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The world is full of waves. Some waves like those in water are obvious. Others, like sound and light waves are less apparent. The wave properties of sound and light are discovered only by careful observations and experiments.

In this section we want to introduce the basic properties of waves with a wave model that emphasizes those aspects of wave behavior common to all waves. Although sound waves, water waves, and radio waves are clearly different, the wave model will allow us to understand many of the important features they share.

The wave model is built around the idea of a traveling wave, which is an organized disturbance that travels with a well-defined wave speed.

Mechanical waves are waves which require a substance through which they can move. We call such a substance the medium of the wave. For instance, the medium of a water wave is the water, the medium of a sound wave is the air, and the medium of a wave on a stretched string is the string.

As a wave passes through a medium, the atoms that make up the medium are displaced from their equilibrium positions. This is a disturbance of the medium. The water ripples in the picture below are a disturbance of the water’s surface.

A wave disturbance is created by a source. The source of a wave might be a rock thrown into water, your hand plucking a stretched string, or an oscillating loudspeaker pushing on the air. Once created, the disturbance travels outward through the medium at the wave speed v. This is the speed with which a ripple moves across the water or a pulse travels down a string.

It is the disturbance which travels through the medium.

The medium as a whole does not travel!

A wave does transfer energy, but it does not transfer some of the medium from here to there.

For instance, the ripples on the pond (the disturbance) move outward from the splash of the rock, but there is no outward flow of water. Likewise, the particles of a string oscillate up and down but do not move in the direction of a pulse traveling along the string. A wave transfers energy, but it does not transfer any material or substance outward from the source.

Transverse and Longitudinal Waves

Most waves fall into two general classes of waves: transverse and longitudinal. In a transverse wave, the particles in the medium move perpendicular to the direction in which the wave travels. In a longitudinal wave, the particles in the medium move parallel to the direction in which the wave travels.

Transverse Wave

Shaking the end of a stretched string up and down creates a wave that travels along the string in a horizontal direction while the particles that make up the string oscillate vertically.

Longitudinal Waves

If you give the first spring a sharp push, a disturbance travels down the chain of springs by compressing and expanding the springs. The masses move in the same direction as the wave.

Water waves have characteristics of both transverse and longitudinal waves. When a water wave passes, individual water molecules describe a circular path as shown to the right, exhibiting both transverse and longitudinal motion.

The rapid motion of the earth’s crust during an earthquake can produce a disturbance that travels through the entire earth. The detection and analysis of these waves is the most important tools we have for understanding the earth’s interior. The two most important types of earthquake waves are S waves (which are transverse) and P waves (which are longitudinal). The P waves are faster, but the S waves are more destructive. Residents of a city a few hundred kilometers from an earthquake will feel the P waves as much as a minute before the S waves, giving them a crucial early warning.

For all of these types of waves, the medium as a whole does not travel through space; the individual particles in the medium undergo back-and-forth motions around their equilibrium positions.

When you drop a pebble in a pond, waves travel outward. But how does this happen? How does a mechanical wave travel through a medium? In answering this question, we must be careful to distinguish the motion of the wave from the motion of the particles that make up the medium. The wave itself is not a particle, so we cannot apply Newton’s laws to the wave. However, we can use Newton’s laws to examine how the medium responds to a disturbance.

No new physical principles are required to understand how a wave moves. The motion of a pulse along a string is a direct consequence of the tension acting on the segment of the string. An external force may have been required to create the pulse, but once started the pulse continues to move because of the internal dynamics of the medium.

A sound wave consists of variations in pressure moving through a medium.

Let’s see how a sound wave in air is created using a loudspeaker. When the loudspeaker moves forward, it compresses the air in front of it, as shown below. When the speaker moves backward, it creates a region of expansion. The compression/expansionmake up the disturbance that travels forward through the air. This is much like the sharp push on the end of a slinky. Thus a sound wave is a longitudinal wave. We usually think of sound waves as traveling in air, but sound can travel through any gas, through liquids, and even through solids.

