Waves
Simple harmonic motion can also be used as an analogy to another topic in physics, which is the subject of waves. Waves travel through space, carrying some kind of information and energy- be it sound waves, light waves, and so on. As the waves travel, they can affect the environment around them, and- depending on the type of wave- can cause the medium they are traveling through to move or react in other ways.
To start with, we need to define what kinds of waves we are dealing with. There are transverse waves, and longitudinal waves. A visual analogy, courtesy of Pearson Prentice Hall, can be seen below:
Transverse waves are waves that travel, with the amplitude perpendicular to the direction of travel. A longitudinal wave, however, has its amplitude in line with the direction of travel. While the above diagram uses slinkies, in reality, the slinky is just a stand-in for whatever the medium is.
Examples of transverse waves include things like: seismic, or secondary, earthquake waves; light and water waves; and "the wave" that is so popular at sports stadiums, while examples of
longitudinal waves include: sound waves; primary earthquake waves; and other types of pressure waves.
It is important to note, for now, that things like sound need a medium to travel through, whereas light can transmit through a vacuum. We will explain why in future units.
Vibrating strings are typically discussed as transverse waves, and sound- what we will be discussing- are modeled as longitudinal waves.
It's important to note that as long as there is no loss of energy, the amplitude of the incoming and outgoing waves is the same, in this case! The only differences between before and after are which way the wave is moving, and whether the wave is on "top" or "bottom."
The wave approaches its strongly-fixed boundary (the wall), after leaving the wall, it has reversed its direction of motion, and is now upside-down! These are called reflections (moving in the opposite direction) and inversions (flipping). This diagram will always describe how a wave interacting with a rigid, fixed boundary, like a wall, will change the motion of a wave. A loosely-fixed string, however, simply reflects; it does not invert. This can be seen below:
It's important to note that as long as there is no loss of energy, the amplitude of the incoming and outgoing waves is the same, in this case, as well!
right-side-up. If you were to sum the energy from both waves, you would find that the total energy of both waves matches the energy from the original wave. We can see a visualization of this in this diagram:
Compare the amplitudes of the waves traveling to the right. Notice how the wave moving through the more-dense medium has had its amplitude decreases substantially. This is because it is harder to cause the thicker medium to vibrate, so the wave's amplitude must decrease
compared to its original size. The wave encountering the thicker medium behaves partly as if it is held tightly, but because the denser medium is free to move, some of the wave's energy is
transmitted.
For a wave on a thicker string crossing to a thinner string, there is reflection and transmission; neither wave is inverted, as shown in this diagram:
Compare the amplitudes of the waves traveling to the right. Notice how the wave moving
its original size. In effect, it is as though the less dense is acting as a "loose attachment" for the more dense medium. So while there is a reflection and some transmission, there is no inversion.
This is just restating what we found when the wave went from a less dense to a more dense medium, just in the reverse direction.
Interference
Now, we can consider what happens when two waves being sent through the same medium encounter one another. Suppose we set up two traveling through a medium, but have one going to the left, and the other to the right. What would the result be? We can study the left side of this figure, which shows the two waves meeting over time.
As we can see, the waves meet, and right at the moment they overlap, the wave shoots up. In fact, the amplitude at the third step is equal to the sums of the amplitudes of both waves. This is called constructive interference, where the waves add together.
The right side of the figure, then, shows two waves traveling towards each other as before, but this time, the second wave is on the opposite side of the medium. When they meet, the combined wave is much shorter than the figure on the left. This, then, is a result of destructive interference, when the waves cancel out some of each other's amplitude when meeting. The amplitude here is equal to the difference of the waves' amplitudes.
do this as long as there is no friction or other damping effects. If we refine the wave so that the length of the wave becomes twice as long as the length of the string, then the outgoing and reflected waves will add constructively in a special way: we get a standing wave. A standing wave is defined to be when two waves move through a medium with the same frequency in opposite directions and interfere with one another. A couple of examples of these waves can be seen in this figure below.
There are a few labels here which we will concern ourselves with later, but we can explain some of these now. The dots I have labeled indicate either nodes (black dots) or antinodes (gray dots). The nodes are where the amplitude of the standing wave reaches zero, and the antinodes are where the amplitude of the standing wave reaches its maximum. The white dot signifies the antinode location for the wave that reaches all the way across the string, while the gray dots signify the antinodes for the wave that has a full wavelength all the way across the string (the inner "bulbs" compared to the outer "melon"). The m here stands for a harmonic, which we will discuss shortly in our "physics of music" section.
