higher the altitude, higher the relative speed of his sound to the ground. Hence, his sound is refracted downward. On the contrary, Dana is talking head wind, making the relative speed of her sound to decrease with increasing altitude. Consequently, her sound is refracted upward. Unit 7 (Part 2) - The Physics of Sound 39

40 (4) Interference of sound An interesting phenomenon of sound interaction is the interference, or superposition, of sound waves. When two sound waves of same frequency interfere, the pattern of constructive and destructive interference that is observed in water waves is formed. Unit 7 (Part 2) - The Physics of Sound 40

41 When two sounds of slightly different frequencies, for example, f 1 and f 2 meet, they mix together, forming a wave whose intensity changes periodically. It is called a beat. Unit 7 (Part 2) - The Physics of Sound 41

42 The mixture is composed of a vibration at the average frequency 1 2 f 1 + f 2 modulated by an envelope of frequency 1 2 f 1 f 2. Human ear is sensitive to sound intensity but not the phase; therefore, we can only hear the ups and downs of the sound. The frequency of beat which is actually audible is determined by the absolute difference of the frequencies of the original waves. f beat = f 2 f 1 Unit 7 (Part 2) - The Physics of Sound 42

43 Example: A tuning fork with a frequency of 440 Hz is played simultaneously with a fork with a frequency of 437 Hz. How many beats will be heard over a period of 10 seconds? The beat frequency is f b = f 2 f 1 = = 3 Hz Therefore in 10 seconds, the number of beats to be heard is Number of beats = 3 Hz 10 s = 30 beats Unit 7 (Part 2) - The Physics of Sound 43

The sound wave is isotropic; in all directions, the wave is identical and is

44 (5) Doppler Effect A stationary source emits a sound which transmits in a circular way. The sound wave is isotropic; in all directions, the wave is identical and is traveling at the same speed. However, when the source is moving, now the wavefronts are no longer anisotropic. Unit 7 (Part 2) - The Physics of Sound 44

Since the speed of sound wave depends only on medium and temperature, if the source is moving toward a stationary observer, the waves in front of the source will become more packed, making the

45 Since the speed of sound wave depends only on medium and temperature, if the source is moving toward a stationary observer, the waves in front of the source will become more packed, making the observer to receive the waves more frequently than the frequency at which the waves are actually produced. According to the general wave equation, as the separation between two successive wavefronts is shortened, the frequency is higher. This phenomenon is known as the Doppler effect. Unit 7 (Part 2) - The Physics of Sound 45

46 Mathematical analysis shows that the frequency heard by an observer, f obs, from a moving source is given by the relationship f obs = v v v s f 0 In this equation, v is the speed of sound, v s is the speed of the source, and f 0 is the actual frequency of the sound. In convention, the negative sign is used when the source is moving toward the observer, while a positive sign is used when the source is moving away from the observer. Unit 7 (Part 2) - The Physics of Sound 46

47 That implies a higher pitch will be heard when an ambulance is approaching you! Unit 7 (Part 2) - The Physics of Sound 47

48 Similar effect will be present when an observer is moving relative to a stationary source. In that situation, the observed frequency is given by f obs = v ± v o f v 0 Positive sign when observer is approaching the source, while negative sign when observer is receding. Unit 7 (Part 2) - The Physics of Sound 48

49 Combining these two equations together we can obtain a general equation for the Doppler effect: f obs = v ± v 0 v v s f 0 Unit 7 (Part 2) - The Physics of Sound 49

50 Example: Suppose you are standing on the corner of 5 th Avenue and 34 th Street waiting for the light to change so you can cross the street. An approaching southbound ambulance is heading your way traveling at 35 miles per hour. If we know that the frequency of the ambulance siren is 700 Hz, what is the frequency that you hear when (a) the ambulance is approaching you? (b) the ambulance is departing from you? assuming the temperature is 15 C. The speed of sound at 15 C is: v = = 340 m/s Unit 7 (Part 2) - The Physics of Sound 50

