## 5 Steps to a 5: AP Physics 1: Algebra-Based 2017 (2016)

### STEP __4__

### Review the Knowledge You Need to Score High

### CHAPTER 17

### Waves and Simple Harmonic Motion

**IN THIS CHAPTER**

**Summary:** This chapter introduces basic properties of waves, especially of sound waves. You”ll define wave speed, frequency, and wavelength, and relate them through *v* = *λf* . Although a wave moves through a material, the pieces of the material themselves do not move. Rather, they tend to oscillate in simple harmonic motion. This chapter describes exactly what that means.

**Definitions**

The **period** is the time for one cycle of simple harmonic motion, or the time for a full wavelength to pass a position.

The **frequency** is the number of cycles, or the number of wavelengths passing a position, in one second.

The unit of frequency is the Hz, which means “per second.”

The **amplitude** is the distance from the midpoint of a wave to its crest, or the distance from the midpoint of simple harmonic motion to the maximum displacement.

The **wavelength** is measured from peak to peak, or between any successive identical points on a wave.

The **spring constant** *k* , measured in units of newtons per meter (N/m), is related to the stiffness of a spring.

A **restoring force** is any force that always pushes an object toward an equilibrium position.

**Nodes** are the stationary points on a standing wave.

**Antinodes** are the positions on standing waves with the largest amplitudes.

You”re probably most familiar with waves on the surface of a lake or pond. Those are transverse waves, which look like the waves on the machine in the following figure. The wavelength of the wave can be measured with a ruler from peak to peak.

**Simple Harmonic Motion**

Simple harmonic motion refers to a back-and-forth oscillation whose position-time graph looks like a sine function. The typical examples are a mass vibrating on a spring, and a pendulum.

**Example:** A cart of mass 0.5 kg is attached to a spring of spring constant 30 N/m on a frictionless air track, as shown. The cart is stretched 10 cm from the equilibrium position and released from rest.

**FACT:** The period of a mass on a spring in simple harmonic motion is given by

Of course, the AP Exam will not likely ask, “What is the period of this oscillation?” Rather, it might ask for the specific effect that doubling the mass of the cart would have on the period. Since the *m* term is in the numerator of the period equation, a bigger mass means a larger (longer) period of oscillation. Since the *m* is under a square root, doubling the mass multiplies the period by the square root of 2.

What if the amplitude of the motion were doubled? How would that affect the period? Since you don”t see an *A* in the equation for the period, the period would not change. This is a general result for all simple harmonic motion and wave problems: The amplitude does not affect the period.^{ }^{1}

**FACT:** The frequency and period are inverses of one another.

Once you know by being told or by doing the calculation that the period of this mass on a spring is 0.81 s, you can use your calculator to do 1 divided by 0.81 s, giving a frequency of 1.2 s.

**FACT:** The amount of restoring force exerted by a spring is given by

The force of the spring on the cart is therefore greatest when the spring is most stretched, but zero at the equilibrium position. And since *a* = *F* _{net} /*m* , the acceleration likewise changes from lots at the endpoints, to nothing at the middle.

This means that you cannot use kinematics equations with harmonic motion. A kinematics approach requires constant acceleration. Instead, when a problem asks for the speed of a cart somewhere, use conservation of energy.

**FACT:** The spring potential energy is given by

The energy stored by the spring is thus largest at the endpoints and zero at the equilibrium position. There, the spring energy is completely converted to the kinetic energy of the cart. Where is the cart”s speed greatest, then? At the equilibrium position, of course, because kinetic energy is ½*mv* ^{2}^{ }—largest kinetic energy means largest speed.

To calculate the value of the maximum speed, write out the energy conversion from the endpoint of the motion to the midpoint of the motion: spring potential energy is converted to kinetic energy. Translated into equations, you get ½*kx* ^{2} = ½*mv* ^{2} . Plug in values, and solve for the speed. Here, you get 0.77 m/s (i.e., 77 cm/s).^{2}

**Example 2:** The pendulum shown in the preceding figure is released from rest at the start position. It oscillates through the labeled positions A, B, C, and D.