In principle, it is possible to have as many nodes packed in this space as possible. However, it becomes more and more difficult to set up a standing wave with the corresponding frequency. In practical terms, the standing waves with fewer nodes/antinodes tend to be the "dominant"
frequency, or the frequency that largely determines what musical note is heard (for example).
We can begin to notice or convince ourselves of some patterns:
First, the number of nodes in a standing wave is always one greater than the number of antinodes in a wave.
And second, for a string (or pipe closed at both ends), like that pictured above, the standing waves will only occur at specific wavelengths or frequencies. The first wavelength is, as described earlier, . The next, also pictured, has a length such that: . The next, where there is one and one half wavelength moving across the length of the string, has a length:
As we know, because sound travels at a constant speed in a given medium, and because a wave's wavelength is inversely related to that same wave's frequency, this gives rise to the harmonics that musicians can sometimes become very familiar with. We will take a closer look at those, and other pattern relationships, later on in the sound unit.
Beats and Resonance
Having discussed wave properties and their superposition, it is important to discuss what happens when two waves interact with one another in special ways.
Suppose we were to strike two tuning forks. One tuning fork may have a frequency of vibration of , while the other may have a frequency of . If we were to listen to them at the same time, what would we hear? The difference in their frequencies is ; we would hear a slow "waaa-waaa-waaa-waaa", with each "waaa" happening over the course of one second. This is an example of what musicians call dissonance, which is very unpleasant to listen to. Musicians tune their instruments so that when they play the same note, they sound the same- that is, the waves do not interfere with one another and form beats. The concertmaster's (usually 1st Violin, Flute, Clarinet, or Oboe) entire job is to supply the tuning note from which all other members tune, so that these beats are not present. When beats are eliminated, we get an effect called consonance.
What happens, though, when those waves do add, and an additional force is supplied? We can get a special effect called resonance. Imagine a child swinging on a swing. If we ignore friction, she should be able to swing back and forth with no extra effort needed. If we want to increase her amplitude, however, we must supply a force. But when should we supply that force? Practical experience might tell you that at the bottom of the swing is not a good time; she would crash into you, and you'd both get hurt, and you'd stop the oscillation. But, suppose we exert a force at the top of her swing. Now, each push adds energy, which increases her amplitude. In principle, if the chains of the swing are rigid, one could continue doing this- pushing on her at precisely the end of her oscillation- and get her to swing all the way around the pole.
In doing this, we are getting the child and swing to resonate. By continuing to exert a force at a frequency that matches the swing's oscillation, we increase the amplitude. For a child, that could mean getting her to clear the cossbar.
For a bridge- as in the case of the Tacoma Narrows Bridge Disaster- this would mean utter destruction and catastrophic material failure.
This bridge was constructed to span a gap of water, but engineers did not take into account the effects of vortex shedding and wind interference in the construction of their bridge. The rate at which the wind pushed on the bridge at certain times would cause the bridge to sway very dramatically, especially when the wind approached a frequency equal to the bridge's natural resonance frequency.
replacement bridge was built, taking great care not to repeat the mistake of forgetting these resonance effects.
In another (possibly mythic) example, Nikola Tesla- the man responsible for many technologies, chiefly the alternatic current we use to power most everything we use in our homes- built an earthquake machine. He calculated what the resonance frequency of Earth would be, and built a hand-held device that could vibrate a plate at that frequency. As the story goes, he applied this machine to the ground, and became so unnerved when the building began to vibrate that he dismantled his device and scrapped his plan for the device, such was his anxiety.
Sound
Understanding waves and how they behave lays the groundwork for understanding the next units, which involve how sound and light waves move and interact with different medium. For now, we will concern ourselves with sound waves, and leave light for its own unit.
Now, we can look briefly at this diagram to help us understand the way sound waves travel better:
On the left, we have an example of one-dimensional waves- these form "planes", hence we sometimes call them "plane waves." We can think of these as water waves on the beach- they have spread out so far from their original source that they are basically moving in straight, paralell lines. The spacing would indicate the waves' wavelength.
