Characteristics of sound:
 Sound is a longitudinal wave—it cannot travel through a vacuum; it consists of
 compressions and rarefactions; the louder the sound, the greater its amplitude
Speed of sound is temperature dependent at 0

C, speed of sound is 331.5 m/s rate of change of speed is [0.6 m/s]x[temp]
AP: if the speed of sound is not given, assume it is 345m/s.



Sound exhibits wave properties—it reflects producing an echo; it interferes constructively and destructively; it refracts or bends; it diffracts, or spreads around barriers
Since sound is a longitudinal wave, the particles of the medium are displaced parallel to the direction of the wave.
The speed of the sound wave is dependent upon the modulus of the material and the density of the material.
A police car is parked by the side of a highway, sounding its 1000 Hz siren. If you are also stationary, you hear 1000 Hz. If you are moving toward the police car, you hear a higher frequency. If you are moving away from the police car, you hear a lower frequency.
Doppler shift change in frequency of waves received by an observer whenever the wave source and/or the observer are in motion toward or away from one another. The Doppler shift was proposed in 1842. It was tested in 1845 by using a locomotive drawing an open car filled with several trumpeters.
"Universal" Doppler formula (one that can be used for any situation):

Where v is the speed of sound, v o
is the speed of the object, and v s
is the speed of the sound source. When the object is moving toward you, use the upper sign; when the object is moving away from you, use the lower sign.
Doppler Shift Physlet
Doppler Shift-Moving Point Source
An example of the Doppler shift
This is my favorite Doppler applet. I hope it opens for you because on some computers it will not. It has a moving source and a stationary detetector.
Simultaneously, it shows the approaching (and receding) wavefront, a picture of the wavelength showing perceived changes, and also plays the sound you would hear. Great Doppler Applet!
Want to see a picture of a real sonic boom? Sonic Boom Picture
Hint: when working Doppler shift problems, associate the word toward with a frequency increase and the words away from or recede with a frequency decrease .
AP Doppler formulas (1-3)
1.
Detector moving; source at rest
2.
where v d
is the velocity of the detector, v is the speed of sound, f ' is the detected frequency, and f is the original frequency.
Here the plus sign indicates that the detector is moving toward the source and the minus sign indicates that the detector is moving away from the source.
Source moving; detector at rest
3.
where v s
is the velocity of the source.
Source and detector both moving:

This can be used for all Doppler calculations. If the detector is stationary, v d
=0 and if the source is stationary, v s
=0.
Doppler problem solving strategy: Establish a coordinant system, decide which direction is positive, and make sure you know the signs of all relevant velocities. A velocity in the direction from the detector and toward the source is positive; a velocity in the opposite direction is negative.
Please note: because of limitations in how I had to construct the image for the "all-inone" Doppler formula, there is a slight error in the denominator. The negative sign should be on the top, rather than the positive sign. I was unable to find a corresponding image to use to make a -/+ and had to use the +/- instead.
The upper signs in the Doppler formula apply is source and/or observer move toward each other; the lower signs apply is they are moving apart. v d
is positive if the detector moves toward the source; if the source moves toward the detector, v s
is positive.
Hubble Picture from March 9, 2004 of the most distant galaxies found.
Terms: o speed
The speed of a wave is given by v =

f o pitch frequency o loudness amplitude o decibels unit for measuring sound level o timbre sound quality o beat what a listener hears when two sound waves of slightly different frequency are played

o resonance a vibrating object induces a vibration of the same frequency in another object
The most famous examples of resonance involve bridge collapses. In most physics textbooks, the Tacoma Narrows Bridge collapse is cited as an example of resonance. Some engineers dispute this. Video of Tacoma Narrows Bridge
Collapse.
Sound can be characterized by its frequency, its wavelength, its speed, and its intensity
(or loundness). Sound waves carry energy that can do work (example: a sonic boom can break windows).
Sound intensity (I)
The intensity of the sound is the power of the wave (or energy/sec) per unit of area (or one square meter. The power of the wave is the amount of energy transported per second. If the sound originates from a point source, you can think of this as a wave front passing through a sphere of area 4

r
2
Sound intensity depends upon distance; if the distance is doubled, sound intensity is reduced by a factor of 4 (This is only valid for a point source with no reflections.)
Intensity level (