Treat a pendulum pretty much the same way as a spring. It”s still in harmonic motion; it still requires an energy approach, not a kinematics approach, to determine its speed at any position.

**FACT:** The period of a pendulum is given by

As always, it”s unlikely you”re going to plug in numbers to calculate a period. Among the gazillion possibilities, you might well be asked to rank the listed positions in terms of some quantity or other. Here are some ideas:

Rank the lettered positions from greatest to least by the bob”s gravitational potential energy. Gravitational potential energy is *mgh* ; the bob always has the same mass and *g* can”t change, so the highest vertical height has the greatest gravitational potential energy. Ranking: D > C = A > B.

Rank the lettered positions from greatest to least by the bob”s total mechanical energy. Total mechanical energy means the sum of potential and kinetic energies. Here, with no nonconservative force like friction acting, and no internal structure to allow for internal energy, the total mechanical energy doesn”t change. The ranking is as follows: (A = B = C = D).

Rank the lettered positions from greatest to least by the bob”s speed. Since gravitational potential energy is converted to kinetic energy, the bob moves fastest when the gravitational potential energy is smallest. Ranking: B > C = A > D.

The gravitational field at the surface of Jupiter is 26 N/kg and on the surface of the Moon, 1.6 N/kg. Rank this pendulum”s period near these two planets and earth. Since *g* is in the denominator of the period equation, the lowest gravitational field will have the greatest period; so *T* _{Moon} > *T* _{Earth} > *T* _{Jupiter} . The ranking by frequency would be just the opposite—because frequency is the inverse of period, a bigger *g* leads to a smaller period but a bigger frequency.

**Waves**

The AP Physics 1 Exam covers only “mechanical” waves, such as sound, or waves on the surface of the ocean. Light waves (i.e., electromagnetic waves) are not covered in detail.

**FACT:** Whenever the motion of a material is at right angles to the direction in which the wave travels, the wave is a **transverse wave.**

**Example 3:** A wave pulse travels to the right through a spring and then extends into a second spring in the preceding figure. The speed of the waves is faster on the right-hand spring.

This is a transverse wave pulse—the coils of the spring travel up and down the page, while the wave itself moves to the right.

**FACT:** The energy carried by a wave depends on the wave”s amplitude.

A good AP-style question might ask you to resketch the diagram so that a pulse of about the same wavelength carries more energy. Make the amplitude—the maximum displacement of the coil above the resting position—bigger, then, because amplitude is related to energy carried by a wave. Keep the pulse about the same length.

**FACT:** When a wave changes materials, its frequency remains the same.

When this waves moves into the new spring, the wave speeds up, but the frequency remains the same. By *v* = *λf* , the wavelengths will also increase, because multiplying the same frequency by the wavelength has to give a bigger value for *v* . The wave will look wider, then, in the new spring.

**FACT:** When a material vibrates parallel to the direction of the wave, the wave is a **longitudinal wave.**

**Example 4:** A wave travels through a spring. A picture of the spring is shown in the preceding figure, with point *C* labeled.

This wave is a longitudinal wave—the disturbance is traveling through the spring, so it is traveling right or left. The coils of the spring itself are spread out and compressed. Since the motion of the spring”s coils is also left-right, parallel to the way the disturbance is traveling, this is a longitudinal wave.

Try drawing a wavelength on the picture. A wavelength is defined as the distance between two identical positions on the wave. From position C to the next spot where the coils are all stretched out is one wavelength.

**Superposition and Interference**

When two waves collide, they don”t bounce or stick like objects do. Rather, the waves interfere—they form one single wave for just a moment, and then the waves continue on their merry way.

**FACT:** In **constructive interference** the crest of one wave overlaps the crest of another. The result is a wave of increased amplitude.

