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THE PHYSICS OF MUSIC AND MUSICAL INSTRUMENTS f f f f 1 3 5 7 DAVID R. LAPP, FELLOW WRIGHT CENTER FOR INNOVATIVE SCIENCE EDUCATION TUFTS UNIVERSITY MEDFORD, MASSACHUSETTS“Everything is determined … by forces over which we have no control. It is determined for the insects as well as for the star. Human beings, vegetables, or cosmic dust – we all dance to a mysterious tune, intoned in the distance by an invisible piper.” – Albert Einstein INTRODUCTION HIS MANUAL COVERS the physics of waves, sound, music, and musical instruments at a level designed for high school physics. However, it is also a resource for those teaching and learning waves and sound from middle school through college, at a mathematical or conceptual level. The T mathematics required for full access to the material is algebra (to include logarithms), although each concept presented has a full conceptual foundation that will be useful to those with even a very weak background in math. MODES OF PRESENTATION Solomon proclaimed that there is nothing new As the student reads through the text, he or she under the Sun and of the writing of books there is no will encounter a number of different presentation end. Conscious of this, I have tried to produce modes. Some are color-coded. The following is a key something that is not simply a rehash of what has to the colors used throughout the text: already been done elsewhere. In the list of references I Pale green boxes cover tables and figures have indicated a number of very good sources, some that are important reference material. classics that all other writers of musical acoustic books refer to and some newer and more accessible works. From these, I have synthesized what I believe Notes Frequency to be the most useful and appropriate material for the interval (cents) high school aged student who has neither a C 0 i background in waves nor in music, but who desires a D 204 firm foundation in both. Most books written on the topic of musical acoustics tend to be either very E 408 theoretical or very cookbook style. The theoretical F 498 ones provide for little student interaction other than G 702 some end of the chapter questions and problems. The A 906 ones I term “cookbook” style provide instructions for B 1110 building musical instruments with little or no C 1200 explanation of the physics behind the construction. f This curriculum attempts to not only marry the best Table 2.8: Pythagorean ideas from both types of books, but to include scale interval ratios pedagogical aids not found in other books. This manual is available as both a paper hard copy as well as an e-book on CD-ROM. The CD- Light yellow boxes highlight derived ROM version contains hyperlinks to interesting equations in their final form, which will be used for websites related to music and musical instruments. It future calculations. also contains hyperlinks throughout the text to sound files that demonstrate many concepts being developed. T m f = 1 2L † 1INTRODUCTION INTRODUCTION INTRODUCTION consider. Investigations are labs really, often requiring Tan boxes show step-by-step examples for students to make measurements directly on the making calculations or reasoning through questions. photographs. Solutions to the “Do you get it?” boxes, Activities, and Investigations are provided in an appendix on the CD-ROM. Finally, projects Example provide students with some background for building If the sound intensity of a screaming baby were musical instruments, but they leave the type of -2 W 1¥10 at 2.5 m away, what would it be at musical scale to be used as well as the key the 2 m instrument will be based on largely up to the student. 6.0 m away? The distance from the source of sound is greater by a PHYSICS AND … MUSIC? 6.0 factor of = 2.4 . So the sound intensity is decreased † 2.5 1 by = 0.174 . The new sound intensity is: 2 (2.4) -2 W -2 W 1.74 ¥10 (1¥10 )(0.174) = 2 † 2 m m † † † Gray boxes throughout the text indicate stopping places in the reading where students are asked, “Do you get it?” The boxes are meant to reinforce student understanding with basic recall questions about the immediately preceding text. These can be used to begin a discussion of the reading with a class of students. Do you get it? (4) A solo trumpet in an orchestra produces a sound intensity level of 84 dB. Fifteen more trumpets join the first. How many decibels are produced? “Without music life would be a mistake.” – Friedrich Nietzsche In addition to the “Do you get it?” boxes, which are meant to be fairly easy questions done individually by students as they read through the text, With even a quick look around most school there are three additional interactions students will campuses, it is easy to see that students enjoy music. encounter: Activities, Investigations, and Ears are sometimes hard to find, covered by Projects. Activities more difficult than the “Do you headphones connected to radios or portable CD get it?” boxes and are designed to be done either players. And the music flowing from them has the individually or with a partner. They either require a power to inspire, to entertain, and to even mentally higher level of conceptual understanding or draw on transport the listener to a different place. A closer more than one idea. Investigations are harder still and look reveals that much of the life of a student either draw on more than an entire section within the text. revolves around or is at least strongly influenced by Designed for two or more students, each one music. The radio is the first thing to go on in the photographically exposes the students to a particular morning and the last to go off at night (if it goes off musical instrument that they must thoroughly at all). T-shirts with logos and tour schedules of 2INTRODUCTION INTRODUCTION INTRODUCTION popular bands are the artifacts of many teens’ most understand the rationale for the development of the coveted event … the concert. But the school bell musical scales one needs a broad foundation in most ringing for the first class of the day always brings elements of wave and sound theory. With that said, with it a