) The units of the intensity level of sound are decibel, or dB, in honor of Alexander Graham Bell. Since the intensity level is based on a log scale, every change of 10 dB means that the sound is 10 times louder; a change of 20 dB means that the sound is 10
2
, or 100 times louder. The human ear is sensitve over the range of 0-120 dB. A whisper is 20 dB; a shout is 90 dB. The threshold of pain is 130 dB.

= (10 dB) log (I/I o
) where I o
is the threshold of sound
Threshold of sound The threshold of sound has the value of I o
= 1 x 10 -12 W/m 2
node region of zero displacement in a standing wave

antinode region of maximum displacement in a standing wave
Sources of musical sound: Most instruments involve more than a single vibrating body. For example, in a violin, both the strings and the violin body vibrate. o o o vibrating strings (guitar, piano, violin) vibrating membranes (drums) vibrating air columns (flute, oboe, organ) o vibrating steel bars (xylophone)
12.
strings produce transverse waves; sound is produced as string compresses and rarefacts air law of strings:



 frequency is increased as string length is decreased frequency is increased as string diameter is decreased frequency is increased as string tension is increased frequency is increased as string density is decreased in a standing wave on a string, each segment is ½ 
13.
Pipes produce standing waves
 closed pipes — an antinode is always at an open end and a node is
 always at a closed end open pipes — an antinode is at each open end
Instruments produce standing waves. In any instrument, several harmonics are excited at the same time and the resultant sound is the superposition of these components. fundamental (1 st
harmonic): o string, length = ½  o closed pipe, length = ¼


o open pipe, length = ½ 
2 nd
harmonic: o string, length =
 o open pipe, length =

3 rd
harmonic: o string, length = 3/2
 o closed pipe, length = ¾  o open pipe, length = 3/2

Here is a trick to remember: Draw the desired harmonic for the string, open pipe, or closed pipe. Determine how much of a wavelength is represented. Set this equal to the length of the pipe and solve for the wavelength. In the pictures above of the harmonics, if it looks like a "v" it is equal to 1/4

. If it looks like two "v's" stuck together to form a closed object (a segment), it is equal to 1/2

.
Notice: There are no even-numbered harmonics in a closed pipe. A closed pipe only produces odd harmonics. In strings and open pipes,


=(n v)/2 l, where n=1, 2, 3, ... In closed pipes,

=(n v)/4 l, where n=1, 3, 5, ...
Where l is the length of the pipe.
In music, harmonics are called overtones.
Beats Suppose two sounds with frequencies very close to one another are played simultaneously. We hear an average of the two sounds. The sound is modulated by a slow, wobbling beat note whose frequency is the difference between the two sound frequencies, or beats. For example, when a 552 Hz and a 564 Hz tone are played simultaneously, we hear 564-552, or 12 beats per second. The beat frequency is 12 Hz.
Beats - you actually HEAR them!
Sound on the AP exam : o o o o o
Typically on multiple choice questions. There are few free response questions that deal with sound/waves.
For a vibrating string, you might be asked to predict how frequency changes if tension is changed.
You might be given a drawing that shows a moving source producing a wave train. They may ask you about the relative speed and direction of movement of the source. They also might ask you to predict what relative frequency an observer detects. You might be asked to predict what factors affect the frequency detected by the observer.
You might be asked to compare characteristics of sound and light waves.
Free response questions - open and closed pipe calculations where you calculate the wavelength, the speed of sound, and predict resonance frequencies.
Interference of Sound Waves
Two speakers which emit identical sinusoidal waves of identical frequencies are another example of sound wave interference phenomena. Suppose the speakers are separated by distance d .
A microphone is placed equidistant from both speakers, on a line perpendicular to the line connecting the speakers as shown below.