**Two wave pulses about to interfere constructively.**

The two wave pulses on a string moving toward each other in the preceding figure will interfere constructively, since their amplitudes are on the same side of the string”s resting position. When the pulses meet, the wave will look like the dark line in the following figure.

**Two wave pulses interfering constructively.**

Then the waves continue on in the direction they were originally traveling (see the following figure).

**Two wave pulses after interfering constructively.**

**FACT:** In **destructive interference** the crest of one wave overlaps the trough of another. The result is a wave of reduced amplitude.

The same principle applies to waves with amplitudes on opposite sides of the string”s resting position, producing destructive interference.

If we send the two wave pulses in the following figure toward each other, they will interfere destructively (next figure following) and then continue along their ways (last figure following).

**Two wave pulses about to interfere destructively.**

**Two wave pulses interfering destructively.**

**Two wave pulses after interfering destructively.**

**Standing Waves**

A **standing wave** by definition is a wave that appears to stay in one place. In some positions, the strings vibrate with large amplitude—these are called **antinodes** . In other positions, the strings don”t vibrate at all—these are called **nodes** .

The reason that a standing wave exists revolves around interference. Waves are traveling back and forth on the string, reflecting off of each end and interfering with each other all willy-nilly. The net effect of all this interference is a pattern of nodes and antinodes.

**Example 5:** A guitar string of length 1 m is plucked.

Generally when you pluck a string, you produce a standing wave of the longest possible wavelength, and thus the smallest possible frequency. The string will look like the preceding picture if you watch it carefully.

**FACT:** The wavelength of a standing wave is twice the node-to-node distance.

In this case, the node-to-node distance is equal to the length of the string: 1 m. So the wavelength here is 2 m.

But what if you put your finger very lightly on the middle of the string, forcing a node to exist there? Then you create a harmonic, like the one shown in the following figure. Now the wavelength of this standing wave is 1 m.

And you can put not just one or two, but any whole number of antinodes on a string. (See the following figure.)

**FACT:** For a string fixed at both ends, or for a pipe open at both ends, the smallest frequency of standing waves is given by

where *v* is the speed of the waves on the string or in the pipe. Other harmonic frequencies for this string or pipe must be whole-number multiples of the fundamental frequency.

When we”re dealing with an open pipe, the speed *v* is generally the speed of sound in air, or about 340 m/s. For a string, the wave speed depends on the tension in the string and the mass of 1 meter worth of string—usually you”re talking a few hundred meters per second (m/s), but that can vary.

So if this guitar string produces a fundamental frequency of, say, 100 Hz, then the harmonics that can be played are 200 Hz, 300 Hz, 400 Hz, etc. This string *cannot* play a frequency of 150 Hz or 370 Hz, at least unless the length of the string or the tension in the string is changed.

**FACT:** The pitch of a musical note depends on the sound wave”s frequency; the loudness of a note depends on the sound wave”s amplitude.

A guitar can generally play any note in a musical scale. But how can that be, if the harmonics are restricted to multiples of the fundamental frequency? Producing harmonics is generally not the way to play a guitar.^{ }^{3}^{ }Rather, the frets on the neck are used to shorten or lengthen the string; since the speed of waves on the string is unchanged, shortening the string lowers the *L* in . The fundamental frequency of the shorter string will be higher, and thus the note played will be higher pitched.

The guitar can be tuned by tightening or loosening the string. A tighter string, for example, will produce a standing wave with higher wave speed. For the same length of string and thus the same wavelength, the frequency will be higher by *v* = λ*f* . A higher frequency means a higher-pitched note.

**Closed-Ended Pipe**

**Example 6:** A 1-m-long PVC pipe is covered at one end and open at the other.

When a pipe is open at one end and closed at the other, the standing wave in this pipe must have a node at one end and an antinode at the other. The wave with the longest possible wavelength is shown in the preceding figure. The wavelength of a standing wave is always twice the node-to-node distance. But the wave is so long that we don”t even see a second node. It turns out that the wavelength of this wave is 4 m (four times the length of the pipe).

