Physics 1240: Sound and Music Scott Parker 1/31/06 Today: Sound sources, resonance, nature of sound waves (begin wave motion) Next Time: Wave motion Outline Last time: Sound sources (string, reed, brass, voice, flute-type) Resonance (vibrations) Nature of sound waves Bit depth, sampling rate CD quality is 16-bit bit depth with a 44,100 Hz sampling rate CD quality requires so much data per second: 44,100 samples/sec x 2 channels x 16 bits x 1 byte/8 bits = 176,400 bytes/sec Dynamic range is the range of sound levels possible. Nyquist frequency = sampling rate / 2 The Nyquist frequency is an upper bounds on the frequencies a given sample can resolve. Musical Sound Sources Bowed String Bowing a string involves a stick-slip action where most of the cycle the string is being pulled on by the horsehairs of the bow, then a shorter part of the cycle the string slips and moves freely. This stick-slip action resonates with the frequency of oscillation of the string. Once the string begins to slip it can move fairly freely until the bow motion begins again to move with the string at the same rate. This is due to the fact that dynamic friction is less than static friction. The wheel spinning phenomena (once your wheels start slipping you loose control). Plucked or hammered strings really require the understanding of waves propagation, so we ll wait until later to discuss plucked/hammered strings. Reed Instruments: (Clarinet, Oboe, Saxophone) The reed vibrates in resonance with the 1
natural modes of the attached tube. The vibrating reed causes pressure pulses at rate in resonance with the tube. When the reed swings open the pressure is a maximum and the source of the pressure is the instrumentalist blowing into the mouthpiece. Then the reed swings closed and the pressure is a minimum, again in resonance with the natural modes in the tube. If you disconnect the mouthpiece it vibrates at quite a higher frequency and the frequency is sensitive to embouchure pressure on the reed. When you connect it to the instrument, you lock in on particular frequencies and it is very clear there is feedback controlling the oscillations of the reed between the reed and resonant cavity. Brass Instruments: The buzzing lips in the mouthpiece plays the role of the reed. The lips are more massive than the reed and the instrumentalist has a little more control over what frequencies are excited. This can be shown by comparing what a brass player can do with his mouthpiece versus the control a woodwind player has over only the mouthpiece. When the lips separate, a puff of air or pressure maximum sent into the horn and when the lips close a pressure minimum. It is the feedback of these oscillations with the pressure anti-node that allows the horn player to play particular tones. The brass mouthpiece/lips, like the reed behaves acoustically like a closed end. A horn without a bell will have overtones which are odd integer multiples of the fundamental (pipe closed at one end). Vocal Folds: The vocal tract is an open cavity including the nasal cavity, mouth cavity, pharynx and larynx. It ends at the vocal folds, which are made of soft skin-like tissue. The voice is like other wind instruments. There is a sound generating device and a resonant cavity. Air is forced from the lungs up through the trachea. The air stream passes through the vocal cords, which vibrate. The sound generation mechanism for the human voice is very similar to brass player's lips buzzing in a mouthpiece. The vocal folds are about 2 cm long. Frequency depends on the tension supplied by small muscles attached to the vocal folds and supporting cartilage. The Frequency range of vocal cord vibrations is approximately 70-200 Hz for men and approximately 140-400 Hz for women. There is an important physical effect called the Bernoulli's effect, which says that when a fluid speeds up, the pressure drops. This makes sense since if a small volume of fluid increases in speed something must be pushing on it so there must be a force on the fluid element, hence a pressure drop. The Bernoulli effect is important in the vocal tract because air flows faster through the folds causing a pressure drop and hence a force pulling the folds together. When, the folds come together, the flow is blocked and the air pressure from the lungs then pushes them apart. This is a simple explanation for how the vocal folds vibrate (especially for large amplitude vibrations of the vocal folds). Flute-type sound source: A jet of air is unstable and wants to mix and equilibrate with the background are. Take, for example, air blown through a straw. The air jet flowing out of the straw will be unstable and will begin to wobble and form a wavelike structure moving with the flow of the jet stream. Eventually, swirls are generated, then turbulence, then farther away from the end of the straw (beginning of the jet stream) there is little left to tell there was a jet stream to begin with. These wobbles of the air stream air the origin of flute-type sound sources or so-called "edge tones". An air stream is blown at a sharp edge. This produces the sound. The stream oscillates above and below the edge. Flutes, 2
flue organ pipes, recorders, whistles, all work using the same principle. Like with the reed (and brass instruments), the oscillating air stream couples with the resonant modes in the tube. However, opposite to the reed, the edge tone does not create pressure minimums and maximums, but rather flow minimums and maximums. When the air jet flows into the pipe the flow is a maximum or inward, when the flow goes outside the pipe the flow in the pipe is a minimum or outward. Therefore, the edge tone behaves as an open end of a pipe. In reality, there is higher pressure near the edge tone source and tuning is necessary to make the flute harmonic. This is done by tapering the pipe slightly near the embouchure hole. This part of the flute is called the head joint. To a good first approximation, you can estimate frequencies of the overtones assuming a flute is a pipe open at both ends. Resonance Resonance is driving an oscillator at its natural frequency of vibration. The classic example is the swing set. You give the child riding the swing little pushes at exactly the right time and he goes a little higher each time. This is driving the swing/child system at it s natural frequency of