Wave crests emitted from the two speakers travel equal distances to arrive at the microphone and thus arrive at the microphone at the same time. According to the principle of superposition, the amplitudes of the two waves add, resulting in constructive interference. If the microphone is moved to another position, destructive interference occurs where the wave from one speaker travels a halfwavelength farther than the wave from the other speaker. According to superposition, the amplitudes of the two waves subtract. o o o
You might be asked to calculate the minimum frequency where destructive interference can occur. Remember - destructive interference occurs every 1/2 wavelength. Thus, the minimum frequency would occur when d = 1/2

. Knowing v=
 f, the speed of sound and d can be used to calculate this minimum frequency.
You might be asked to graph how intensity varies with horizontal distance.
Remember, intensity follows an inverse square relationship.
You might be asked to graph how intensity varies with vertical distance.
Remember, this looks like double slit diffraction pattern. At the midpoint, the intensity is the greatest. As you move outwards vertically, a minimum next occurs. As you continue to move out vertically, another maximum occurs, but it will not be as intense as the first one. This is followed by another minimum and so forth.
Sound Sample Problems
1.
A student shouts across a canyon and hears his echo 4 sec later. If the speed of sound is 343 m/s, how wide is the canyon?
2.
A 440 Hz frequency sound is heard. What is its wavelength is the temperature is
20

C? If it is -10

C?
3.
A train blows a 400 Hz horn as it approaches an intersection at 36 m/s. What frequency is heard by a stationary observer? What is heard by the stationary observer if it now is receding?
4.
A 5000 Hz sound wave is directed toward a an object moving at 3.5 m/s toward the stationary source. What is the frequency detected by the moving object?

5.
What is the intensity level of a sound whose intensity is 2.0 x 10
-6
W/m
2
?
6.
A stone is dropped in a mine shaft 15 m deep. The speed of sound is 343 m/s.
How long does it take to hear the echo?
7.
A closed tube resonates in its fundamental. What is the length of the tube if the wavelength is 78 cm? What would be its length if it were an open tube?
8.
A guitar string 1.2 m long vibrates with a wave speed of 720 m/s. What is the frequency of the first three harmonics?
9.
An open pipe 0.5 m long is used to produce the first three harmonics. What are their frequencies if the speed of sound is 340 m/s?
10.
Repeat number seven using a closed pipe 0.3 m long. Remember, the first three harmonics are the fundamental, third harmonic, and the fifth harmonic.
11.
A student uses a water-filled tube 100 cm high to determine the speed of sound. A fundamental is found to occur when the water level is 83 cm high and a 512 Hz tuning fork is used to produce the sound. What is the experimentally determined speed of sound?
If no information is given, assume the speed of sound to be 345 m/s.
1.
An explosion occurs at a distance of 6 km from a person on a day when the temperature is 14

C. How long after the explosion did the person hear it? Ans:
17.65 sec
2.
A car is moving at 20 m/s along a straight road with its 500 Hz horn sounding.
You are standing at the side of the road. What frequency do you hear as the car is approaching? Ans: 530.77 Hz
3.
A car is moving at 20 m/s along a straight road with its 500 Hz horn sounding.
You are standing at the side of the road. What frequency do you hear as the car is receding from you? Ans: 472.60 Hz
4.
When a car approaches, sounding its horn, a stationary observer detects a frequency of 550 Hz. When it is receding from you, a stationary observer detects a frequency of 500 Hz. What is the speed of the car? Ans: 16.43 m/s
5.
The bellow of a territorial hippopotamus has been measured at 115 dB avove the threshold of hearing. What is the sound intensity? Ans: 0.32 W/m
2
6.
The sound intensity level of a jet engine is 138 dB while at a rock concert it is 115 dB. Find the ratio of the sound intensity of the jet engine to the sound intensity at the rock concert. Ans: 199.53:1
7.
A cliff is 12 m high. You drop a stone from the top of the cliff. How long does it take for you to hear the sound of its "thud?" Ans: 1.60 sec
8.
A student uses a water-filled tube to determine the speed of sound. The tube is 1 m tall. He detects a resonance position when the water level is 55 cm tall that corresponds to the third harmonic when a 575 Hz tuning fork is used. What is the wavelength of the sound wave? What speed of sound did he experimentally determine? Ans: 0.60 m; 345 m/s

