Chapter 5: Oscillations and Waves — Review and Exercises

Chapter contents: Chapter 5: Oscillations and Waves

Glossary

amplitude

the maximum magnitude of an oscillating quantity’s departure from its equilibrium value

antinode

the location of maximum amplitude in standing waves

audible range

frequencies between 20 Hz and 20,000 Hz

beat frequency

the frequency of the amplitude fluctuations of a wave

bow wake

V-shaped disturbance created when the wave source moves faster than the wave propagation speed

constructive interference

reinforcement where overlapping wave disturbances add in the same direction

damping force

a frictional force which converts the mechanical energy of the oscillatory motion into thermal energy

destructive interference

partial or complete cancellation where overlapping wave disturbances have opposite directions

Doppler shift

the change in wave frequency due to relative motion of source and observer

frequency

the number of complete cycles of a periodic process per unit time; measured in hertz, or cycles per second

fundamental frequency

the lowest frequency of a periodic waveform

hearing

the perception of sound

hertz

the SI unit of frequency, equal to one cycle per second

infrasound

sound waves below audible frequencies (less than 20 Hz)

longitudinal wave

a wave in which the disturbance is parallel to the direction of propagation

natural frequency

the frequency at which a simple harmonic oscillator oscillates if it is set in motion without a driving force

nodes

the points where the string does not move; more generally, nodes are where the wave disturbance is zero in a standing wave

oscillate

to vary repeatedly about an equilibrium value or position

overtones

frequencies of vibration above the fundamental frequency; when they are integer multiples of the fundamental, they are also harmonics

period

time it takes to complete one oscillation

periodic motion

motion that repeats itself at regular time intervals

pitch

the perception of the frequency of a sound

resonance

enhanced oscillation of a system driven periodically at a frequency near one of its natural frequencies

simple harmonic motion

the oscillatory motion in a system where the net force can be described by Hooke’s law

simple harmonic oscillator

a device that implements Hooke’s law, such as a mass that is attached to a spring, with the other end of the spring being connected to a rigid support such as a wall

sonic boom

a constructive interference of sound created by an object moving faster than sound

sound

a disturbance of matter that is transmitted from its source outward

standing wave

a wave pattern with fixed nodes and antinodes, produced by interference of waves traveling in opposite directions

superposition

the addition of individual wave disturbances to obtain the resulting disturbance where waves overlap, for waves obeying the superposition principle

transverse wave

a wave in which the disturbance is perpendicular to the direction of propagation

ultrasound

sound waves above audible frequencies (greater than 20,000 Hz)

wave

a disturbance that moves from its source and carries energy

wave velocity

the speed at which the disturbance moves; also called wave speed, propagation velocity, or propagation speed

wavelength

the distance between successive points at the same phase of a periodic wave, such as adjacent crests

Section Summary

5.2: Period and Frequency in Oscillations

Periodic motion is a repetitious oscillation.
The time for one oscillation is the period $T$ .
The number of oscillations per unit time is the frequency $f$ .
These quantities are related by
$f = \frac{1}{T}$ .
(5.2.1)

5.3: Simple Harmonic Motion: A Special Periodic Motion

Simple harmonic motion is oscillatory motion for a system that can be described only by Hooke’s law. Such a system is also called a simple harmonic oscillator.
Maximum displacement is the amplitude $X$ . The period $T$ and frequency $f$ of a simple harmonic oscillator are given by
$T = 2π \sqrt{\frac{m}{k}}$ and $f = \frac{1}{2π} \sqrt{\frac{k}{m}}$ , where $m$ is the mass of the system.
Displacement, velocity, and acceleration in simple harmonic motion as a function of time are represented by sinusoidal functions in time, that is functions that are similar to $\sin (t)$ and $\cos (t)$ .

5.4: Forced Oscillations and Resonance

A system’s natural frequency is the frequency at which the system will oscillate if not affected by driving or damping forces.
A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate.
The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies.

5.5: Waves

A wave is a disturbance that moves from the point of creation with a wave velocity $v_{w}$ .
A wave has a wavelength $λ$ , which is the distance between adjacent identical parts of the wave.
Wave velocity and wavelength are related to the wave’s frequency and period by $v_{w} = \frac{λ}{T}$ or $v_{w} = f λ$ . However, wave velocity is a property of the medium and remains constant as wavelength or frequency changes.
A transverse wave has a disturbance perpendicular to its direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.

5.6: Wave Interference: Standing Waves and Beats

Superposition is the combination of two waves at the same location.
Constructive interference occurs when two waves are superimposed in phase (both disturbances in the same direction).
Destructive interference occurs when two waves are superimposed exactly out of phase (two disturbances in opposite directions).
A standing wave is one in which two waves superimpose to produce a wave that forms a standing pattern of nodes and antinodes, locations of no displacement and of maximum displacement.
Standing waves form at specific resonance frequencies, with the lowest frequency being the fundamental frequency, and higher frequencies being called overtones or harmonics.
Beats occur when waves of similar frequencies $f_{1}$ and $f_{2}$ are superimposed. The resulting amplitude oscillates with a beat frequency given by
$f_{B} = | f_{1} - f_{2} |$ .
(5.6.1)

5.7: Sound

Sound is a disturbance of matter that is transmitted from its source outward.
Sound is one type of wave.
Hearing is the perception of sound.