The motion of a sound wave in air is determined by the physics of the gas. Once created, the wave will propagate forward, its motion being entirely determined by the properties of the air.

Wave Speed Is a Property of the Medium

The above discussion of waves on a string and sound waves in air show that the motion of these waves is due to the properties of the medium. A careful analysis of these waves would lead to the important and somewhat surprising conclusion that the wave speed does not depend on the shape or size of the pulse, how the pulse was generated, or how far it has traveled.

The speed of a wave depends only on the properties of the medium.

So what properties of a string determine the speed of waves traveling along the string? The only likely candidates are the strings mass, length, and tension. Because a pulse doesn’t travel faster on a longer and thus more massive string, neither the total mass m nor the total length L is important. Instead, the speed depends on the mass-to-length ratio, which is called the linear density.

Linear density characterizes the type of string we are using. A fat string has a larger value of than a skinny string made of the same material. Similarly, a steel wire has a larger value of than a plastic string of the same diameter.

Wave Speed Is a Property of the Medium

- How does the speed of a wave on a string vary with the tension and the linear density?
- A string with a greater tension will exert larger forces on the segments of the
- string. The string will respond more rapidly, so the wave will move at a higher
- speed. Wave speed increases with increasing tension.
- A string with greater linear density has more inertia. It will respond less
- rapidly, so the wave will move at a lower speed. Wave speed decreases with
- increasing linear density.
- A full treatment yields the following equation for the wave speed:

FT = tension in the string

μ = linear mass density = m/L

Describing waves, both pictorially and mathematically, takes a bit more thought than describing particles and their motion. Until now, we have been concerned with quantities, such as position and velocity, which depend only on time. In other words, if we asked “Where is the object?” we would need only to specify what time t it is. The object’s position x depends only on the time t. We write this mathematically as x(t).

Functions of the single variable t are all right for a particle, because a particle is in only one place at a time. But a wave is not “localized” like a particle is; a wave is spread out throughout space at each instant of time. To describe a wave mathematically requires a function that specifies what the medium is doing not only at an instant of time (when) but also at a point in space (where). In other words, we need a function of twovariables.

In what follows, we’ll consider a wave on a string.

Consider the wave pulse shown below moving along a stretched string. For now we’ll look at somewhat artificial triangular-shaped pulse to clearly show the edges of the pulse.

The graph shows the strings vertical displacement y as a function of position x along the string, at a particular instant of time.

This is a snapshot graphof the wave, much like what you might take with a camera. It’s a picture of the actual physical wave form at an instant in time.

As the wave moves, we can take more snapshots. If we place these next to each other, we get a clear picture of the wave’s motion, as shown below. Note that the wave’s shape stays the same as it moves.

Another way of showing several snapshots is to simply show them separately, one below the next. This shows us how the wave moves along the string as time passes. An example is shown below.

The pictures to the left show a sequence of snapshots, showing how the wave moves along the string as time goes from t1 to t2 and so on.

The wave pulse moves forward a distance x = vt during the time interval t. That is, the wave moves with a constant speed v.

Notice also the motion of the point indicated on the string. The wave moves horizontally, while this point moves up and down. As we mentioned, the motion of the medium is distinct from the motion of the wave.

Since the point on the string moves perpendicularly to the wave motion, this is a transverse wave.

Each of these shots is a y-vs-x graph at a certain time t.

As a different way of picturing the wave, suppose we follow the dot marked on the string and make a graph showing how the displacement of this dot changes with time. The result, shown at right, is a y-vs-t graph at a certain position (the position of the dot).

We might call this a point history graph, since it tells the history of that particular point in the medium.

Important note. Notice that the above graph is reversed from the snapshots on the previous page. Why? Because as the wave moves toward the dot, the steep leading edge causes the dot to rise quickly. On the y-vs-t graph, earlier times (smaller values of t) are to the left, and later times are to the right. Thus the leading edge of the wave appears on the left side of the point’s history graph.