**FACT:** For a pipe closed at one end, or for a string fixed at one end but free at the other, the smallest frequency of standing waves is given by

where *v* is the speed of the waves on the string or in the pipe. Other harmonic frequencies for this string or pipe must be *odd* multiples of the fundamental frequency.

The speed of waves in this pipe is the speed of sound, or about 340 m/s. The fundamental frequency is 85 Hz. The other frequencies that this pipe can produce are only the *odd* multiples of 85 Hz: 255 Hz, 425 Hz, etc.

**Beats**

**FACT:** Beats are rhythmic interference that occurs when two notes of unequal but close frequencies are played.

If you have a couple of tuning forks of similar—but not identical—frequency to play with, or if you have a couple of tone generators at your disposal, you might enjoy generating some beats of your own. They make a wonderful “wa-wa” sound, which is due to a periodic increase and decrease in intensity, or loudness. The frequency of the “wa-wa” is equal to the *difference* between the two frequency generators.

**Doppler Effect**

Whenever a fire engine or ambulance races by you with its sirens blaring, you experience the **Doppler effect** . Similarly, if you enjoy watching auto racing, that “Neeee-yeeeer” you hear as the cars scream past the TV camera is also attributable to the Doppler effect.

**FACT:** The Doppler effect is the apparent change in a wave”s frequency that you observe whenever the source of the wave is moving toward or away from you.

To understand the Doppler effect, let”s look at what happens as a fire truck travels past you (see the following two figures).

**When the fire truck moves toward you, the sound waves get squished together, increasing the frequency you hear.**

**As the fire truck moves away from you, the sound waves spead apart, and you hear a lower frequency.**

As the fire truck moves toward you, its sirens are emitting sound waves. Let”s say that the sirens emit one wave pulse when the truck is 50 meters away from you. It then emits another pulse when the truck is 49.99 meters away from you. And so on. Because the truck keeps moving toward you as it emits sound waves, it appears to you that these waves are getting scrunched together.

Then, once the truck passes you and begins to move away from you, it appears as if the waves are stretched out.

Now, imagine that you could record the instant that each sound wave hits you. When the truck is moving toward you, you would observe that the time between when one wave hits and when the next wave hits is very small. However, when the truck is moving away from you, you would have to wait a while between when one wave hits you and when the next one does. In other words, when the truck is moving toward you, you register that the sirens are making a higher frequency noise; and when the truck is moving away, you register that the sirens are making a lower frequency noise.

That”s all you really need to know about the Doppler effect. Just remember, the effect is rather small—at normal speeds, the frequency of, say, a 200 Hz note will only change by a few tens of Hz, not hundreds of Hz.

** Practice Problems**

** Note: **Additional drills regarding simple harmonic motion graphs are included in

__Chapter 18__.

** 1 .** In a pipe closed at one end and filled with air, a 384-Hz tuning fork resonates when the pipe is 22-cm long; this tuning fork does not resonate for any smaller pipes.

(a) State three other lengths at which this pipe will resonate with the 384-Hz tuning fork.

(b) The end of the pipe that was closed is now opened, so that the pipe is open at both ends. Describe any changes in the lengths of pipe that will resonate with the 384-Hz tuning fork.

(c) The air in the closed pipe is replaced with helium. Describe an experiment that would use the pipe to determine the speed of sound in helium.

** 2 .** The wave shown in the previous figure travels in a material in which its speed is 30 m/s.

(a) What is the wavelength of this wave?

(b) Calculate the frequency of this wave.

(c) On the diagram, draw a different wave that has a larger frequency but carries less energy than the one pictured.

** 3 .** Consider the following questions about electromagnetic and mechanical waves. Justify your answer to each.

(a) Which of the following types of wave can be transmitted through space, where there is no air? Choose all that apply. Justify your answer briefly.

