One of the properties of matter is that is can support transfers of mechanical energy, whether the matter is solid, liquid, or gas, without any net movement of the molecules involved. Such transfers are referred to as "wave motion", or a transitory displacement of atoms within the matter that passes on kinetic energy.
This chapter discusses the physics of mechanical wave motion. Light also involves wave motion, but not of a mechanical sort, and will be discussed in the following chapter.
[6.1] WAVE MOTION / STANDING WAVES
[6.2] ACOUSTICS & MUSICAL SOUND / SONAR
[6.3] REFLECTION, REFRACTION, & DIFFRACTION / STANDING WAVES
[6.4] THE DOPPLER SHIFT
[6.5] WAVE INTERFERENCE
[6.6] MECHANICAL RESONANCE
There are two kinds of waves: "compression" or "longitudinal" waves, and "transverse" waves. In a compression wave, the molecules move back and forth in the direction of the wave motion. Sound is a compression wave. A more easily visualized compression wave can be produced by taking a "Slinky" toy, holding it in both hands, and shaking it with a side-to-side motion.
In a transverse wave, the molecules move back and forth at a right angle to the direction of motion. A rope being shaken with an up-and-down motion generates a transverse wave. Waves on water or other fluids, interestingly, have both compression and transverse components, with the molecules of the fluid oscillating in a circular pattern.
In reality, the speed of sound is far from constant. For one thing, an increase in temperature increases the speed of sound. For example, at 20 degrees Celsius, the speed of sound increases to 344 meters per second or 1,238 KPH. At higher temperatures, the molecules are moving more quickly and can transmit sound energy more quickly.
Moisture in the air also increases the speed of sound, since the water molecules are on the average lighter than the air molecules they replace, and lighter molecules have less inertia and move more quickly. In addition, the speed of sound also slows with altitude, since the air is less dense and the molecules cannot transmit energy to each other as quickly. The speed varies inversely to the square root of the density.
In general terms, then, the speed of sound:
- Increases with temperature.
- Decreases as the size of the molecules involved grows greater.
- Increases with density.
Sounds generally move much faster in dense liquids and solids than in gases. However, in solids sounds are also dependent on the elasticity of the material, increasing with elasticity.
The speed of sound in water is about 1,525 meters per second or 5,490 KPH (3,409 MPH) at normal temperatures, increasing greatly with temperature. Of course, sound is a compression wave, and this is not the same as the velocity of the transverse surface waves that travel across bodies of water, which surfers can "ride" for their amusement. This would not be fun for most people if the speed of such waves was any real percentage of 5,490 KPH.
The speed of sound in copper is about 3,353 meters per second or 12,070 KPH (7,497 MPH) at normal temperatures, but oddly decreases with temperature, because the metal's elasticity falls off.
Waves can have different spacings, or "wavelengths", between successive
regions of maximum density (for a compression wave) or peaks (of a transverse
wave). The wavelength is related to the "frequency" of the wave, or the
number of times it oscillates in a second while passing through a fixed
location, by the simple relationship:
frequency = velocity / wavelengthFor example, for a sound wave propagating at 344 meters per second and a wavelength of ten centimeters, the frequency is 3,440 "cycles per second" or "hertz (Hz)". For a wavelength of 20 centimeters, the frequency falls to 1,720 Hz.
Waves also have a "period", which is the time between passages of consecutive
crests troughs of the wave. The period is given by the inverse of the
period = 1 / frequency = wavelength / velocityFinally, waves have a "phase", in that the peaks of one wave may not coincide in time with the peaks of another wave of the same frequency. Suppose you have two wave tanks, side by side, and generate waves of the same frequency in both. At first the two wave generators are set up to operate in perfect synchronization, but then one is momentarily paused or delayed so that the peaks of its waves don't match up with the other. The two waves are now "out of phase".
Phase is measured in degrees. When the two waves are in phase, they have a "phase difference" of 0 degrees. As one of the generators is delayed so that the phase falls behind that of the other, the phase difference increases. When the two waves are completely out of step or "antiphase", with the peaks of one coinciding with the troughs of the other, the phase difference is 180 degrees. If the generators is delayed again, its wave output will eventually come back into phase with the other, and the "phase shift" will have gone through a full cycle of 360 degrees.
Please remember that phase is a comparative concept. One wave by itself doesn't really have a phase, but multiple waves being compared, or a single wave compared to a specific instant in time, does have a phase.