The graph below shows a snapshot graph of a wave on a string that is moving to the right. A point on the string is noted. Which of the choices is the history graph for the subsequent motion of this point?

Waves can come in many different shapes. One very common shape is that described by the sinusoidal functions, sine and cosine. The way to produce a sinusoidal wave is to move an end of a string up and down in simple harmonic motion.

In this case, many of the quantities carry over directly—the amplitude of the SHM is the amplitude of the wave; the same goes for the frequency and period.

The graph on the left shows a sinusoidal wave moving through a medium. The source of the wave is at x = 0. Notice how the wave moves with steady speed toward larger values of x at later times t.

The graph below shows a point’s history graph for a point on a string as a sinusoidal wave passes by. The periodT of the wave, shown on the graph, is the time interval for one cycle of the wave to pass by a particular point.

The period is related to the frequency f by T = 1/f.

The figure above on the left shows a snapshot graph for a sinusoidal wave. The two graphs above correspond. On the left we see that the wave stretches out in space, and moves to the right with speed v. The amplitudeA of the wave is the maximum value of the displacement. The crests of the wave (the high points) have displacement ycrest = A. The troughs (the low points) have displacement ytrough = -A.

A sinusoidal wave repeats itself both in space and in time. The wavelength is the distance spanned by one cycle of the wave (a cycle being a full repeat of the shape). The wavelength in the left picture above is from crest to crest, but it could just as well be drawn from trough to trough.

Wavelength is the spatial analog of period. The period T is the time in which the disturbance at a single point in space repeats itself. The wavelength is the distance in which the disturbance at one instant of time repeats itself.

Since the waveform travels with constant speed v and advances a distance of one wavelength, λ, in a time interval of one period, T, we can write for the wave speed the following equation:

- What is the wavelength?
- 0.5 m
- 1 m
- 2 m
- 4 m

- For this sinusoidal wave:
- What is the frequency?
- 50 Hz
- 100 Hz
- 200 Hz
- 400 Hz

The Principle of Superposition

When two or more waves are simultaneously present at a single point in space, the displacement of the medium at that point is the sum of the displacements due to each individual wave.

When wave pulses meet, they assume a shape that is the sum of the individual pulse shape.

That is, the amplitudes add together.

The two pulses move apart after overlapping, and they continue on unaffected as though they hadn’t run into another wave.

This principle applies to all types of waves.

In a standing wave, the individual points on the string oscillate up and down, but the wave itself does not travel.

The reflections at the ends of the string cause two waves of equal amplitude and wavelength to travel in opposite directions. These are the conditions that cause a standing wave. Superposition creates a standing wave.

The possible modes for standing waves

The condition for standing waves of wavelength λ to exist on a string of length L fixed at both ends:

In words: integer multiples of half-wavelengths must fit onto the string.

We can rearrange:

Here, f1 is the fundamental frequency.

The mode number, n, tells us how many antinodes are on the string.

Constructive and Destructive Interference

We say that two sound sources are in phase if they move outward together and move inward together.

Suppose you are directly in the middle of two speakers which are in phase with one another. What is the wave form like at your position?

Since a condensed region leaves from both speakers at the same time, and both travel the same distance, these two condensations will meet and precisely overlap at your ears.

The same goes for when rarefactions leave the speakers.

When two waves always meet condensation-to-condensation and rarefaction-to-rarefaction (crest-to-crest and trough-to-trough) they are said to be exactly in phase and to exhibit constructive interferenceat you location.

The conditions for constructive interference:

If d1 is the distance from source 1 to the observer, and d2 is the distance from source 2 to the same observer, the difference in the path lengths must be an integer multiple of the source wavelength.

Δd = nλ( n = 0, 1, 2, 3, . . .)