9.
An open pipe is used to determine the speed of sound to be 335 m/s using a 324 Hz tuning fork. What is the wavelength of the sound wave? If this is the position of the third harmonic, how long is the pipe? Ans: 1.034 m; 1.551 m
10.
Repeat for a closed pipe. Ans: 1.034 m;
0.7755 m
11.
A wire 30 m long supports a radio antenna tower. The mass of the wire is 90 kg and its fundamental vibrational frequency is 20 vibrations in 10 sec. What is the speed of the wave on the wire? What is the tension in the wire? Ans: 120 m/s; 43200 N
12.
The elastic modulus of a substance is 1.63 x 10
8
and its density is 880 kg/m
3
.
What is the speed of sound in this medium? Ans: 430.38 m/s
Sound is perceived when fluctuations in air pressure cause structures inside our ears to vibrate. These air pressure fluctuations can be quite small or large and can occur slowly or rapidly. We refer to the rate at which pressure fluctuates cyclically from higher to lower to higher and so forth as its frequency. Typically we express frequency in cycles per second or equivalently Hertz. The following figure is a graph of two "cycles" of fluctuation. This figure shows the Amplitude of air pressure variations relative to mean air pressure (in no particular units) as a function of
Time (expressed in milliseconds or thousandths of a second). Thus, 0 on the Pressure scale corresponds to the mean air pressure. In this figure the pressure starts at the average air pressure, increases to a value of 100 at a time corresponding to about 1.25 msec, decreases to -100 at 3.75 msec and returns to zero at 5.0 msec before starting the second cycle.
The length of each cycle in time is called the period of the waveform because the shape of the waveform repeats periodically at this interval.

Since the period of this waveform is 5.0 msec, there would be 200 periods or cycles in one second. The frequency of this sound is thus 200 cycles per second or 200 Hertz (which we will abbreviate as Hz hereafter). More generally, the frequency of a periodic waveform is the inverse of its period; F = 1/P or in this example, 200 = 1.0 / 0.005. If you would like to hear what this 200 Hz waveform sounds like, click on the graph with your mouse or pointer.
In addition to the frequency of a sound, we can describe its amplitude . In general, small variations in pressure produce weak (or quiet) sounds while large variations produce strong (or loud) sounds. The next figure shows another sound which is lower in amplitude that the previous example because the pressure varies less extremely over time. This figure shows a sound which also differs in frequency from the sound illustrated in the previous figure. Note that frequency and amplitude vary independently. Although the amplitude is lower in this figure, the pressure fluctuations are more rapid than in the previous figure; six cycles occur within ten msec so this tone has a frequency of 600 Hz. Consequently, this sound is higher in frequency but lower in amplitude than the sound depicted in the first figure.
One other property called phase is important in describing the physical properties of sound. To illustrate what is meant by phase, the next figure shows two 200 Hz sinusoids, one drawn with a solid line and the other drawn with a dotted line. The two sinusoids are identical except that they are differently aligned with respect to the time axis. These two sinusoids are said to differ in phase while having the same amplitude and frequency. This is a good moment to point out that the notion of `beginning' and `ending' needs some qualification here. The figures drawn on this page have waveforms which obviously begin and end within the limits of the graph. However, they represent snippets of functions which do not have beginning and ending points. Thus, the phase differences shown in the present figure do not reflect the notion that one function started at a different time than the other. Rather, the phase differences represent the way the two