Given a sine wave of a given amplitude, velocity, and period, then that wave
can be represented by the formula:
amplitude * SIN( 360 * time / period )-- or equivalently:
amplitude * SIN( 360 * time * frequency ) amplitude * SIN( 360 * time * velocity / wavelength )
amplitude * SIN( 360 * time / period + phase )
The sounds we hear can be characterized by three parameters: the loudness of the sound, its "pitch", and its "quality". The loudness of the sound obviously depends on the level of energy being pumped into the sound waves, but interestingly our perception of loudness is roughly proportional to the cube root of the energy of that sound. In simpler terms, to make a sound seem twice as loud, we have to pump eight times the energy into it.
The loudness of sounds is measured in a logarithmic scale known as "decibels", defined as ten times the exponent of 10 for the loudness value. The threshold of hearing is defined as 0 decibels. 10 decibels is ten times as loud, while 20 decibels is 100 times as loud, and 30 decibels is 1,000 times as loud. Typical sounds of traffic are about 70 decibels, while the noise of a nearby jetliner is about 140 decibels.
Pitch is simply the frequency of the sound. Somebody who talks in a deep gravel voice has a low-pitched voice, while most children tend to have high-pitched voices. The concept of pitch is of course fundamental to music, with musical "notes" of given frequencies arranged on a musical "scale".
The classical scheme defining a musical scale involves dividing the range of
musical tones into "octaves", named because each octave contains the eight
whole notes "A", "B", "C", "D", "E", "F", "G", and "A", centered on the
"Middle C" note at 264 hertz. Each "octave" of eight notes below Middle C
halves the frequency, while each octave above doubles it, as follows:
_______________________ -C''' 33 hertz -C'' 66 hertz -C' 132 hertz MIDDLE C: 264 hertz +C' 528 hertz +C'' 1,056 hertz +C''' 2,112 hertz +C'''' 4,224 hertz _______________________There are different ways to define the tones in each octave. The most common, for Western music at least, is the "equitonic" scale. Examination of a piano keyboard shows that there are twelve keys in each octave, with seven white keys defining the "whole" notes and five black keys defining "sharp notes" and "flat notes". In the equitonic scale, the ratio from one of the keys to the next is the same for each key.
Since an octave means a doubling of frequencies, that means that this
interval is the twelfth root of 2, or about 1.05946. The frequency values
are rounded off to integers in practice. The following table gives the tones
for the first octave above Middle C, with a "#" indicating a sharp note and a
"-" indicating a flat note:
______________________ C 262 hertz C# / D- 277 hertz D 294 hertz D# / E- 311 hertz E 330 hertz F 349 hertz F# / G- 370 hertz G 392 hertz G# / A- 415 hertz A 440 hertz A# / B- 466 hertz B 494 hertz C' 524 hertz ______________________Other scales, such as the more traditional "diatonic" scale, do not use equal intervals, but this makes playing music in different keys more troublesome.
The quality of a sound describes the kind of sound we hear, regardless of tone. Most people can distinguish a note played from a piano from a note played from a flute, even if they're the same tone. This intuitively obvious, but not so easy to define in detail.
One of the important aspects of musical quality is the fact that, in general, a musical tone will not consist of a single frequency. This will be true for a simple electronic tone, but if you press the Middle C key on a piano, it will not only produce a sound with a frequency of 262 hertz, it will also simultaneously produce a range of other sounds of lesser amplitudes and higher frequencies, mostly integer multiples such 524 hertz and 1,048 hertz. These different sounds are the "components" of the actual tone.
The basic and most significant component of the tone, in this case 262 hertz, is called the "fundamental", while the component at the higher frequencies are called "harmonics". The set of these sounds is referred to as an "audio spectrum".
Different instruments generate different audio spectra for the same tone, which is one reason they sound distinct. There are other features to the quality of a musical tone, but further discussion would be leaving the domain of physics and crossing over into musical theory.
Some animals can hear well above the human range of hearing. For example, bats, whales, and dolphins have a hearing range several times that of humans. These animals use their vocalizations to perform ranging and location. They emit high-pitched sounds and then listen for echoes bounced off a target. The longer the interval between producing the sound and the return of its echo, the farther the target, and these animals also have ear structures that allow them to determine the direction of the echo.
Humans developed an electronic approach to the same procedure after the First World War to hunt submarines, ultimately naming the scheme "sonar", for "sound navigation and ranging".
This is the simplest scenario of a reflection, and in the real world it's a bit more complicated. First, if the flat wall is at an angle to the direction of the propagation of the sound, it will be reflected away at the same angle relative to the wall.
For example, if the wall is at an angle of 45 degrees to the direction of propagation of the sound wave, the reflection will be at an angle of 45 degrees relative to the wall, or 90 degrees relative to the original direction of the sound. If the wall is made up of many different elements at many different angles relative to the propagation of the sound wave, it will generate reflections in many different directions, scattering the sound wave.