On the right, we have a source represented by the black dot, and sound spreads out from the source. As with the case on the left, the spacing in between the waves indicate the wavelength of the waves. Many times, we simplify the sound as spreading out in a circle, but in reality, it spreads out in something more closely resembling a sphere. As the energy contained within the sound waves spreads out over a larger and larger area, so the amplitude of the wave diminishes.
air, for example, the speed of sound is , while in water or metal, the speed jumps to about or , respectively. We can calculate a wave's speed by combining two
measurements: one of the distance between peaks of a wave's oscillation, and one of how often a wave repeats in one second. A more concise term for each of these phrases would be
"wavelength" and "frequency."
As long as the wave stays in one medium, its speed will remain constant. And, knowing that its speed will remain constant, we can always find a wave's frequency and wavelength (provided we know one or the other):
Now, we can always qualitatively tell the intensity of sound- stand next to speakers at a rock concert, and you're bound to hear the phrase "That's so intense!" at some point in the night. In a similar way, some things can be intensely quiet. Neither of these measures are mathematical, but we can actually quantify these intensities. First, we start with how powerful the source is, and allow the sound to spread out, either at the edge of a circle or on the edge of a sphere. For short distances, either will suffice; for longer distances, the spherical form will yield better results, but requires remembering the expression for the surface area of a sphere.
As sound spreads out from a source, it creates a pressure wave, vibrating the air molecules. How much the air compresses can be measured and quantified by introducing a unit called a decibel. The definition of a decibel would be a unit of measure that expresses the relative intensity of one sound compared to another.
The mathematical definition of a decibel (dB) is:
Where we have to define , which we refer to as the "threshold of hearing,” or
the absolute softest a sound can be and still be heard. This threshold is “soft,” meaning that each individual person may have a slightly different threshold. The true purpose of this number, in reality, is that we want our scale to have a mark, but having would give an undefined expression. Instead, we can establish this number as a baseline, and any sound with this intensity will result in an intensity of , as .
engine from about ten meters away. 130 dB is the limit at which, without ear protection, the listener experiences pain. And so on!
To find the change in the decibel level between one sound to another- say, for instance,
comparing the amount of noise in a room with a loudspeaker working, to one without- we use the following formula:
And using a property of logarithms:
This discussion leads to a very important and relevant point. We are probably all familiar with the common earbud, made popular for listening to music and other audio/visual files on the go, without having to carry large, obnoxious headphones with us. These earbuds are rated to handle sounds of very high intensity, mostly due to efforts to block out ambient conversation. However, studies have shown that many young adults turn the music in their earbuds up to about 80-100 dB, which is in the range that hearing loss due to extended hearing begins to occur. When doctors, parents, and others tell you that you are damaging your hearing when they can hear the music from your earbuds, they are not lying- this is physics!
The Doppler Effect
Now, the intensity of a sound can change depending on your position, as we may have already experienced; it is much easier to understand someone when you are closer to them. However, if you are continuously changing your position (as in, if you are moving), or if the source of the sound is continuously changing its position (as in, it is moving), the pitch that you hear will be different than the pitch that you hear when it is stationary.
The top diagram shows what is happening for a source at rest; the waves spread out in an even, measured way. The pitch that you hear is precisely the pitch that is emitted from the source. Once the source begins to move, however- say, to the right- the waves that are emitted in front of it are slightly closer to the waves it previously emitted. As a result, for the waves ahead of the moving source, the wavelength is shortened, and for the waves behind the source, the
wavelength is made longer. If the speed of sound is constant in the same medium, and we change the wavelength, the frequency must respond- in front, we hear a higher frequency than emitted, and in back, we hear a lower frequency then emitted.
This effect explains why we often may hear a siren or a car horn rolling by going: "... nnnnnnnNNNNNNNNNNYYYOOWWwwwwwwwww..." Initially, the intensity as heard by us is low, owing to the relative distance between us and the car. As it gets closer, the intensity grows, but the pitch stays the same. When it is directly next to us, we hear near the true frequency and the highest intensity. And finally, as it goes past us, the intensity slowly diminishes, and we hear a lower pitch.
It is important to note that while the volume changes, as long as the velocity does not change, the frequency that you perceive will not until the source changes its direction of travel relative to you. In fact, we can actually calculate the magnitude of this effect! There are three different cases to consider, here: 1) the source moves as you stand still; 2) you move as the source stands still; and 3) both you and the source move. (Your relative directions of travel do factor into these equations and derivations.)