stiff dose of reality. the approach here will be to understand music and musical instruments first, and to study the physics of waves and sound as needed to push the understanding of the music concepts. The goal however is a deeper understanding of the physics of waves and sound than what would be achieved with a more traditional approach. SOUND, MUSIC, AND NOISE Do you like music? No, I guess a better question is, what kind of music do you like? I don’t think anyone dislikes music. However, some parents consider their children’s “music” to be just noise. Likewise, if the kids had only their parent’s music to listen to many would avoid it in the same way they avoid noise. Perhaps that’s a good place to start then – the contrast between music and noise. Is there an objective, physical difference between music and H. L. Mencken writes, “School days, I believe, noise, or is it simply an arbitrary judgment? are the unhappiest in the whole span of human After I saw the movie 8 Mile, the semi- existence. They are full of dull, unintelligible tasks, autobiographical story of the famous rapper Eminem, new and unpleasant ordinances, brutal violations of I recommended it to many people … but not to my common sense and common decency.” This may mother. She would have hated it. To her, his music is paint too bleak a picture of the typical student’s just noise. However, if she hears an old Buddy Holly experience, but it’s a reminder that what is taught song, her toes start tapping and she’s ready to dance. often lacks meaning and relevance. When I think back But the music of both of these men would be to my own high school experience in science, I find considered unpleasant by my late grandmother who that there are some classes for which I have no seemed to live for the music she heard on the memory. I’m a bit shocked, but I realize that it would Lawrence Welk Show. I can appreciate all three be possible to spend 180 hours in a science classroom “artists” at some level, but if you ask me, nothing and have little or no memory of the experience if the beats a little Bob Dylan. It’s classroom experience were obviously not easy to define lifeless or disconnected the difference between noise from the reality of my life. and music. Certainly there is Middle school and high the presence of rhythm in the school students are a tough sounds we call music. At a audience. They want to be more sophisticated level there entertained … but they is the presence of tones that don’t have to be. What they combine with other tones in really need is relevance. an orderly and ... “pleasing” They want to see direct way. Noise is often associated connections and immediate with very loud and grating applications. This is the sounds – chaotic sounds which reason for organizing an don’t sound good together or introduction to the physics are somehow “unpleasant” to of waves and sound around listen to. Most would agree the theme of music and that the jackhammer tearing musical instruments. up a portion of the street is It’s not a stretch either. noise and the sound coming Both music and musical from the local marching band instruments are intimately is music. But the distinction connected to the physics of is much more subtle than that. waves and sound. To fully If music consists of sounds appreciate what occurs in a with rhythmic tones of certain musical instrument when it frequencies then the makes music or to jackhammer might be 3INTRODUCTION INTRODUCTION INTRODUCTION considered a musical instrument. After all, it the vast appreciation of this type of music. Defining pummels the street with a very regular frequency. And the very earliest music and still prominent in many if noise consists of loud sounds, which are unpleasant cultures, this musical sound stresses beat over to listen to, then the cymbals used to punctuate the melody, and may in fact include no melody at all. performance of the marching band might be One of the reasons for the popularity of rhythm-only considered noise. I suppose we all define the point music is that anyone can immediately play it at some where music becomes noise a bit differently. Perhaps level, even with no training. Kids do it all the time, it’s based on what we listen to most or on the naturally. The fact that I often catch myself generation we grow up in or … make a break from. spontaneously tapping my foot to an unknown beat But we need to be careful about being cavalier as I or lie in bed just a bit longer listening contentedly to was just now when I talked about “pleasing” sounds. my heartbeat is a testament to the close connection ® John Bertles of Bash the Trash (a company dedicated between life and rhythm. to the construction and performance of musical Another aspect of music is associated with more instruments from recycled materials: or less pure tones – sounds with a constant pitch. ) was quick to caution Whistle very gently and it sounds like a flute playing a single note. But that single note is hardly a song, me when I used the word “pleasing” to describe and certainly not a melody or harmony. No, to make musical sound. Music that is pleasing to one person the single tone of your whistle into a musical sound may not be pleasing to others. Bertles uses the you would have to vary it in some way. So you could definition of intent rather than pleasing when change the way you hold your mouth and whistle discussing musical sound. He gives the example of a again, this time at a different pitch. Going back and number of cars all blaring their horns chaotically at forth between these two tones would produce a an intersection. The sound would be considered noise cadence that others might consider musical. You to most anyone. But the reason for the noise is not so could whistle one pitch in one second pulses for three much the origin of the sound, but the lack of intent seconds and follow that with a one second pulse of to organize the sounds of the horns. If someone at the the other pitch. Repeating this pattern over and over intersection were to direct the car horns to beep at would make your tune more interesting. And you particular times and for specific periods, the noise