(i) Visible light

(ii) Radio waves

(iii) Gamma rays

(iv) Sound waves

(b) (multiple choice) A tuning fork vibrating in air produces sound waves. These waves are best classified as

(A) Transverse, because the air molecules are vibrating parallel to the direction of wave motion

(B) Transverse, because the air molecules are vibrating perpendicular to the direction of wave motion

(C) Longitudinal, because the air molecules are vibrating parallel to the direction of wave motion

(D) Longitudinal, because the air molecules are vibrating perpendicular to the direction of wave motion

(c) (multiple choice) Radio waves and gamma rays traveling in space have the same

(A) Frequency

(B) Wavelength

(C) Period

(D) Speed

(d) Which type of wave exhibits the Doppler effect? Choose all that apply. Justify your answer briefly.

(i) Visible light

(ii) Radio waves

(iii) Gamma rays

(iv) Sound waves

** 4 .** The driver of a car blows the horn as the car approaches you.

(a) Compared to the horn”s pitch heard by the driver, will the pitch observed by you be higher, lower, or the same?

(b) The car passes you, while the driver continues to blow the horn. After the car passes, you notice that the horn doesn”t sound as loud as it did when it was near you. Is this observation a result of the Doppler effect?

(c) The car recedes from you after passing you, still producing sound waves from the horn. Discuss how the amplitude, period, and frequency of the sound waves that you would measure compare to the amplitude, period, and frequency of the sound waves that the driver would measure.

** 5 .** The period of a mass-on-a-spring system is doubled, while still using the same spring.

(a) By what factor does the frequency of the mass-on-a-spring system increase or decrease?

(b) By what factor does the mass attached to the spring increase or decrease?

** Solutions to Practice Problems**

** 1 .** (a) Because 22 cm is the shortest length of pipe that resonates, 384 Hz is the fundamental frequency, the one that produces waves without any nodes inside the pipe. There must be an antinode at one end and a node at the other (because it”s closed at one end and open at the other). As the pipe length is increased, the wavelength of the sound wave doesn”t change, because the frequency of the tuning fork and the speed of sound don”t change. The pipe will next resonate when once again there is an antinode at one end and a node at the other. (This time, though, there will be another node inside somewhere, too.) Since the antinode-to-node distance was 22 cm, we need to add that distance twice to get another full “hump” of a standing wave inside the pipe. Add 44 cm to the pipe to get resonance at a pipe length of 66 cm; add another 44 cm to get resonance at 110 cm; and add another 44 cm to get resonance at 154 cm.

(b) Now only standing waves with nodes at both ends of the pipe will resonate. The wave of 22 cm has a node at one end and an antinode at the other—it will no longer resonate. But 44 cm is the node-to-node distance, so 44 cm will resonate. Whereas 66 cm used to resonate in the closed pipe, it will not in the open pipe because an antinode is at one end. Instead, any multiple of 44 cm will resonate because adding 44 cm adds a full node-to-node distance, ensuring a node at each end.

(c) We know the frequency of the tuning fork. To use the equation *v* = λ*f* , we need the wavelength of the wave as well. Play the tuning fork and shorten the pipe until once again we find the shortest pipe length that resonates with the tuning fork. That”s the fundamental, with an antinode at one end, a node at the other, and no nodes in between. The wavelength of the sound wave is four times this shortest resonating pipe length.

** 2 .** (a) The wavelength is measured from peak to peak or trough to trough. That”s 3.0 m.

(b) Use *v* = λ*f* … (30 m/s) = (3.0 m)*f* … *f* = 10 Hz.

(c) The energy carried by a wave depends on the wave”s amplitude; this wave should have a smaller amplitude. Since the wave speed doesn”t change, a bigger frequency means a smaller wavelength by *v* = λ*f* .

** 3 .** (a) Correct answers: (i), (ii), and (iii). Only electromagnetic waves can be transmitted through a vacuum. That includes gamma rays, visible light, and radio waves, which are all parts of the electromagnetic spectrum. Sound waves require a material to be transmitted.