functions are aligned with respect to each other at all times, including those which lie outside the bounds of the present graph.
The physical properties of amplitude, and frequency correspond to the sensory/perceptual qualities of loudness and pitch . It is often useful to maintain a clear distinction between the physical properties of sound and the perceptual correlates of those properties. For one thing, the perceptual domains of pitch and loudness are bounded by the capabilities of our auditory systems whereas the physical properties of sound are not. The normal young human auditory system is sensitive to a range of frequencies from about 20 Hz to about
20,000 Hz. The amplitude range is substantially broader, beginning at a level so low that we can almost "hear" the fluctuations in air pressure due to random motion of air molecules near the ear drum and extending to the threshold of pain at about 10 million times that level.
A second important difference between the perceptual properties of sound and its physical properties is that even within the bounds of the perceptual system, the relationship between the perceived and physical properties of sound is generally nonlinear. For example, if we increase the amplitude of a sound in a series of equal steps, the loudness of the sound will increase in steps which seem successively smaller. Similarly, increasing the frequency of a sound in equal steps will lead to perceived increases in pitch that seem to grow smaller with each step. Here's an example. Click on any of the following numbers to hear a tone of the corresponding frequency. Note that as you go through these tones in 25 Hz steps, the steps sound like they are getting closer together.
For instance, compare the step between 200 and 225 Hz with the step from 400 to 425
Hz. The step from 200 to 225 sounds larger than the step between 400 and 425 Hz even thought both are exactly 25 Hz.
200Hz 225Hz 250Hz 275Hz 300Hz 325Hz 350Hz 375Hz 400Hz 425Hz 450Hz 475Hz
Whole Series
We often describe sounds using scales that reflect equal perceptual differences. For frequency, one such scale is the Mel scale. Equal Mel steps will correspond to equal changes in pitch, but not equal changes in frequency. Similarly, for loudness, it is most convenient to describe sound over the enormous range of perceptible amplitudes in logarithmic units called Decibels and abbreviated dB . On the decibel scale, 0.0 dB corresponds to about the normal threshold of hearing and 130 dB to the point at which sound becomes painful. Moreover, each 1 dB step corresponds to approximately a Just
Noticeable Difference in loudness, that is, the smallest change in loudness that is noticeable about 50% of the time.
The third physical property of sound, its phase is less directly related to perceived sound quality. In most work related to speech perception, phase is entirely disregarded.
However, phase is important in describing how complex sounds can be constructed from the simple sinusoidal sounds we've discussed so far.

Despite their differences in amplitude and frequency, the sounds shown and heard above depict simple sounds because the pressure fluctuations associated with these sounds are sinusoidal. That is, the pressure variations over time follow the form of a trigonometric sine or cosine function. Most sounds in nature are not so simply described; their shape, rather than being sinusoidal, is of some other form, typically one for which we have no name. Fortunately, it turns out that such complex sounds can be described mathematically as combinations of simple sounds.
Consider for example, the sound illustrated in the next figure which simply alternates between a region of constant high pressure and a region of constant low pressure. This particular waveform does have a name, it is called a square wave because of its boxy shape. This square wave is very similar to the 200 Hz sine wave shown in the first figure in that it too repeats a single pattern two hundred time a second. Moreover, (if you haven't already listened to it, you should now) it has the same pitch as the 200 Hz sine wave, but a different timbre.
This complex square wave can be described as the summation of a set of simple sinusoids. In other words, the square wave can be formed by adding together sinusoids of the appropriate amplitude, frequency, and phase. Not surprisingly, the first and strongest sinusoid needed to form the square wave in our example is a sine wave of 200 Hz. This first component corresponds to what is called the Fundamental Frequency (hereafter abbreviated as F0 ) and is the frequency which gives rise to the pitch we normally hear when listening to a complex sound. Thus, the common F0 accounts for the pitch similarity between the 200 Hz sine wave and the 200 Hz square wave. To construct a square wave we need, in addition to a 200 Hz sine wave, a sequence of higher frequency sine wave components. The components in this sequence are called overtones or harmonics , and by definition, can only occur at integer multiples of F0. Since F0 in this example is 200 Hz, the harmonics can only occur at 400 Hz, 600 Hz, 800 Hz, and so forth. However, the square wave is a special case in that all of the even-multiple harmonics (i.e., the ones at 2F0, 4F0, 6F0, etc.) have zero amplitude so they contribute nothing to the shape of the square wave.
Using only the odd-numbered harmonics then, we can construct a square wave by adding sine waves at F0, 3F0, 5F0, and so forth. For our example 200 Hz square wave, this means we need sine waves at 200 Hz, 600 Hz, 1000 Hz, 1400 Hz, and on. In addition to