could add more pitches for even more sophistication. would evolve into something more closely related to You could even have a friend whistle with you, but music. And no one can dispute that whether it’s with a different pitch that sounded good with the one Eminem, Buddy Holly, Lawrence Welk, or Bob you were whistling. Dylan, they all create(d) their particular recorded If you wanted to move beyond whistling to sounds with intent. making music with other physical systems, you could bang on a length of wood or pluck a taut fiber or blow across an open bamboo tube. Adding more “There are two means of refuge from pieces with different lengths (and tensions, in the case the miseries of life: music and cats.” of the fiber) would give additional tones. To add more – Albert Schweitzer complexity you could play your instrument along with other musicians. It might be nice to have the sound of your instrument combine nicely with the sound of other instruments and have the ability to BEGINNING TO DEFINE MUSIC play the tunes that others come up with. But to do Music makes us feel good, it whisks us back in this, you would have to agree on a collection of time to incidents and people from our lives; it rescues common pitches. There exist several combinations of us from monotony and stress. Its tempo and pace jive common pitches. These are the musical scales. with the natural rhythm of our psyche. Here we have to stop and describe what the pitch The simplest musical sound is some type of of a sound is and also discuss the various rhythmical beating. The enormous popularity of the characteristics of sound. Since sound is a type of stage show Stomp and wave, it’s additionally necessary to go even further the large screen Omnimax movie, Pulse back and introduce the idea of a wave. gives evidence for 4“It is only by introducing the young to great literature, drama and music, and to the excitement of great science that we open to them the possibilities that lie within the human spirit – enable them to see visions and dream dreams.” – Eric Anderson CHAPTER 1 WAVES AND SOUND SK MOST PEOPLE to define what a wave is and they get a mental image of a A wave at the beach. And if you really press them for an answer, they're at a loss. They can describe what a wave looks like, but they can't tell you what it is. What do you think? If you want to get energy from one place to another you can do it by transferring it Figure 1.1: Most people think of the ocean when with some chunk of matter. For example, if asked to define or describe a wave. The recurring you want to break a piece of glass, you don't tumult is memorable to anyone who has spent time have to physically make contact with it at the beach or been out in the surf. But waves occur yourself. You could throw a rock and the most places and in many different forms, transferring energy you put into the rock would travel on energy without transferring matter. the rock until it gets to the glass. Or if a (like the one that hit the coast of the East Indies in police officer wants to subdue a criminal, he doesn't 2 August 1883) would have 25 or 625 times more have to go up and hit him. He can send the energy to energy than the four-foot wave. the criminal via a bullet. But there's another way to Streaming through the place you're in right now transfer energy – and it doesn't involve a transfer of is a multitude of waves known as electromagnetic matter. The other way to transfer energy is by using a waves. Their wavelengths vary from so small that wave. A wave is a transfer of energy without a millions would fit into a millimeter, to miles long. transfer of matter . If you and a friend hold onto both They’re all here, but you miss most of them. The ends of a rope you can get energy down to her simply only ones you're sensitive to are a small group that by moving the rope back and forth. Although the stimulates the retinas of your eyes (visible light) and rope has some motion, it isn't actually transferred to a small group that you detect as heat (infrared). The her, only the energy is transferred. others are totally undetectable. But they’re there. A tsunami (tidal wave generally caused by an earthquake) hit Papua New Guinea in the summer of WAVES, SOUND, AND THE EAR 1998. A magnitude 7 earthquake 12 miles offshore Another type of wave is a sound wave. As small sent energy in this 23-foot tsunami that killed in energy as the tsunami is large, we usually need an thousands of people. Most people don't realize that ear to detect these. Our ears are incredibly awesome the energy in a wave is proportional to the square of receptors for sound waves. The threshold of hearing is the amplitude (height) of the wave. That means that if -12 2 somewhere around 1¥10 Watts /meter . To you compare the energy of a 4-foot wave that you understand this, consider a very dim 1-watt night- might surf on to the tsunami, even though the light. Now imagine that there were a whole lot more tsunami is only about 6 times the height, it would 2 people on the planet than there are now – about 100 have 6 or 36 times more energy. A 100-foot tsunami † 5WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND times more. Assume there was a global population of 1 trillion (that's a million, million) people. If you split the light from that dim bulb equally between all those people, each would hold a radiant -12 power of 1¥10 Watts. Finally, let's say that one of those people spread that power over an area of one † square meter. At this smallest of perceptible sound intensities, the eardrum vibrates less distance than the diameter of a hydrogen atom Well, it's so small an amount of power that you can hardly conceive of it, but if you have pretty good hearing, you could detect a sound wave with that small amount of power. That's Figure 1.2: The ear is an astonishing receptor for sound waves. At not all. You could also the smallest of perceptible sound intensities, the eardrum vibrates detect a sound wave a less distance than the diameter of a hydrogen atom If the energy in thousand times more a single 1-watt night-light were converted to acoustical energy and powerful, a million times divided