Not all the energy of the sound wave will be reflected from the wall. Some of it will be absorbed by the wall as well. This combination of reflection and transmission will actually happen any time a sound wave makes a transition from one distinct acoustic medium to another, or in other words propagates through a "discontinuity".
This means that the direction of propagation of the wave in the second medium will be shifted from that of the first. This property is known as "refraction".
Neither refraction nor diffraction are all that important in the study of sound waves, but they are of great interest in the study of light waves, as will be discussed in the next chapter.
You can time your shakes to synchronize in phase with the reflections so that fixed sections of the rope oscillate up and down, with the boundaries between the sections remaining effectively stationary. This is known as a "standing wave", as opposed to "traveling waves", or waves in normal unconstrained propagation.
Of course, the number of half-wavelengths on the rope have to be an integer value, since otherwise there would be no synchronization between the input waves and the reflected waves. The stationary points on the wave are referred to as "nodes".
A plucked string on a musical instrument, such as a guitar, sets up standing waves. Similarly, a police whistle generates standing waves with the wavelength given by the size of the whistle -- the smaller the whistle, the shorter the wavelength and the sharper the tone. The whistle acts as a "resonant chamber" that selectively produces that tone from the air blown through it by the policeman. A slide whistle allows the player to vary the tone continuously by moving a rod connected to a plug that varies the size of the resonant chamber.
The most popular example of the Doppler shift is the sound of a moving train. If a train approaching at high speed blows a whistle, the frequency of the whistle is higher than it would be if the train were at rest. Similarly, if the train is moving away, the frequency of the whistle is lower.
This is easy to see mathematically. Suppose we designate the speed of sound as S and the velocity of an approaching train as V. Suppose the train is emitting a very short sharp blast on its whistle on an interval given by time T.
If the approaching train emits a blast at time 0, then by the time it emits its second blast at time T, it will have moved closer by the distance V * T, and the sound of the second blast will not have to travel as far as the sound of the first.
The amount of decrease in the measured blast period will be given by the ratio of the train's velocity to the speed of sound. For example, suppose the blast period T is one second. If the train's velocity V is a tenth of the speed of sound, then in one second it will travel a tenth as far as sound does in one second, and the measured blast period Tm will be reduced to 0.9 seconds.
In more general terms:
Tm = T * ( 1 - V/S )In sum, the blast period observed from a moving train to the blast period observed from the stationary train is then:
Tm = T * ( 1 - fraction_soundspeed )
Tm = T * ( 1 + fraction_soundspeed )
This analysis applies to continuous waveforms as well. Suppose the train was also emitting a low-frequency audio signal in the form of a sine wave, and the blast was synchronized so that it was emitted at the beginning of each sine cycle. Clearly the two signals will remain in lockstep, and the sine wave will be Doppler-shifted exactly as the blast is.
An interesting example of interference at work is to produce two sound waves of frequencies that are almost, but not quite, the same. As the two sounds move in and out of phase, they will eerily grow louder and quieter on a regular cycle. This is known as a "beat", though it has little to do with the "beats" that time musical compositions.
Suppose Dexter generates two sounds, one at a frequency of 300 hertz and the other at a frequency of 302 hertz. This means that they will both complete an integer number of cycles -- 150 and 151 respectively -- every half a second, and so the sound will vary in volume through a full cycle twice a second. The actual frequency of the sound heard will be 301 hertz.
Wave interference is the basis of "active noise cancellation" systems. These systems include three elements: a microphone, a digital processor, and loudspeakers. The microphone samples external sounds and feeds them to the processor, which then calculates how to drive an antiphase signal to the loudspeakers that cancels out the external sound. Of course, this works best when the sounds are predictable, for example the constant rumble of a large ventilation fan.
Now suppose that the boat is moving through the water. It then sets up a continuous series waves that propagate circularly from every point that the boat moves through. If the boat is moving slower than the speed of the waves, this creates a Doppler shift, with the wavelength compressed in front of the boat and expanded behind the boat.
The interesting thing is what happens when the boat exceeds the speed of the waves in water. All the waves in the front of the boat merge together, leaving behind a high-amplitude, vee-shaped bow shock wave behind the boat. Water skiers are very familiar with this bow shock wave, since they like to hop over it or even surf on it for a while. The angle of the vee grows narrower as the speed of the boat increases.
Supersonic aircraft also produce a bow shock wave that results in the propagation of an abrupt high-pressure wave in a conical footprint under the aircraft's flight path, causing a "sonic boom" that can rattle windows. Such sonic booms were common in the late 1950s and the 1960s, before the complaints of the citizenry forced the military to impose more stringent flight rules on their pilots.