51 The speed of the ambulance is: v s = = m/s 3.6 When the ambulance is approaching, the frequency is: f obs = = 734 Hz When the ambulance is departing, the frequency is: f obs = = 669 Hz Unit 7 (Part 2) - The Physics of Sound 51

52 Example: A particular bat emits ultrasonic waves with a frequency of khz. The bat is traveling at m/s toward a moth, which is flying away from the bat at 8.00 m/s. The speed of sound is m/s. Assuming the moth could detect the waves, what frequency waves would it observe? Since the bat is moving toward the moth and the moth is moving away from the bat, the general Doppler effect equation gives f obs = = 58.1 khz The frequency heard by the moth is 58.1 khz. Unit 7 (Part 2) - The Physics of Sound 52

53 (6) Sonic boom When a source is moving fast, it produces a pile of wavefronts in front of it. Faster the source is moving, bigger this pile of wavefronts. Since sound waves are indeed compression of air molecules, such a thick pile implies a huge compression of air, causing a wall of air, or sound barrier, which will hinder the source from speeding up further. The sound barrier is reached when the source is moving at the speed of sound. At this situation, we say that the Mach number is equal to 1. Mach number is defined as the ratio of the speed of object to the speed of sound at a particular temperature, i.e., M = speed of object speed of sound Unit 7 (Part 2) - The Physics of Sound 53

54 Technically, we can classify the speed of object in terms of Mach number. Mach number Term 0 Stationary < 0.8 Subsonic Transonic Supersonic > 5 Hypersonic When an object is traveling close to the speed of sound (i.e., M 1), the air pressure causes disturbance and other aerodynamic effects that drag the object. Normal designs of aircrafts are very difficult to exceed this sound barrier. Unit 7 (Part 2) - The Physics of Sound 54

When a jet is able to break through this sound barrier, it can go faster than the speed of sound. Sounds waves are being produced continually yet they are left behind the jet.

55 When a jet is able to break through this sound barrier, it can go faster than the speed of sound. Sounds waves are being produced continually yet they are left behind the jet. These circular sound waves overlap and interfere constructively, forming a loud cone of noise called the shock wave. It grows bigger but weaker when it expands. Unit 7 (Part 2) - The Physics of Sound 55

Water in the atmosphere condenses because of the air pressure drop and forms cloud around the jet traveling

56 Since sound wave is isotropic in three dimensions, the interference of wavefronts results in a cone-like wave trailing the jet. This cone can sometimes be visible because of humidity. Water in the atmosphere condenses because of the air pressure drop and forms cloud around the jet traveling at transonic speed. In this situation, the cone is called a vapor cone, sonic collar or sonic egg. Unit 7 (Part 2) - The Physics of Sound 56

57 Associated with shock waves is a sonic boom which is a sudden change of air pressure when a supersonic jet is passing by. This sudden change of air pressure is heard like an explosion with a great release of sound energy. The shock wave intersects the surface in a hyperbolic curve. All people at different points on this curve will hear the sound at the same time. This sonic boom curve sweeps the surface when the supersonic jet moves forward, and cause damage to whatever it encounters on the surface. Unit 7 (Part 2) - The Physics of Sound 57

58 There are many examples of shock waves and sonic boom that we can observe in daily lives besides supersonic aircrafts. For example: explosive detonation (i) low explosive: subsonic combustion, speed at hundreds of m/s (ii) high explosive: supersonic combustion, speed at thousands of m/s It is called Octahydro-1,3,5,7- tetranitro-1,3,5,7-tetrazocine, or octogen, or HMX. An extremely powerful military explosive. Detonation velocity up to 9100 m/s! Unit 7 (Part 2) - The Physics of Sound 58

Sound Waves. Sound waves are longitudinal waves traveling through a medium Sound waves are produced from vibrating objects.

Sound Waves. Sound waves are longitudinal waves traveling through a medium Sound waves are produced from vibrating objects. Sound Waves Sound waves are longitudinal waves traveling through a medium Sound waves are produced from vibrating objects Introduction Sound Waves: Molecular View When sound travels through a medium, there