up into equal portions for every person in the world, it more powerful, and even a would still be audible to the person with normal hearing. billion times more out of a simple sound. He didn’t have sight so he had powerful. And, that's before it even starts to get to compensate with his other senses. He got so much painful out of what I would have considered a very simplistic I have a vivid fifth grade memory of my good sound. For him the world of sound was rich and friend, Norman. Norman was blind and the first and diverse and full. When I think of sound, I always only blind person I ever knew well. I sat next to him think first of Norman. He’s helped me to look more in fifth grade and watched amazed as he banged away deeply and to understand how sophisticated the world on his Braille typewriter. I would ask him questions of sound really is. about what he thought colors looked like and if he What about when more than one wave is present could explain the difference between light and dark. in the same place? For example, how is it that you He would try to educate me about music beyond top- can be at a symphony and make out the sounds of 40 Pop, because he appreciated and knew a lot about individual instruments while they all play together jazz. But when it came to recess, we parted and went and also hear and understand a message being our separate ways – me to the playground and him to whispered to you at the same time you detect the wall outside the classroom. No one played with someone coughing five rows back? How do the sound Norman. He couldn’t see and so there was nothing for waves combine to give you the totality as well as the him. About once a day I would look over at Norman individuality of each of the sounds in a room? These from high up on a jungle gym of bars and he would are some of the questions we will answer as we be smacking one of those rubbery creepy crawlers continue to pursue an understanding of music and against the wall. He would do it all recess … every musical instruments. recess. I still marvel at how much Norman could get 6WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND TWO TYPES OF WAVES Waves come in two basic types, depending on their type of vibration. Imagine laying a long slinky on the ground and shaking it back and forth. You fi would make what is called a transverse wave (see Figure 1.3). A transverse wave is one in which the medium vibrates at right angles to the direction the Figure 1.3: A transverse wave moves to the energy moves. If instead, you pushed forward and right while the medium vibrates at right pulled backward on the slinky you would make a angles, up and down. compressional wave (see Figure 1.4). Compressional waves are also known as longitudinal waves. A compressional wave is one in which the medium vibrates in the same direction as the movement of energy in the wave Certain terms and ideas related to waves are easier fi to visualize with transverse waves, so let’s start by thinking about the transverse wave you could make with a slinky. Imagine taking a snapshot of the wave Figure 1.4: A compressional wave moves to from the ceiling. It would look like Figure 1.5. Some the right while the medium vibrates in the wave vocabulary can be taken directly from the same direction, right to left. diagram. Other vocabulary must be taken from a mental image of the wave in motion: Crest Wavelength Rest Amplitude position Trough Amplitude Wavelength Figure 1.5: Wave vocabulary vocabulary CREST: The topmost point of the wave medium or energy grows, so does the amplitude. This makes greatest positive distance from the rest position. sense if you think about making a more energetic slinky wave. You’d have to swing it with more TROUGH: The bottommost point of the wave intensity, generating larger amplitudes. The medium or greatest negative distance from the rest relationship is not linear though. The energy is position. actually proportional to the square of the amplitude. So a wave with amplitude twice as large actually has WAVELENGTH (l): The distance from crest to four times more energy and one with amplitude three adjacent crest or from trough to adjacent trough or times larger actually has nine times more energy. from any point on the wave medium to the adjacent The rest of the vocabulary requires getting a corresponding point on the wave medium. mental picture of the wave being generated. Imagine your foot about halfway down the distance of the AMPLITUDE (A): The distance from the rest slinky’s stretch. Let’s say that three wavelengths pass position to either the crest or the trough. The your foot each second. amplitude is related to the energy of the wave. As the 7WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND FREQUENCY (f): The number of wavelengths to somewhat like a playground swing. The playground pass a point per second. In this case the frequency swing, the tuning fork, and most physical systems would be 3 per second. Wave frequency is often will act to restore themselves if they are stressed from spoken of as “waves per second,” “pulses per second,” their natural state. The “natural state” for the swing, or “cycles per second.” However, the SI unit for is to hang straight down. If you push it or pull it and frequency is the Hertz (Hz). 1 Hz = 1 per second, so then let go, it moves back towards the position of in the case of this illustration, f = 3 Hz. hanging straight down. However, since it’s moving when it reaches that point, it actually overshoots and, in effect, stresses itself. This causes another attempt PERIOD (T): The time it takes for one full to restore itself and the movement continues back and wavelength to pass a certain point. If you see three forth until friction and air resistance have removed all wavelengths pass your foot every second, then the 1 the original energy of the push or pull. The same is time for each wavelength to pass is of a second. 