Notes on: The Physics of Sound

Sound is the physiological and psychological response to pressure variations in the air. Such pressure variations are typically caused by the vibration of an elastic body such as a stereo speaker, a violin, or the vocal chords–to cite just a few possibilities. To understand how sound is created and detected, it is useful to begin with the simplest vibrating system–a mass attached to a spring. A typical setup is shown in the figure to the right. The point "O" is a reference point and the position of the bottom of the mass is reckoned relative to "O."

When the mass just hangs at rest from the spring it is said to be at its "equilibrium position." By displacing the mass either up or down and letting go, we can produce a motion that repeats again and again (up and down). This motion is called harmonic oscillation. A graph of position of the mass versus time looks like the following.

This rhythmically undulating set of data traces out what is called a sine curve. The time necessary for this motion to exactly repeat is called the period of the oscillation. The units of period are (usually) seconds. For the data shown, the period is about 1.1 seconds. The reciprocal of the period is called the frequency of the oscillation. The units of frequency are cycles per second, also known as hertz (Hz). The frequency for the data shown is about 1/1.1 Hz = 0.9 Hz. The maximum distance the mass gets from the equilibrium position is called the amplitude of the oscillation. The units of amplitude are (usually) meters. The amplitude for the data shown is about 0.03 m. A remarkable thing about masses on springs is that the period (or, equivalently, the frequency) doesn’t depend (very much) on the amplitude. That’s also true for lots of other vibrating systems–such as a pendulum (recall Galileo in church), for example. The frequency does depend on the stiffness of the spring and how much mass is attached, however. The stiffer the spring the higher the frequency; the heavier the mass the lower the frequency.

For the data on the graph friction isn’t very important, because the amplitude remains the same for the two plus cycles shown. If we had a much longer data set we would observe the amplitude getting smaller and smaller, because friction converts the oscillations into heat (in the air). The following figure is an example in which friction is substantially greater than for the oscillations in the previous figure. The actual data are the diamonds. An oscillation of the same frequency and the same starting amplitude but without friction is shown also for comparison.

A damped oscillator (one in which friction is important) can be "pumped up" (just like a child pumps a swing), provided it is vibrated by an outside agent of some kind. The outside driver puts energy back into the oscillator. For the undamped oscillator you can have about any old amplitude you like. But, that’s not true for the driven, damped oscillator. The size of the amplitude of the driven, damped oscillator is fixed by how large the driving force is, what the driving frequency is, and how much friction there is. If the rate of replacement of energy is "just right," the amplitude of the driven, damped oscillator can become quite large. A large amplitude response at just the right frequency of energy replacement is called resonance. A typical resonance response graph is depicted in the following figure. The vertical axis is the amplitude you when the oscillator is damped and driven. The horizontal axis is the driving frequency. Don’t worry about the units of either, this is just an illustrative sketch of what can happen.

The curve with the sharp peak shows a typical resonance for low friction. The flatter curve shows what happens to resonance when there’s lots of friction: resonance gets squashed and spread out. In other words, increasing friction makes the resonance response much less dramatic. The frequency for which the amplitude is a maximum is called the natural or resonant frequency.

If you hang two masses from two springs, there are two motions that show resonance. One is when the masses move together in the same direction, the other when the two masses move in opposite directions ("beat against each other"). The former has a lower resonant frequency than the latter. If you hang three masses from three springs, there are three resonant frequencies–the lowest corresponding to all masses moving together in the same direction. In general, if there are N masses and N springs, there are N resonant frequencies, with the lowest corresponding to all N masses moving together in the same direction.

Any macroscopic body, such as a string or a tube of air, is really made up of tiny masses called atoms. The atoms of any body interact with each other as if they were attached by tiny atomic springs. Thus, any body will have MANY resonant frequencies (because a typical number of atoms in a macroscopic body is about 10²⁵ or so–that’s a 1 followed by 25 zeroes!). The lowest frequency resonance (the "fundamental") occurs when all atoms are moving together in a synchronized fashion, as in the top part of the figure to the right. The next lowest frequency (the "first overtone") occurs when half of the atoms on one side of the body are moving opposite to the half on the other side, as in the bottom part of the figure. The figure shows a snapshot at one instant. The material in the string is represented by the dots. For comparison purposes, horizontal lines have been added to indicate the equilibrium position of the string material in each case. As time goes on, the string material vibrates up and down, except at special places where the string never moves, called nodes. Because the ends of the string are fixed, the ends are always nodes. The fundamental vibration has no intermediate nodes. Higher frequency vibrations (the overtones) do have intermediate nodes–one for each level of overtone. (The first overtone has one intermediate node, the second has two, the third has three, etc.)