(b) Correct answer: (C). Sound waves are longitudinal, by definition. Also by definition, “longitudinal” means that the particles of the material are vibrating parallel to the direction that the wave travels.

(c) Correct answer: (D). The speed of light—or any electromagnetic wave—in a vacuum is 300 million m/s. Radio and gamma rays have different frequencies, and thus different wavelengths and periods.

(d) Correct answers: (i), (ii), (iii), (iv). All waves exhibit the Doppler effect. When the source of the wave moves toward an observer, the observer observes waves of higher frequency. For sound waves, this means you”d hear a higher pitch; for visible light, this means you”d see a “blue shift.” Gamma and radio waves would also be observed to have a higher frequency.

** 4 .** (a) By definition, when a source of waves approaches an observer, the observer observes waves of a higher frequency. Pitch of a sound is related to the sound wave”s frequency.

(b) The Doppler effect says nothing about loudness, only about frequency. The horn sounds less loud because the energy created by the wave source spreads out over a larger and larger space as the wave gets farther from the source. Since the energy carried by a wave is related to its amplitude, and since amplitude is related to loudness for a sound wave, you hear a softer noise.

(c) The horn won”t seem as loud (as discussed in [b]), so the amplitude is smaller than what the driver hears. The frequency you hear will be lower based on the Doppler effect. Since period is the inverse of frequency, a smaller frequency means a larger period.

** 5 .** (a) Frequency is the inverse of the period. When the period doubles, the frequency is cut in half.

(b) The relevant equation is . The spring constant doesn”t change because it”s the same spring. Since the mass term is in the numerator and square rooted, the mass should quadruple. Then, square rooting the factor of four increases the whole fraction by a factor of two.

** Rapid Review**

- The period of a mass on a spring in simple harmonic motion is given by .
- The frequency and period are inverses of one another.
- The amount of restoring force exerted by a spring is given by
*F*=*kx*. - The spring potential energy is given by
*PE*_{spring}= ½*kx*^{2}. - The period of a pendulum is given by .
- Whenever the motion of a material is at right angles to the direction in which the wave travels, the wave is a
**transverse wave.** - The energy carried by a wave depends on the wave”s amplitude.
- When a wave changes materials, its frequency remains the same.
- When a material vibrates parallel to the direction of the wave, the wave is a
**longitudinal wave.** - In
**constructive interference**the crest of one wave overlaps the crest of another. The result is a wave of increased amplitude. - In
**destructive interference**the crest of one wave overlaps the trough of another. The result is a wave of reduced amplitude. - The wavelength of a standing wave is twice the node-to-node distance.
- For a string fixed at both ends, or for a pipe open at both ends, the smallest frequency of standing waves is given by , where
*v*is the speed of the waves on the string or in the pipe. Other harmonic frequencies for this string or pipe must be whole-number multiples of the fundamental frequency. - The pitch of a musical note depends on the sound wave”s frequency; the loudness of a note depends on the sound wave”s amplitude.
- For a pipe closed at one end, or for a string fixed at one end but free at the other, the smallest frequency of standing waves is given by , where
*v*is the speed of the waves on the string or in the pipe. Other harmonic frequencies for this string or pipe must be*odd*multiples of the fundamental frequency. - Beats are rhythmic interference that occurs when two notes of unequal but close frequencies are played.
- The
**Doppler effect**is the apparent change in a wave”s frequency that you observe whenever the source of the wave is moving toward or away from you.

^{1}^{ }There are exceptions for simple and physical pendulums once the amplitude reaches large enough values, but the AP will not likely ask much about these situations.

^{2}^{ }Don”t forget to convert the maximum distance from equilibrium to 0.1 meters before plugging into the equation.

^{3}^{ }Harmonics (other than the fundamental) are not the notes you hear. The *overtones* (the harmonics higher than the fundamental) are what give the note its *quality* , which makes a guitar sound like a guitar or a violin sound like a violin. After all, the strings themselves are about the same, so did you ever wonder how it was you are able to tell the difference between the sounds of stringed instruments?