oscillation. Which is to a very good approximation a simple pendulum. When we bow a string, we are driving the string at it s natural frequencies. The slip-grip mechanism gives very small nudges to the string, but eventually the string oscillates quite a bit. Likewise, when a trombonist buzzes into the trombone with his mouthpiece. The natural modes of vibration of the air column in the horn are driven at their natural frequency. Sound Waves Air is what is called a compressible fluid. It is spongy, you can smash it, then let it expand back. If you push on it, it decreases in volume. For example, when you pump air into your bicycle tire, air reduces in volume and increases in pressure. Alternately, if it is allowed to expand it will. E.g. when a balloon expands when connected to a high pressure source. Sound waves are small compressions and expansions (rarefactions) of air. Convincing you that sound is simply pressure disturbances propagating from a source to the listener will not be easy. 1) We cannot see the air around us, that is, the medium of sound waves. If we can't see the medium how are we going to see the waves which propagate within it? 2) Sound produces pressure fluctuations that are one-millionth of the background of the air pressure. What is pressure anyway? Pressure is the amount of force applied per unit area. p = F / A Newtons/m 2 or Pascals (Pa). 3
Background air pressure is 10 5 N/ m 2. The background air pressure is a lot. We are buried beneath a sea of air, the earth's atmosphere. Why is the background air pressure so huge? Well, you can think of earth as a giant spherical fish tank, and we have a 50 km or so of air on top of us pressing down. Air weighs something. A lot less than water, but the atmosphere is pretty high, so it weighs a lot! I've drawn a 1m x 1m square on the board for perspective. 4.5 N= 1 lb (you do not need to know this specific fact). If we took a 100,000 N weight (22,000 lb) and spread it uniformly of the 1m x 1m surface (say put it on a VERY STRONG piece of plywood). This would be 10 5 N/ m 2, or equivalent to atmospheric pressure. This is a heck of a lot of weight or force, and pressure! A 1000 N person (220 lbs) standing on this square piece of plywood would only exert 1000 N/m 2! Or, only 1% of atmospheric air pressure! It is important to understand that in p=f/a, the area plays an equal role. A woman can do a lot of damage to hardwood floor when wearing heels with a very small surface area, e.g. 2 cm 2 (0.0002m 2 ). Your eardrum is very small (0.3cm), so very small forces are involved in hearing (nature has miniaturized your ear's components to save space for other important things). Sound waves of musical interest have an amplitude of 0.01 to 1 N/ m 2. That is the pressure fluctuates between say: 100,000 + 1 N/m 2 to 100,000-1 N/m 2 We will talk more about the amplitude, intensity and level of sound waves, in a later chapter. However, this is a very small change in pressure. How many percent would this be? 0.001 % And, this would be for a loud sound! This is one of the reasons it is hard to observe sound waves directly. One important aspect of physical nature (or physics) is the similarities or strong analogies between vastly different phenomena. Wave motion is such an example. Sound waves, water waves, radio waves, light waves, they all follow very similar patterns and rules. A personal example in my own research in the area of plasma physics. Waves in space that accelerate particles to high energies and cause the aurora are virtually identical (at least from a theoretical point of view) as the waves that cause heat to escape in a magnetic bottle, and keep us from harnessing fusion energy, the energy generated by stars! 4
Experiment: Slinky waves We study waves on a coiled spring because we can more clearly see what is going on. The longitudinal (or compressional) waves we generate here, are very similar to sound waves. The pulse moves at a fixed speed (what we call propagates). Generally speaking, there are two types (or classes) of waves. 1) Transverse waves: motion of the medium is perpendicular to the direction of propagation of the wave. Light waves, radio waves, all electromagnetic waves, water waves, waves on a string. 2) Longitudinal waves: the motion of the medium is along the direction of propagation. Sound waves, earth quakes, all acoustic waves regardless of the medium (e.g. sound waves propagating through water, steel, etc.) Sound Speed Giving a sudden pulse on the slinky, is like hitting a drum or the bang of a fire cracker. Except for sound waves, the sound expands as a spherical shell. Really, it is much more complicated than that because neglecting there are floors and walls that reflect the sound. It would expand like a sphere if we climbed a tall pole and hit a drum, but that is not typical behavior. Anyway, the pulse propagates with a fixed speed, the speed of sound. v = 344 m/s = 770 mph All sound waves propagate at this speed regardless of pitch (or frequency). The finite speed of sound is very important in music. For example, speakers not aligned at the stage, even a few feet, will give the listener an uneasy feeling (in the high frequency range). Sound travels about 1 foot in a millisecond. The speed of sound increases slightly with temperature (0.6 m/s per degree Centigrade). 344 m/s is at 20 degrees C. Propagation of sound waves through liquids and solids is typically much faster, e.g. water v = 1500m/s steel: v = 6000 m/s The speed of an electronic signal is the speed of light or 3x10 8 m/s, which for audio purposes is instantaneous. v = d / t 5
Example problem: How long does it take a sound to travel 34.4 m? (approximately 113 ft, or roughly the distance from the chalk board to back of the class) t = d / v t = 34.4 / 344 = 0.1 sec 0.1 sec is very noticeable. 0.01 sec or 10 milliseconds is where delays become very noticeable. The is a simple fix for live performances, as done here, is to put the speaker system right on stage. Through experience, a listener expects a delay consistent with the distance from the source. Real problems occur when the direct sound from the source is delayed relative to a sound coming from a speaker. A good place to hear this problem is a sports arena, like the Pepsi Center. The speed of sound plays an important role in architectural acoustics and sound reenforcement. 6