3 true for the tuning fork. It’s just that the movement Period and frequency are reciprocals of each other: (amplitude) is so much smaller that you can’t visibly see it. But if you touched the fork you could feel it. Indeed, every time the fork moves back and forth it † 1 1 smacks the air in its way. That smack creates a small T = and f = f T compression of air molecules that travels from that point as a compressional wave. When it reaches your ear, it vibrates your eardrum with the same frequency SPEED (v) Average speed is always a ratio of as the frequency of the motion of the tuning fork. † † distance to time, v = d /t . In the case of wave speed, You mentally process this vibration as a specific an appropriate distance would be wavelength, l. The tone. corresponding time would then have to be the period, T. So the wave † speed becomes: l v = or v = lf t † SOUND WAVES † If a tree falls in the forest and there’s no one there to hear it, does it make a sound? It’s a common question that usually evokes a philosophical response. I could argue yes or no convincingly. You will too later on. Most people have a very strong opinion one way or the other. The problem is that their opinion is usually not based on a clear understanding of what sound is. I think one of the easiest ways to understand sound is to look at something that has a simple mechanism for making sound. Think about how a tuning fork makes sound. Striking one of the forks causes you to immediately hear a Figure 1.6: The energy of the vibrating tuning fork tone. The tuning fork begins to act violently splashes water from the bowl. In the air, the energy of the tuning fork is transmitted through the air and to the ears. 8WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND mph). But at room temperature (about 20°C) sound travels at: Ê ˆ m m /s v = 331 + 0.6 20°C Á ˜ ( ) s Ë °C ¯ m = 343 s † (This is the speed you should assume if no temperature is given). † Figure 1.7: The front of a speaker cone faces upward with several pieces of orange paper lying on top of it. It was one of aviation’s greatest Sound is generated when an electric signal causes the accomplishments when Chuck Yeager, on speaker cone to move in and out, pushing on the air Oct. 14, 1947, flew his X-1 jet at Mach and creating a compressional wave. The ear can detect 1.06, exceeding the speed of sound by 6%. these waves. Here these vibrations can be seen as they Regardless, this is a snail’s pace compared to cause the little bits of paper to dance on the surface of the speed of light. Sound travels through air the speaker cone. at about a million times slower than light, which is the reason why we hear sound A sound wave is nothing more than a echoes but don’t see light echoes. It’s also the reason compressional wave caused by vibrations . Next time we see the lightning before we hear the thunder. The you have a chance, gently feel the surface of a speaker lightning-thunder effect is often noticed in big cone (see Figure 1.7). The vibrations you feel with stadiums. If you’re far away from a baseball player your fingers are the same vibrations disturbing the who’s up to bat, you can clearly see the ball hit air. These vibrations eventually relay to your ears the before you hear the crack of the bat. You can consider message that is being broadcast. So, if a tree falls in that the light recording the event reaches your eyes the forest and there’s no one there to hear it, does it virtually instantly. So if the sound takes half a second make a sound? Well … yes, it will certainly cause more time than the light, you’re half the distance vibrations in the air and ground when it strikes the sound travels in one second (165 meters) from the ground. And … no, if there’s no one there to batter. Next time you’re in a thunderstorm use this mentally translate the vibrations into tones, then method to estimate how far away lightning is there can be no true sound. You decide. Maybe it is a striking. Click here for a demonstration of the effect philosophical question after all. of echoes. CHARACTERIZING SOUND All sound waves are compressional waves caused Do you get it? (1.1) by vibrations, but the music from a symphony varies Explain why some people put their ears on railroad considerably from both a baby’s cry and the whisper tracks in order to hear oncoming trains of a confidant. All sound waves can be characterized by their speed, by their pitch, by their loudness, and by their quality or timbre. The speed of sound is fastest in solids (almost 6000 m/s in steel), slower in liquids (almost 1500 m/s in water), and slowest in gases. We normally listen to sounds in air, so we’ll look at the speed of sound in air most carefully. In air, sound travels at: Ê ˆ m /s m v = 331 + 0.6 Temperature Á ˜ s Ë °C ¯ The part to the right of the “+” sign is the † temperature factor. It shows that the speed of sound increases by 0.6 m/s for every temperature increase of 1°C. So, at 0° C, sound travels at 331 m/s (about 740 9WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND distinguish between similar sounding words Do you get it? (1.2) that use these letters. Furthermore, the vowels are generally loud (they use about The echo of a ship's foghorn, reflected from the cliff 95% of the voice energy to produce). The on a nearby island, is heard 5.0 s after the horn is consonants are left with only 5% to go around sounded. How far away is the cliff? for all of them. But it is mostly the consonants that give speech its intelligibility. That is why many older people will say, ‘I can hear people talking. I just can't understand what they are saying.’” One important concept in music is the octave – a doubling in frequency. For example, 40 Hz is one octave higher than 20 Hz. The ear is sensitive over a frequency range of about 10 octaves: 20 Hz Æ 40 Hz Æ 80 Hz Æ 160 Hz Æ 320 Hz Æ 640 Hz Æ 1,280 Hz Æ 2,560 Hz Æ 5,120 Hz Æ 10,240 Hz Æ 20,480 Hz. And within that range it can discriminate between thousands of differences in sound frequency. Below about 1,000 Hz the Just Noticeable The pitch of sound is the same as the Difference (JND) in frequency is about 1 Hz (at the frequency of a sound wave. With no hearing losses or loudness at which most music is played), but this defects, the ear can detect wave frequencies between 20 rises sharply beyond 1,000 Hz. At 2,000 the JND is Hz and 20,000 Hz. (Sounds below 20 Hz are about 2 Hz and