Here, Δd= |d2 – d1| is the path-length difference.

Now suppose you move to a location that is not in the middle of the speakers.

If the distance from one speaker to you and the distance from the other speaker to you differ by a special amount, you will actually hear nothing at all!

What must the difference in path lengths be?

We call the situation when crest overlaps with trough destructive interference.

Both constructive and destructive interference arise from the fact that overlapping waves follow the superposition principle.

If you are at a position of destructive interference, the waves at your position are exactly out of phase.

The conditions for destructive interference:

If d1 is the distance from source 1 to the observer, and d2 is the distance from source 2 to the same observer, the difference in the path lengths must be out of phase by half a wavelength.

Δd = (2n + 1)λ/2( n = 0, 1, 2, 3, . . .)

Here, Δd= |d2 – d1| is the path-length difference.

Circular or Spherical Wave Fronts

Plane wave fronts

Quantities such as loudness (and brightness) depend not only on the rate of energy transfer, or power, but also on the area that receives that power.

A bright, tightly focused laser beam pointer emits 1 mW of light power into a beam that is 1 mm in diameter. What is the intensity of the laser beam?

A plane wave, a circular wave, and a spherical wave have the same intensity. Each of the waves travels the same distance. Afterward, which wave has the largest intensity?

A. The plane wave B. The circular wave C. The spherical wave

Suppose you are listening to one person singing. If nine people join in, so a total of 10 people are singing in unison, will the song be 10 times as loud as the soloist?

No! Even though the intensity will be 10 times as great, the loudness will not be.

In fact, you will perceive the sound of 10 people singing together to be approximately twice as loud as the sound of one person.

Now, let’s add another 90 people, for a total of 100 singers. This will increase the intensity by another factor of 10, but it will seem to you only twice as loud as the 10 person ensemble, and four times as loud as the original soloist.

In general, increasing the sound intensity by a factor of 10 results in an increase in perceived loudness by approximately a factor of 2.

This allows our ears to be sensitive over a huge range of intensities. The quietest sound you can detect and the loudest sound you can safely hear differ by a factor of 1,000,000,000,000!

The hair cells of the cochlea are responsible for sensing sound.

These tiny hairs give our ears remarkable sensitivity.

Motion by as little as .5 nm can produce an electrical response!

Loudness of Sound—The Decibel Scale

Since sound intensity doesn’t correspond to loudness (10 more singers aren’t 10 times louder), we need a different quantity to capture loudness.

The loudness of sound is measured by a quantity called the sound intensity level.

Because of the wide range of intensities we can hear (10-12 – 10 W/m2), and because the difference in perceived loudness is much less than the actual difference in intensity, the sound intensity level is measured on a logarithmic scale.

The units of sound intensity level (i.e. loudness) are decibels, which have the symbol dB.

Loudness of Sound—The Decibel Scale

The lowest intensity the human ear can hear is about I0 = 1 10-12 W/m2.

We call this the threshold of hearing.

It makes sense to place the zero of our loudness scale at this intensity.

To create a loudness scale, we define the sound intensity level to be

This equation uses the base-10 logarithm. Logarithms are a way of keeping track of the exponents of very large or very small numbers. The base of 10 is usually assumed, and so we usually simply write log.

For example, log (1000) = log (103) = 3 and log (.0001) = log (10-4) = -4

Notice that = (10 dB) log (I0/I0) = (10 dB) log(1) = 0 dB.

If the sound intensity level is = 0 dB, does this mean there is no sound?

Notice that the sound intensity level increases by 10 dB each time the actual intensity increases by a factor of 10. This is the essence of a log scale.

Earlier we noted that a factor of 10 in the sound intensity corresponds to twice the loudness. From the graph we can see that loudness doubles when the intensity level increases by 10 dB.

Aperson shouting at the top of her lungs emits about 1 W of energy as sound waves. What is the sound intensity level 1 m from this person?