1. Moving Source
Suppose the source is moving. We know the wavelength will change depending on the object's speed and direction of travel. Originally, this wave has the form , where is the true wavelength, is the speed of the wave, and is the period of the wave. So, we write:
Where is the shifted wavelength. The positive here denotes moving away from you (as it lengthens the wavelength), and negative denotes moving toward you (as it shortens the wavelength). We eventually want to be able to quantify the pitch of the sound, so we're
interested in the frequency of the observed waves. Remembering we can write as (as the speed of the wave must stay constant), we can substitute all these together to find:
If we rearrange to solve for the shifted frequency, we can find:
Which can be written as:
All you need to know, then, is the direction of travel, the original frequency of the sound, and how fast the source is moving toward or away from you, and you can find the shifted frequency! It is important to note, as well, that this proves that the frequency you hear will be a constant, so long as the source's velocity remains constant.
Moving on:
2. Stationary Source, Moving Receiver
In a very, very similar way, we can find the frequency you observe if you move toward or away from a stationary source. If we denote as the velocity of the receiver, we know that you will encounter the waves emitted by the source at slightly different times than a stationary observer. There's a certain bit of handwaving I'm going to go through, here- consult page 554 of your professor's notes if you need a diagram or other reference.
We know that the source emits waves of wavelength at regular intervals, dependent on the period . You will encounter the wave at slightly different times, , in the distance you cover, . In one In that same time, the wave will have moved . Writing this in terms of the distances involved- whether you move toward or away from the source- we can have:
Substituting in:
Let's condense this a bit:
Rearranging, we can find:
This proves, then, that the frequency we hear from a source will be a constant, provided we are moving with a constant velocity.
3. Moving Source and Receiver
It turns out this situation is "simply" the product of both of the above effects. In essence, it can be shown that:
If the two are moving, but spreading farther apart relative to one another, a lower frequency is observed; for objects moving towards one another as they move, the frequency shifts up.
It is good to notice that we can use this relationship for any situation in which the Doppler Effect applies. If the source is not moving, then the denominator collapses to 1, and we are left with the appropriate Case 2. If the receiver is not moving, then the numerator collapses to 1, and we get the appropriate Case 1.
The Doppler Effect is also useful for electromagnetic (light) waves; "red shift" means that the source and receiver are moving apart, while "blue shift" means that the source and receiver are moving towards one another. It is how Edwin Hubble first figured out that the universe is
expanding; no matter where he pointed his telescope, everything appeared "red-shifted," meaning that the wavelengths of light reaching his telescope appeared longer than they actually should be, based on other verified measurements. No matter where he pointed his telescope, he saw red shifts, and therefore concluded that the universe must be expanding. Moreover, he found that the universe is expanding at an accelerating rate.
To this day, our best explanation for why this is happening is called dark energy. What's dark energy?
The Sound Barrier
Now, suppose a source of sound is moving at such a fast pace that the waves emitted ahead of the source are moving slower than the source itself. This would mean that we have broken the sound barrier.
This diagram shows the source increasing in speed until- in Case 3- it has actually broken the sound barrier. The implication of traveling faster than the sound you are emitting means that you can pass by a location- screaming all you like- and your screams will not be heard by any
receiver until you are actually past the receiver. The "booms" we are familiar with result from differences in pressure created from accumulating air in front of you, pushing it out of the way and leaving lower-pressure areas behind you, and the booms radiate outward. They are most intense directly in line with your direction of motion, and decrease rapidly in intensity as a receiver's lateral distance is increased.
Sound in Music: Standing Waves in Pipes and Strings
We can transition from these waves we have talked about to setting up a new kind of wave: a standing wave. A standing wave results when a wave pulse is transmitted through a medium, reflected back, and interferes with the source wave. These waves have a very common
application, in the making of music. How waves interfere controls what pitches are heard, which in turn determines what sounds an instrument is capable of making.
There are very specific frequencies allowed to a musical instrument (or any pipe or string, really), that are cause as a result of standing waves being set up inside the instrument. They are available based on how the object is classified: as a pipe closed at both ends/string fixed at both ends; or as a pipe open at one end; or as a pipe open at both ends.
There are aspects of this wave called nodes and antinodes. The nodes are where the amplitude of the standing wave reaches zero, and the antinodes are where the amplitude of the standing wave reaches its maximum. The m here stands for a harmonic, which we will discuss shortly.