at 4,000 Hz the JND is about 10 Hz. classified as subsonic; those over 20,000 Hz are (A JND of 1 Hz at 500 Hz means that if you were ultrasonic). However, many older people, loud concert asked to listen alternately to tones of 500 Hz and attendees, and soldiers with live combat experience 501 Hz, the two could be distinguished as two lose their ability to hear higher frequencies. The good different frequencies, rather than the same). It is news is that the bulk of most conversation takes interesting to compare the ear’s frequency perception place well below 2,000 Hz. The average frequency to that of the eye. From red to violet, the frequency of range of the human voice is 120 Hz to approximately light less than doubles, meaning that the eye is only 1,100 Hz (although a baby’s shrill cry is generally sensitive over about one octave, and its ability to 2,000 - 3,000 Hz – which is close to the frequency discriminate between different colors is only about range of greatest sensitivity … hmm, interesting). 125. The ear is truly an amazing receptor, not only Even telephone frequencies are limited to below its frequency range, but also in its ability to 3,400 Hz. But the bad news is that the formation of accommodate sounds with vastly different loudness. many consonants is a complex combination of very The loudness of sound is related to the high frequency pitches. So to the person with high amplitude of the sound wave. Most people have some frequency hearing loss, the words key, pee, and tea recognition of the decibel (dB) scale. They might be sound the same. You find people with these hearing able to tell you that 0 dB is the threshold of hearing losses either lip reading or understanding a and that the sound on the runway next to an conversation by the context as well as the actual accelerating jet is about 140 dB. However, most recognition of words. Neil Bauman, a hearing expert people don’t realize that the decibel scale is a at , offers the following logarithmic scale. This means that for every increase information: of 10 dB the sound intensity increases by a factor of ten. So going from 60 dB to 70 dB is a ten-fold “Vowels are clustered around the increase, and 60 dB to 80 dB is a hundred-fold frequencies between about 300 and 750 Hz. increase. This is amazing to me. It means that we can Some consonants are very low frequency, hear sound intensities over 14 orders of magnitude. such as j, z, and v – at about 250 Hz. Others This means that the 140 dB jet on the runway has a such as k, t, f, s, and th are high frequency 14 14 loudness of 10 times greater than threshold. 10 is sounds and occur between 3,000 and 8,000 100,000,000,000,000 – that’s 100 trillion It means Hz. All of these consonants are voiceless ones our ears can measure loudness over a phenomenally and are produced by air hissing from around large range. Imagine having a measuring cup that the teeth. Most people, as they age, begin could accurately measure both a teaspoon and 100 losing their hearing at the highest frequencies trillion teaspoons (about 10 billion gallons). The ear first and progress downwards. Thus, the is an amazing receptor However, our perception is above consonant sounds are the first to be skewed a bit. A ten-fold increase in loudness doesn’t lost. As a result, it is most difficult to 10WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND sound ten times louder. It may only sound twice as means that if we defined the faintest sound as “1”, we loud. That’s why when cheering competitions are would have to use a scale that went up to done at school rallies, students are not very excited by 100,000,000,000,000 (about the loudest sounds you the measure of difference in loudness between a ever hear. The decibel scale is much more compact freshmen class (95 dB) and a senior class (105 dB). (0 dB – 140 dB for the same range) and it is more The difference is only 10 dB. It sounds perhaps twice closely linked to our ears’ perception of loudness. as loud, but it’s really 10 times louder. (Those lungs You can think of the sound intensity as a physical and confidence grow a lot in three years) measure of sound wave amplitude and sound intensity level as its psychological measure. The equation that relates sound intensity to sound Do you get it? (1.3) intensity level is: When a passenger at the airport moves from inside a waiting area to outside near an airplane the decibel level goes from 85 dB to 115 dB. By what factor has I 2 the sound intensity actually gone up? L = 10 log I 1 L ≡ The number of decibels I is greater than I 2 1 † I ≡ The higher sound intensity being compared 2 I ≡ The lower sound intensity being compared 1 2 Remember, I is measured in Watts /meter . It is like the raw power of the sound. The L in this equation is what the decibel difference is between these two. In normal use, I is the threshold of 1 † -12 2 hearing, 1¥10 Watts /meter . This means that the decibel difference is with respect to the faintest sound that can be heard. So when you hear that a busy The quality of sound or timbre is the intersection is 80 dB, or a whisper is 20 dB, or a class subtlest of all its descriptors. A trumpet and a violin † cheer is 105 dB, it always assumes that the could play exactly the same note, with the same pitch comparison is to the threshold of hearing, 0 dB. (“80 and loudness, and if your eyes were closed you could dB” means 80 dB greater than threshold). Don’t easily distinguish between the two. The difference in assume that 0 dB is no sound or total silence. This is the sounds has to do with their quality or timbre. simply the faintest possible sound a human with The existence of supplementary tones, combined with perfect hearing can hear. Table 1.1 provides decibel the basic tones, doesn’t change the basic pitch, but levels for common sounds. gives a special “flavor” to the sound being produced. If you make I twice as large as I , then 2 1 Sound quality is the characteristic that gives the DL 3dB. If you make I ten times as large as I , 2 1 identity to the sound being produced. then DL = 10dB . These are good reference numbers to tuck away: DETAILS ABOUT DECIBELS It was mentioned