Here, the train, the observers, and the air are all stationary with respect to the earth.

Each blue sphere represents a wave front. Since the pattern is symmetric, listeners standing in front of or behind the train detect the same number of wave fronts per second.

Cycles per second is frequency. Thus both listeners receive sound waves of the same frequency. With sound, frequency corresponds to pitch, so both hear the same pitch.

Now with the train moving to the right, the wave fronts ahead of the train are closer together, resulting in a decreased wavelength.

This “bunching-up” occurs because the moving train gains ground on a previously emitted wave front before emitting the next one.

The observer in front receives more wave fronts per second than she did when the train was at rest. Thus she hears a higher frequency and so a higher pitch.

Behind the train, the wave fronts are farther apart than when the train was stationary. This means the wavelength of the sound waves behind the train are longer than when the train was at rest.

Thus the listener behind the train receives fewer wave fronts per second and so hears a lower frequency. So he will hear a lower pitch.

Every T seconds, a wave front is emitted. The wave fronts are separated by a distance .

The waves travel with a speed given by v = fs, where fs is the frequency of the sound emitted by the source.

Also, we have Ts = 1/fs.

Now suppose the train (the source) is moving with a speed vs.

The train emits a wave front at t = 0s.

Before emitting the next, the train moves closer to the observer in front by a distance vsT.

The observer in front thus receives wavelengths of ’ = – vsT.

The observer in front thus receives wavelengths of ’ = – vsT.

Key point: The observer in front still observes the sound to be travelling with the speed of sound v, just as before, when the train was stationary.

So the frequency the observer in front measures is: fo = v/ ’ = v/ ( – vsT)

Let’s now substitute = v/fs and Ts = 1/fs:

This is the equation for the frequency measured by a stationary observer when the source of sound is moving toward them.

If the source is moving away from the observer, the wavelength this listener observes is ’ = +vsT.

This is the equation for the frequency measure by a stationary observer when the source of sound is moving away from them.

In either case, we define the Doppler shiftto be the

difference of the observer and source frequencies:

Doppler shift = f0 – fs

Notice that the Doppler shift depends on the ratio of

the speed of the source, vs, to the speed of sound, v.

There is also a Doppler effect when the sound source is stationary but the observer is moving relative to the source.

When both the source and observer are at rest, there is no Doppler shift, and the listener hears the emitted frequency.

The observer moves with a speed vo toward the stationary source, and covers a distance vot in a time t.

During this time, the moving observer encounters all the wave fronts that he would have if he were stationary, plus an additional number.

The additional number of wave fronts will be the distance he’s moved, vot, divided by the wavelength .

Additional number: vot/

Additional number per time: vo/

Total number of wave fronts per time: “stationary amount” + vo/

Thus the observer will hear a higher frequency, since frequency has to do with the number of wave fronts (or cycles) encountered per time.

We can write this in the following way: fo = fs + vo/ = fs( 1 + vo/fs )

Observer moving toward a stationary source.

Total number of wave fronts per time: “stationary amount” – vo/

Thus the observer will hear a lower frequency, since frequency has to do with the number of wave fronts (or cycles) encountered per time.

We can write this in the following way: fo = fs – vo/ = fs( 1 – vo/fs )

Observer moving away from a stationary source.

The physics behind the Doppler effect for a moving observer is different than that for the moving source.

When the source moves and the observer is stationary, the wavelength changes.

When the observer moves and the source is stationary, the wavelength does not change. Instead, a moving observer intercepts a different number of wave fronts per second than does a stationary observer.

However, in both cases, when there is relative motion towards one another, the observer hears a higher frequency.

And when there is relative motion away from each other, the observer hears a lower frequency.

If both the source and the observer are moving, we can combine our previous results. This gives us the following:

Numerator (observer moving):

+ is used for observer moving toward.

– is used for the observer moving away.

Denominator (source moving):

– is used for source moving toward.

+ is used for the observer moving away.

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