Resonant vibrations in an extended body are also said to be standing waves. A (harmonic) wave is a disturbance in something (the ocean, a guitar string, the air, ... ) that has the shape of a sine curve. This sine curve pattern changes orientation as time goes on. That is, parts of it that start out as "up" become "down," and vice versa. The rearrangement of this pattern repeats over and over. A wave is therefore a space-time structure–it varies periodically in space as well as in time. The distance one has to travel along the wave at a fixed time for the pattern to exactly repeat is called the wavelength of the wave. Wavelength is measured (usually) in meters. The time one has to wait at a fixed point in space for the pattern to exactly repeat is called the period of the wave. The reciprocal of period is frequency. Frequency is measured in cycles per second or hertz. The product of wavelength times frequency is called the wave speed, measured in meters per second. If the nodes of a wave move the whole pattern moves, and the wave is called a traveling wave. The speed with which the nodes move is the wave speed. A wave whose nodes don’t move is called a standing wave. Two such waves are depicted in the previous figure. Even though a standing wave doesn’t travel anywhere, the relation between wavelength, frequency, and wave speed is still the same. Wave speed in an elastic body is determined by the density of mass and the force between pieces of the body. For a string, the more tightly strung it is the faster the wave speed. In general, the thicker the string the slower the wave speed. Wave speed in air is determined by the makeup of air (nitrogen, principally, and oxygen) and the forces the air molecules exert on each other. Though, the wave speed in air can change a bit with temperature and humidity, it is roughly 350 meters per second.

For the standing waves shown in the previous figure, both have the same wave speed (assuming it is the same string in both cases). Since the wavelength of the first overtone is half a big as the wavelength of the fundamental (you have to imagine the wave of the fundamental as if it continued beyond one of the ends–the length of the string equals half a wavelength for the fundamental and a whole wavelength for the first overtone), the frequency of the first overtone has to be twice as large as the frequency of the fundamental. In that case, the overtone is called the second harmonic–"second" because its frequency is two times the fundamental. If you bow a string whose fundamental frequency is 400 Hz at just the right rate you can create an overtone with frequency 800 Hz. A string with two ends fixed allows all harmonics. The second overtone corresponds to the third harmonic, the third overtone to the fourth harmonic, and so on. (Recall the string demonstration I did in class.)

Tubes of air also permit standing waves. Some tubes are open at both ends (woodwind instruments, for example), some closed at both ends (certain bizarre percussion instruments), and some open at one end and closed at the other (organ pipes, pop bottles, the auditory canal in your ear). Open-open and closed-closed tubes are like strings. They permit all harmonics. Open-closed tubes, however, only permit odd harmonics. In all cases–including strings and vibrating membranes–the fundamental has a wavelength that is comp[arable to the size of the tube, string, or membrane. "Big vibrating body" means "long fundamental wavelength," and that, in turn, means "low frequency."

Sound is typically generated by exciting standing waves in a body (vocal chords, musical instruments, membrane of a speaker, and so on)–the transmitter–and detected by exciting standing waves in the auditory canal of the ear or in the membrane of a microphone–the receiver. The sound emitted by the transmitter is carried by a traveling wave through the air to the receiver. The situation looks like the following cartoon.

The pattern of gray and light areas represents a traveling wave moving through the air. The gray regions might be regions of higher density and higher air pressure, while the light areas might be regions of lower density and lower air pressure. These regions travel through the air at about 350 m/s. The reason standing waves are used in the transmitter and the receiver is that they give large amplitude vibrations. For the transmission part that means large amplitude traveling waves are emitted. For the receiving part that means a large signal is received.

The receiver of your ear is a complex system that involves a small tube that is effectively "open" at both ends. "Open" for tubes means that the air molecules at the end can undergo large amplitude motion. They can’t if the end is a rigid wall, but they can if the end is a flexible membrane, for example. That’s exactly what happens in the ear. The auditory canal is capped at one end by the "ear drum"–the so-called timpanic membrane. The timpanic membrane is flexible and can be driven into fairly large vibrations. Thus, the auditory canal is an open-open tube. Because it is open at both ends, the fundamental standing wave in the auditory canal has a wavelength twice as long as the canal. The canal is about 5 cm long, so the fundamental wavelength in this tube is about 10 cm, 0.1 m. If we divide 0.1 m into 350 m/s, we get a fundamental frequency of about 3500 Hz. You would expect the best hearing to occur at about that frequency, and that is indeed true. The auditory response is not extremely sharp, however. The response is more like the "gooey" flat response curve in resonance graph (three figures up), than the sharply peaked one. That’s good, because it allows us to hear many frequencies besides the fundamental. In fact, normal hearing ranges from about 20 Hz to perhaps 12,000 to 15,000 Hz.

The auditory system is very complicated and exactly how it works isn’t yet unraveled. The background pressure of the air on every surface exposed to it is about 15 pounds per square inch. A square foot of a roof has about 15 pounds per square inch x 144 square inches per square foot = 2,160 pounds pushing down on it from the air. Fortunately, most of the time the same air pressure pushes back so the roof stays intact. Tornadoes wreak havoc because the air pressure in one might drop very rapidly by about 5% or so. If the air inside a building can’t change pressure as rapidly as the outside then there will be more than 100 pounds per square foot pushing out, and the whole building can essentially explode. Remarkably, in normal conversation the change in air pressure associated with a sound wave is about 1 part in 10,000,000. That is, the ear can detect exceedingly small pressure variations in the air–provided they occur at the right frequencies. The human auditory system perceives loudness in a very nonlinear fashion. In particular, if a sound intensity (energy of the sound) is increased by a factor of 10 we perceive the loudness to increase by a factor of 2. This is an amazing result: it says that if the energy in a sound wave is increased by a factor of 10 billion, we perceive the loudness as increasing by only a factor of 1000! (The range of energies we can perceive from the least audible sound to just before pain and auditory damage sets in is about a 10 billion folding.)