earlier that the sensitivity of † the human ear is phenomenally keen. The threshold † of hearing (what a young perfect ear could hear) is -12 2 1¥10 Watts /meter . This way of expressing sound Double Sound Intensity +3dB wave amplitude is referred to as Sound Intensity (I). It is not to be confused with Sound Intensity Level (L), measured in decibels (dB). The reason why 10 ¥ Sound Intensity = +10dB † loudness is routinely represented in decibels rather 2 than Watts /meter is primarily because the ears don’t hear linearly. That is, if the sound intensity doubles, it doesn’t sound twice as loud. It doesn’t Click here to listen to a sound intensity level reduced really sound twice as loud until the Sound Intensity is by 6 dB per step over ten steps. Click here to listen † † about ten times greater. (This is a very rough to a sound intensity level reduced by 3 dB per step approximation that depends on the frequency of the over ten steps. sound as well as the reference intensity.) If the sound intensity were used to measure loudness, the scale would have to span 14 orders of magnitude. That 11WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND Frequency Sound Sound Relative Sound Sound (Hz) Intensity Sound W Ê ˆ W Source of sound Intensity Intensity Intensity ( 2 ) Á ˜ m Level (dB) Intensity 2 Level (dB) Ë m ¯ -8 50 43 13,000 2.0 ¥10 -12 Threshold of hearing 0 1¥10 -9 100 30 625 1.0 ¥10 -10 Breathing 20 1¥10 -11 † 200 19 49 † 7.9 ¥10 -8 Whispering 40 -11 1¥10 500 11 8.1 1.3¥10 † -6 -11 Talking softly 60 1,000 10 6.3 † 1¥10 1.0 ¥10 † -12 -4 2,000 8 3.9 Loud conversation 80 6.3¥10 † 1¥10 † -12 3,000 3 1.3 -2 2.0 ¥10 Yelling 100 † 1¥10 † -12 4,000 2 1 1.6 ¥10 Loud Concert 120 † 1 † -12 5,000 7 3.1 5.0 ¥10 Jet takeoff 140 † 100 † -12 6,000 8 3.9 6.3¥10 † -11 † 7,000 11 8.1 1.3¥10 Table 1.1 Decibel levels for typical sounds † -10 † 8,000 20 62.5 1.0 ¥10 † -10 9,000 22 100 † 1.6 ¥10 FREQUENCY RESPONSE OVER THE † -9 14,000 31 810 1.3¥10 AUDIBLE RANGE † We hear lower frequencies as low pitches and † Table 1.2: Sound intensity and sound higher frequencies as high pitches. However, our † intensity level required to perceive sensitivity varies tremendously over the audible sounds at different † frequencies to be range. For example, a 50 Hz sound must be 43 dB equally loud. A comparison of relative before it is perceived to be as loud as a 4,000 Hz sound intensities arbitrarily assigns sound at 2 dB. (4,000 Hz is the approximate 4,000 Hz the value of 1. frequency of greatest sensitivity for humans with no hearing loss.) In this case, we require the 50 Hz sound to have 13,000 times the actual intensity of the Example 4,000 Hz sound in order to have the same perceived The muffler on a car rusts out and the decibel level intensity Table 1.2 illustrates this phenomenon of increases from 91 dB to 113 dB. How many times sound intensity level versus frequency. The last louder is the leaky muffler? column puts the relative intensity of 4,000 Hz arbitrarily at 1 for easy comparison with sensitivity at The “brute force” way to do this problem would be to other frequencies. start by using the decibel equation to calculate the If you are using the CD version of this sound intensity both before and after the muffler rusts curriculum you can try the following demonstration, out. Then you could calculate the ratio of the two. which illustrates the response of the human ear to It’s easier though to recognize that the decibel frequencies within the audible range Click here to difference is 22 dB and use that number in the decibel calibrate the sound on your computer and then click equation to find the ratio of the sound intensities here for the demonstration. directly: I 2 L = 10 log I 1 I I 2 2 22dB = 10 log fi 2.2 = log I I 1 1 † Notice I just dropped the dB unit. It’s not a real unit, just kind of a placeholder unit so that we don’t have † to say, “The one sound is 22 more than the other sound.” and have a strange feeling of “22 … what?” I 2.2 2.2 2 10 = fi I = 10 I fi I = 158I 2 1 2 1 I 1 So the muffler is actually 158 times louder than † before it rusted out. † 12WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND Do you get it? (1.4) Example A solo trumpet in an orchestra produces a sound If the sound intensity of a screaming baby were -2 intensity level of 84 dB. Fifteen more trumpets join W 1¥10 at 2.5 m away, what would it be at 2 m the first. How many decibels are produced? 6.0 m away? The distance from the source of sound is greater by a 6.0 factor of = 2.4 . So the sound intensity is decreased † 2.5 1 by = 0.174 . The new sound intensity is: 2 2.4 ( ) -3 W -2 W † 1.74 ¥10 1¥10 0.174 = ( ) 2 ( 2 ) m m † † † Another way to look at this is to first consider that the total power output of a source of sound is its 2 sound intensity in Watts /meter multiplied by the area of the sphere that the sound has reached. So, for Do you get it? (1.5) example, the baby in the problem above creates a What would it mean for a sound to have a sound -2 2 sound intensity of 1¥10 W /m at 2.5 m away. intensity level of -10 dB † This means that the total power put out by the baby is: † Power = Intensity ¥ sphere area Ê ˆ W 2 -2 fi P = 1¥10 4p 2.5m = 0.785 W Á ˜ ( ) 2 Ë m ¯ † Now let’s calculate the power output from the † information at 6.0 m away: Another factor that affects the intensity of the sound you hear is how close you are to the sound. Ê ˆ W -3 2 P = 1.74 ¥10 4p (6.0m) = .785 W Á ˜ Obviously a whisper, barely detected at one meter 2 Ë ¯ m could not be heard across a football field. Here’s the way to think about it. The power of a particular sound goes out in all directions. At a meter away It’s the same of course, because the power output from the source of sound, that power has to cover an † depends on the baby, not the position of the observer. 