having harmonics of the correct frequencies, the amplitude relations among the harmonics must be correct or we will not construct the desired waveform.
For a square wave, the 3rd harmonic (at 600 Hz) should be 1/3 the amplitude of the fundamental. This is exactly the sinusoid shown in the second figure above. Here is the waveform that results from adding a 200 Hz sine wave with a 600 Hz sine wave at 1/3 the amplitude. Already, the combined waveform is beginning to take on some features of the square wave with a more extended portion near its most positive and negative values (albeit still with much fluctuation).
Continuing to build a square wave by adding sinusoids, the third component needed (the fifth harmonic at 1000 Hz) should be 1/5 the amplitude of the fundamental, and the fourth sine wave, corresponding to the 7th harmonic (at 1400 Hz) should be 1/7 the amplitude of the fundamental. These are shown in the next set of figures along with the square wave approximations when we sum all harmonics up to and including the given harmonic.
As you can see, with the addition of each subsequent harmonic, the complex waveform more nearly approaches the shape of a square wave. The addition of each higher frequency harmonic reduces the amplitude (and increases the frequency) of the small ripples in the more stationary parts of the square wave. To achieve the shape of a true square wave with absolutely no ripple in the stationary parts would require the summation of an infinite number of sinusoids. But we don't have time for that in the present tutorial.
So far all the discussion of sound has centered on its description as fluctuations in air pressure over time. The representation of sound in the time domain is important to understand, but in some ways it is also awkward. For instance, the frequency of a sound is one of its most important physical properties, but determining frequency from a waveform requires making measurements of time intervals and then doing arithmetic.
Indeed, for many complex waveforms, where multiple sinusoids of various frequencies are simultaneously present, it is often unclear where the

intervals to be measured begin and end. The frequency domain provides an alternative description of sound in which the time axis is replaced by a frequency axis. In the frequency domain, sounds are represented in a frequency by amplitude and/or phase diagram.
The next figure is a frequency domain representation of the 200 Hz sine wave we saw in the first figure. In the frequency domain, this sound is represented by a line at a point on the frequency axis corresponding to 200 Hz and with a length corresponding to its amplitude. Figures like this are called line spectra .
There are several things to note in this figure. First, the Y axis is labeled Amplitude rather than pressure because the axis now provides a measure of the strength of the pressure changes: neither absolute pressure, nor the direction of relative pressure change is represented. In fact, pressure need not be the physical measure on which amplitude is based here. With sound, we often measure the voltage fluctuations produced by a microphone rather than pressure per se . Consequently, amplitude is a better, more general, term. Second, note that the Amplitude axis has no values less that zero. In this spectral representation, called a magnitude spectrum amplitudes cannot be less than zero-
-it is not possible to have negative amounts of sound energy. A third feature to note is the labeling of the Frequency axis which is in units of Kilohertz or thousands of cycles per second.
One of the most convenient features of frequency domain representations of sound is that sounds of many different frequencies can be plotted simultaneously on the same figure.
This figure, for instance, shows all of the components we used above to start an approximation to a square wave. In this figure, each line is one of the harmonics of the
200 Hz fundamental frequency of the square wave. The height of each harmonic line indicates the amplitude of the sinusoid at that frequency. This figure does not show us anything about the phase relationships among the harmonics which were obvious in the time-domain figures earlier. Try clicking on each line in the line spectrum; if you're