This seems puzzling at first, since it leads to the question: which is for real? The spectrum of component sounds? Or the single messy-looking sound waveform? The answer is: they both are. The tone could be interpreted in either way depending on which was more convenient to the task at hand.
A branch of applied mathematics known as "Fourier analysis" has been developed to translate between the two aspects of the sound, as well as other, similar wave phenomena. The sound can be inspected either in the "frequency domain", in the form of a spectrum of simple sine waves, or in the "time domain", in the form of a messy single waveform. Further discussion of Fourier analysis is far beyond the scope of this document.
The most vivid example is that of a tuning fork. If it is struck with a mallet, it vibrates at a certain frequency, and this frequency rises as the arms of the tuning fork become shorter, since then they can vibrate more rapidly.
Another example of mechanical resonance that was of great practical use for centuries is the "pendulum", the swinging weight on a rod that times a grandfather clock. The weight on the end of the pendulum swings back and forth, with maximum kinetic energy and speed at the bottom of its arc and maximum potential energy as it momentarily halts and reverses direction at the ends of its arc.
Although a detailed analysis of pendulum motion is beyond the scope of this
document, very interestingly in principle the period of a pendulum only
depends on the acceleration of gravity and the length of the rod as follows:
period = 2PI * SQRT ( acceleration_of_gravity / length_of_rod )This neat property allowed them to count the ticks for a clock. As there are always some losses, the clock included a spring-driven mechanism to provide a little force that kept the pendulum in motion. As the length of the rod tended to change with temperature from summer to winter, sophisticated clocks often had rods assembled from segments of dissimilar metals so their length would remain more constant.
If a pendulum is made of a weight hung from a wire so that it can swing back and forth in any direction in the horizontal plane, then if the pendulum is placed at the North or South Pole, its direction of motion will remain unchanged as the Earth spins beneath it, or from an observer's point of view the direction of its swing will pivot around in a full circle every 24 hours.
If such a "Focault pendulum" is taken to lower latitudes, the time required for it to shift its direction of swing will increase. At the equator, its direction of swing will not change at all, because then the change in the Earth's direction is in the pendulum's vertical plane, not its horizontal plane. Large Focault pendulums are often displayed in science museums, at least at high latitudes. However, this is drifting away from the topic of mechanical resonance.
A pendulum can also be constructed as a weight attached to a spring, bobbing up and down. As the motion of this pendulum tends to run down quickly, this scheme is not used much or at all in clocks or other machines, but the mathematical analysis of such a system is straightforward, and it is used as a tool in teaching the theory behind mechanical resonance.
As an idealized spring has no losses, a third element called a "dashpot" is added to incorporate losses into the system. Analysis of such a "spring-mass-dashpot" system is another thing beyond the scope of this document, though it will be quickly encountered by readers who go on to college-level basic physics or engineering courses.
Skyscrapers tend to sway slightly in high winds, acting as inverted pendulums fixed to the ground, and will oscillate back and forth with a resonant period of several seconds. Under certain wind conditions, such oscillation could be a threat to the building, since the motions could build up, like a kid "pumping up" a playground swing, until they bring the house down.
The most famous example of destructive structural resonance occurred on 7 November 1940. The Tacoma Narrows Bridge, a new 850 meter (2,800 foot) long, slender suspension bridge in the northwest US state of Washington, proved to oscillate badly in crosswinds and was nicknamed "Galloping Gertie". When the structure was hit with a steady 68 KPH (42 MPH) wind on that day, it began to oscillate at its resonant frequency, the wind twisting the roadbed one way and then the other as it turned up and down, and collapsed a little over an hour later. Nobody was hurt, but in the aftermath of the accident some other suspension bridges were reinforced to damp out such oscillations.
The Citicorp skyscraper in New York City, mentioned in an earlier chapter, incorporates an "oscillation damper" at its top to reduce its sway and damp out oscillation. The damper consists of a 365 tonne (400 ton) concrete slab that is moved back and forth under control of motion sensors attached to the the building's structure. If the building begins to sway one way, the slab is shifted in the other direction to compensate.
Implementing the damper was cheaper than adding wind bracing to the building's structure. It is also possible in principle to build dampers using water tanks, with large tanks at the four sides of top of the building and cross-connected by pipes at the bottoms of the tanks. Under normal circumstances, the water level will remain the same in all tanks, but if the building sways, inertia will cause the water to slosh across the pipes to oppose the sway. The tank system is design so that its own resonant frequency is the same as that of the building, with the antiphase resonances of the building and the tank system tending to cancel each other out.