2 area equal to the area of a sphere (4πr ) with a radius This means we can always equate the power outputs 2 of one meter. That area is 4π m . At two meters away that are measured at different locations: the same power now covers an area of 2 2 4π(2 m) = 16π m , or four times as much area. At three meters away the same power now covers an area 2 2 2 2 P = P fi I 4pr = I 4pr ( ) ( ) of 4π(3 m) = 36π m , or nine times as much area. 1 2 1 ( 1 ) 2 ( 2 ) So compared to the intensity at one meter, the intensity at two meters will be only one-quarter as much and the intensity at three meters only one-ninth 2 2 fi I r = I r 1 1 2 2 as much. The sound intensity follows an inverse † square law, meaning that by whatever factor the distance from the source of sound changes, the intensity will change by the square of the reciprocal † of that factor. 13WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND Do you get it? (1.6) It’s a good idea to make sure that you keep the chronic sound you’re exposed to down under 80 dB. If you were working 1.0 m from a machine that created a sound intensity level of 92 dB, how far would you need move away to hear only 80 dB? (Hint: remember to compare sound intensities and not sound intensity levels.) 14WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND ACTIVITY ORCHESTRAL SOUND level at 10 m, we could start by finding the sound intensity at 10 m: 67W W I = = 0.056 . 2 2 m 4p 10m ( ) The sound intensity level would then be: † Ê ˆ W 0.056 Á ˜ 2 m L = 10 log = 107.5dB . Á ˜ W -12 Á ˜ 1¥10 2 Ë ¯ m You can hear plenty of sound in a concert hall Table 1.3 lists the sound intensity level of various where an orchestra is playing. Each instrument instruments in an orchestra as heard from 10 m away. vibrates in its own particular way, producing the † You can use these decibel levels to answer the unique sound associated with it. The acoustical power questions throughout this activity. coming from these instruments originates with the musician. It is the energy of a finger thumping on a Orchestral Sound Intensity piano key and the energy of the puff of air across the Instrument Level (dB) reed of the clarinet and the energy of the slam of cymbals against each other that causes the Violin (at its quietest) 34.8 instrument’s vibration. Most people are surprised to Clarinet 76.0 learn that only about 1% of the power put into the instrument by the musician actually leads to the Trumpet 83.9 sound wave coming from the instrument. But, as you know, the ear is a phenomenally sensitive receptor of Cymbals 98.8 acoustical power and needs very little power to be Bass drum (at its 103 stimulated to perception. Indeed, the entire orchestra loudest) playing at once would be loud to the ear, but actually generate less power than a 75-watt light bulb An Table 1.3: Sound intensity levels orchestra with 75 performers has an acoustic power of (measured at 10 m away) for various about 67 watts. To determine the sound intensity musical instruments 1. How many clarinets would it take to equal the acoustic power of a pair of cymbals? 15WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND 2. If you had to replace the total acoustic power of a bass drum with a single light bulb, what wattage would you choose? 3. How far would you have to be from the violin (when it’s at its quietest) in order to barely detect its sound? 4. If the sound emerges from the trumpet at 0.5 m from the trumpet player’s ear, how many decibels does he experience during his trumpet solo? 16WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND 5. If the orchestra conductor wanted to produce the sound intensity of the entire orchestra, but use only violins (at their quietest) to produce the sound, how many would need to be used? How about if it were to be done with bass drums (at their loudest? 6. The conductor has become concerned about the high decibel level and wants to make sure he does not experience more than 100 dB. How far away from the orchestra must he stand? 7. If he doesn’t use the one bass drum in the orchestra, how far away does he need to stand? 17WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND WAVE INTERFERENCE T’S INTRIGUING THAT at a lively party pulses move toward each other. In the second scene with everyone talking at once, you can hear the two pulses have reached the same spot in the the totality of the “noise” in the room and medium and the combined amplitude is just the sum then alternately distinguish and concentrate on of the two. In the last scene, the two wave pulses the conversation of one person. That is move away from each other, clearly unchanged by Ibecause of an interesting phenomenon of their meeting in the second scene. waves called superposition. Wave superposition When it comes to music, the idea of interference occurs when two or more waves move through the is exceptionally important. Musical sounds are often same space in a wave medium together. There are two constant frequencies held for a sustained period. Sound important aspects of this wave superposition. One is waves interfere in the same way other waves, but that each wave maint ains its own identity and is when the sound waves are musical sounds (sustained constant pitches), the resulting superposition can unaffected by any other waves in the same space . This sound either pleasant (consonant) or unpleasant is why you can pick out an individual conversation (dissonant). Musical scales consist of notes (pitches), among all the voices in the region of your ear. The which when played together, sound consonant. We’ll second aspect is that when two or more waves are in use the idea of sound wave interference when we the same medium, t he overall amplitude at any point begin to look for ways to avoid dissonance in the on the medium is simply the sum of the individual building of musical scales. wave amplitudes at that point . Figure 1.8 illustrates both of these aspects. In the top scene, two wave Figure 1.8: Wave superposition. Note in the middle drawing that the wave shape is simply the arithmetic sum of the amplitudes of each wave. Note also in the bottom drawing that the two waves have the same shape and amplitude as they had before encountering each other. 18WAVES AND SOUND WAVES AND SOUND WAVES AND SOUND ACTIVITY WAVE INTERFERENCE In each of the following two cases, the wave pulses are moving toward each other. Assume that each wave pulse moves one graph grid for each new graph. Draw the shape the medium would have in each of the blank graphs below. 19