click, you should hear a sine wave at the appropriate frequency and amplitude. Next click in the figure but not on one the spectral lines; you should of hear the complex sound which results from summation of the four spectral components in the figure. See if by listening carefully you can hear any of the individual tones in the complex sound. car efu l wh ere you
Notice that the amplitude reduces very quickly with each successive harmonic in this spectrum. In fact, the apparent differences in amplitude are actually much larger than the differences we would hear when listening to each of these tones. In this next figure, amplitude is expressed in dB rather than in linear units. The amplitude relations among the harmonics expressed in dB are much closer to the loudness relations we hear among the harmonics. This figure doesn't play any tones: they'd be exactly the same as the last figure--only the scaling of the Amplitude axis is different--that's the point.
Line spectra exactly represent periodic signals like sine waves and square waves, but these are a special case in that sounds we encounter in nature are never truly periodic.
First, most sounds we encounter are bounded in time and/or may be periodic only within certain temporal bounds. Further, many important sounds like the voiced sounds of speech are only approximately periodic since they vary slightly from one period to the next. We refer to these sounds as quasiperiodic . Let's take another look at the spectrum of the four-harmonic square wave approximation we've been using, but this time treating it in the way sounds are most often actually handled for study in the laboratory. First, because we are normally interested in looking at the spectrum of a sound at a particular point in time, we will apply what's called an analysis window to the sound. This makes the sound fade in and back out again gradually. When we first window the sound and then determine its frequency components, we get this kind of a figure. The axes and frequency scale are the same as the previous figure, but the amplitude scale is different in this figure. Previously, the amplitude scale was set to arbitrary units, but now, amplitude is based on the units used in the digitized and windowed sound.
The most important (and probably most obvious) difference between this figure and the last however is that the harmonic lines now look like pointed bars. These are still called harmonics, but they no longer represent pure tones, instead, they represent the presence of sound energy at many frequencies quite close to the true harmonic frequencies. If you listen to the sounds underlying this figure by clicking on the harmonics or outside the

harmonics to hear the complex tone, you'll hear the way the tones fade in and out rather than starting and ending abruptly. We call spectra like this harmonic spectra rather than line spectra.
The difference between line spectra and the broader bars of harmonic spectra illustrates an important general difference between sounds represented in the time domain and in the frequency domain. Sounds which extend for long times and with great consistency in the time domain have very narrow profiles in the frequency domain. A sinusoid extending forever at a fixed frequency has the narrowest possible profile (a line) in the frequency domain. On the other hand, sounds which are narrowly defined in time, that is, have a brief temporal extent, exhibit a broader frequency profile. Thus, sinusoids which fade in and out as in the last example, have a broader distribution in frequency.
Carrying this trend to its logical conclusion, the shortest possible sound (a single pressure spike; like a hand clap but even shorter in duration) would have the broadest possible frequency profile. In fact, a pure impulse sound (i.e., a sound that is of zero amplitude at all times except for one infinitesimal instant when its amplitude is non-zero) would spread out in frequency to the point of having a perfectly flat spectrum. Of course, this would no longer be called a harmonic spectrum, it would be a continuous spectrum.
Continuous spectra are associated with sounds that are not periodic, that is, with aperiodic sounds. An impulse is the paradigm exemplar of an aperiodic sound, but other more commonly encountered aperiodic sounds are the hissing sounds of fricatives in speech, and generally any sounds which do not have an identifiably tonal quality.
To summarize, we have discussed three kinds of spectra:
Line Spectra
Associated with strictly periodic signals or sounds that are (at least theoretically) unbounded in time.
Harmonic Spectra
Associated with quasiperiodic sounds or signals that are bounded in time.
Continuous Spectra
Associated with aperiodic sounds.
Before finishing with this discussion of sound represented in the frequency domain, let's look at two more spectra. These are associated with actual speech sounds.