This has a node at and standing wave. The node will only appear following the relationship we showed earlier, where:
Where
This means that for any wavelength that satisfies the above, that wave is allowed to exist in the string! It means that only certain wavelengths are allowed, depending on the length of the string. This is the reason that pipe organs all have different lengths; the different lengths allow the notes of the musical scale to exist. If all the pipes were the same length, you would have a very boring pipe organ. (The same principle in general also applies to guitars, violins, cellos, and other stringed instruments.)
We can take a look at the frequencies corresponding to these wavelengths. For example, it is known that a guitar's G-string will have waves moving across its length at a speed of about . (While this is faster than sound, we must remember that this is the speed that the wave is traveling on the string, and not the speed at which the sound wave leaving the string is traveling; wind instruments will have wave speeds equal to the speed of sound.) Knowing that the speed of a wave is related thus:
We can then relate this to the frequency of each harmonic:
We can then construct a table to find these frequences. We will use for the full length of a string as we construct this table:
Musical Note
If you were to translate this to a musical scale, as in the rightmost column, you would see that the musical pitch we hear increases with each harmonic. In fact, note what happens for
. All of them are the same musical note, just separated by one octave (which is what those subscripts mean). This means that as a general rule, when you double the frequency that is being played, you have moved up one octave on the musical scale.
Now, you may have noticed a hand-wave: we discussed here that pipe organs and guitar strings have the same general principle governing their harmonics. However, pipe organs are allowed to change their pipes' length; guitars and stringed instruments, however, tend to have all strings cut to the same length. What allows them to have different frequencies is increasing or decreasing the tension in the string. The tension is the factor that governs how fast waves move up and down the length of the string, which in turn determines the frequencies available to each string.
Musicians use harmonics to their advantage often without realizing it. Whenever a string player tunes their instrument- be it guitars, violins, etc- they tune all their strings to the next-thickest string. For example, in ascending order, a guitar has strings E, A, D, G, B, E. The A string is tuned to the low E string by shortening the length of the E string until it "sounds" like an A. Then the length of the true A string is adjusted until there are no "beats" heard (meaning the waves are in phase, and the same frequency, meaning there is no constructive/destructive interference). The D is tuned to A, the G to D, the B to G, and finally the E to B. By shortening the length of the string, the musician determines what frequencies are available on the string, allowing him/her to tune their instrument to the ensemble as well as individually.
You may notice that different musical instruments function very similarly, simply seeking to lengthen or shorten their respective pipes or strings in order to generate different frequencies. We can turn our attention to some other familiar musical instruments, namely the brass, most
The wavelengths available are:
The reason for this comes from algebra. Say, for , we look at the corresponding wave in the diagram. From to , there is only one-quarter of a full wave present. This means that the wave must travel a full to reach one cycle. In other words:
And for , it appears that three-quarters of a full wave are present. This means that the wave must travel to reach one cycle, or:
Continuing, we can begin to see a pattern emerge- that 4 tends to always be present in the numerator, and the denominator forms the pattern of for each successive harmonic. So, we can convince ourselves that this is the harmonic pattern for waves in pipes open at one end.
All brass instruments and most woodwind instruments follow this pattern, because the
mouthpiece or reed close off one end, and the sound (mostly) leaves through the bell at the end of the instrument. The purpose of the buttons or slides, then, is to change the length of the pipe; as discussed for pipes closed at both ends, if you change the length of the pipe, you change what frequencies are allowed to exist in that pipe.
It is important to note that while this pattern is very different from the other, but the principles governing the frequencies, octaves, etc all still apply just as equally as before.
The wavelengths available are:
This should be a little easier to convince oneself of: the pipe open at both ends is just a phase-shift away from being a pipe or string closed/fixed at both ends. An example of an instrument like this would be the flute, piccolo, or other similar instruments.
Now, then, attempt the following problems involving sound:
1.) An organ pipe, open at one end, has a length of 3 m. Remembering that speed of sound in air is , find the wavelength and frequencies of the 1st, 2nd, and 3rd harmonics. (For extra brownie points, look up the approximate notes of these frequencies!)
2.) A reckless driver drives by you at a speed of . He constantly blares his horn to get people out of the way, and his horn emits a frequency of .
a) As the car moves toward you, what frequency do you hear?
b) Right as the car is physically next to you, what frequency do you hear? c) As the car moves past you, what frequency do you hear?
d) At each step, what frequency does the maniac driving the car hear?