Time signal or waveform
The description of a sound in the time domain as fluctuations in some physical property like pressure over time. Often, because the pressure fluctuations have been transduced by a microphone or other measurement instrument, we have converted pressure fluctuations to voltage fluctuations over time.
Fourier transform or spectrum
The description of a sound in the frequency-domain as the amplitude or extent of fluctuation occurring at different frequencies.
Line Spectrum
The kind of spectrum that is found for sounds that are purely periodic, that is, for sounds that repeat the same pattern infinitely. Each line in a line spectrum is a harmonic of the fundamental period of the waveform and represents a sinusoid at a particular frequency and amplitude. Line spectra are the ideal case of harmonic spectra.
Harmonic Spectrum
Similar to a line spectrum except that sounds giving rise to harmonic spectra are not purely periodic, but only approximately so.
Such sounds produce an harmonic spectrum in which the lines have some discernible width. As sounds deviate increasingly from true periodicity, their spectra deviate increasingly from line spectra to approach a continuous spectrum. For example, any sound that has finite duration is not strictly periodic. Many natural sounds, like the human voice, are quasi-periodic in that the sound deviates in a variety of ways from one period to the next.
Continuous Spectrum
A spectrum exhibiting non-zero amplitude for one or more broad regions of the continuous frequency spectrum. This is the kind of spectrum that is found for aperiodic sounds, that is, sounds that do not repeat any pattern at all. The ``ideal'' aperiodic sound is an

impulse, that is, a sound consisting of a single instantaneous pressure spike. The impulse is a sound which has equal amplitude at all frequencies.
Period or T0
The duration of a single complete cycle of a periodic waveform. We sometimes notate the period of a signal as T0.
Fundamental Frequency or F0
The fundamental frequency is 1.0/T0, that is, the inverse of the period. Normally, we express F0 in units of cycles per second or Hz.
This can be slightly confusing since we often express T0 in units of msec. You must remember to multiply T0 by 1000.0 if it is expressed in msec to arrive at F0 expressed in Hz. For complex sounds, F0 will normally be the frequency of the first, or lowest frequency harmonic.
Harmonic
A line (or near-line) in the spectrum of a periodic (or nearperiodic) signal that can occur at any integer multiple of the fundamental frequency. In a harmonic spectrum, the harmonics are
Pitch spaced F0-Hz apart.
The perceptual correlate of frequency. Normally, the pitch of a complex sound is a function of its F0. Equal steps in pitch are roughly equal to logarithmic steps in frequency.
Loudness
The perceptual correlate of amplitude. Equal steps in loudness are roughly equal to logarithmic steps in amplitude.
Decibel (dB)
A logarithmic scale of amplitude which is roughly associated with our perception of loudness. Zero Decibels is near the threshold for hearing and each Decibel increment in amplitude is roughly one Just
Noticeable Difference in loudness. The formula for computing decibels is:
Decibels = 20.0 * log(Amplitude/Reference)

where Reference is generally something like the smallest perceptible amplitude fluctuation.
Hertz (Hz)
Frequency expressed in cycles per second.
Mel Scale
A logarithmic scale of frequency based on human pitch perception.
Equal intervals in Mel units correspond to equal pitch intervals.
Bark Scale
A logarithmic scale of frequency based on human frequency resolution. Sounds which are separated by more than about one
Bark unit are generally resolvable as separate sounds and do not interact with each other at a sensory level.