5.8: Speed of Sound, Frequency, and Wavelength

The relationship of the speed of sound $v_{w}$ , its frequency $f$ , and its wavelength $λ$ is given by
$v_{w} = f λ$ ,
(5.8.1)
which is the same relationship given for all waves.
In air, the speed of sound is related to air temperature $T$ by

$v_{w} = (331 m/s) \sqrt{\frac{T}{273 K}}$ .
(5.8.2)
$v_{w}$ is the same for all frequencies and wavelengths.

5.9: Doppler Effect and Sonic Booms

The Doppler shift is change in observed frequency of a wave (for example, a sound wave) due to relative motion of the source and the observer, with approaching source and/or observer increasing the observed frequency and receding source and/or observer decreasing the observed frequency.
A sonic boom is constructive interference of sound created by an object moving faster than sound.
A shock wave is a type of bow wake created when any wave source moves faster than the wave propagation speed.

Questions and Exercises

5.2: Period and Frequency in Oscillations

Exercise 1

Give examples of natural phenomena that occur with constant frequency (or period) over a length of time.

Exercise 2

Which of the following can express a period, and which can express a frequency: seconds, hertz, beats per minute, and minutes per cycle? Explain how the units distinguish the two quantities.

Exercise 3

If an oscillation occurs more rapidly, how does the period of oscillation change? How does the frequency change?

Exercise 4

A mass oscillates vertically on a spring. One student times its motion from its highest point until it next reaches its highest point. Another times its motion from the equilibrium position until it next reaches the equilibrium position. Do both measurements give one period? Explain, including how the direction of motion matters.

Exercise 5

A pendulum completes 12 full oscillations in 24 s. (a) What is its period? (b) What is its frequency? (c) Why might timing several oscillations give a more reliable measurement of the period than timing just one?

Exercise 6

A health practitioner counts 18 heartbeats in 15 s. Assuming a steady rhythm, find (a) the heart rate in beats per minute, (b) the frequency in hertz, and (c) the time between successive beats.

5.3: Simple Harmonic Motion: A Special Periodic Motion

Exercise 7

What conditions must be met to produce simple harmonic motion?

Exercise 8

A mass on a spring moves between positions 4 cm above and 4 cm below equilibrium. (a) What is its amplitude? (b) Starting at its highest point, how far does it travel during one complete cycle? (c) What is its displacement over that cycle?

Exercise 9

An object undergoes simple harmonic motion. (a) Where in its motion is its speed greatest? (b) Where is its speed zero? (c) Is its acceleration zero when its speed is zero? Explain in terms of the restoring force.

Exercise 10

Give an example of a simple harmonic oscillator, specifically noting how its frequency is independent of amplitude.

Exercise 11

A mass on a spring undergoes simple harmonic motion. At a larger amplitude, the mass travels a greater distance during each cycle, yet its frequency remains the same. How is this possible? Explain in terms of the mass’s speed and the restoring force.

Exercise 12

Explain why you expect an object made of a stiff material to vibrate at a higher frequency than a similar object made of a spongy material.

Exercise 13

As you pass a freight truck with a trailer on a highway, you notice that its trailer is bouncing up and down slowly. Is it more likely that the trailer is heavily loaded or nearly empty? Explain your answer.

Exercise 14

Some people modify cars to be much closer to the ground than when manufactured. Should they install stiffer springs? Explain your answer.

Exercise 15

A cart oscillates on a horizontal spring with negligible friction. Describe how its kinetic energy and the spring’s potential energy change as it moves (a) from a turning point to equilibrium and (b) from equilibrium to the opposite turning point. What happens to their sum?

5.4: Forced Oscillations and Resonance

Exercise 16

A mass on a spring oscillates at 2 Hz when displaced and released. It is then driven by a repeating force whose frequency can be adjusted. (a) Distinguish the natural frequency from the driving frequency. (b) With weak damping, near what driving frequency would you expect the largest oscillations? Explain.

Exercise 17

Give additional examples of resonance phenomenon from everyday life.

Exercise 18

What “damps” the resonance of a forced oscillation? What are the dynamics of “damping” and what physical quantity is involved?

5.5: Waves

Exercise 19

Give one example of a transverse wave and another of a longitudinal wave, being careful to note the relative directions of the disturbance and wave propagation in each.

Exercise 20

A ribbon is tied to one point on a stretched horizontal rope. A transverse wave travels along the rope. Does the ribbon travel along the rope with the wave? Describe its motion and explain the distinction between wave speed and the speed of the ribbon.

Exercise 21

Compare the wave shown in Figure 5.5.3 with the position-versus-time graph in Figure 5.3.4. (a) Which shows repetition over distance, and which shows repetition over time? (b) Identify an interval representing one wavelength in the wave picture and one period in the position graph.

Exercise 22

What is the difference between propagation speed and the frequency of a wave? Does one or both affect wavelength? If so, how?

Exercise 23

During one period, how far does a crest of a periodic traveling wave move? Use your answer to explain the relationship between wave speed, wavelength, and period.

Exercise 24

Given a wave of frequency

f

traveling with wave speed

v_{w}

, what happens to the wave speed if the frequency doubles? What happens to the wavelength?

5.6: Wave Interference: Standing Waves and Beats

Exercise 25

Two pulses travel toward one another on a rope. At one instant, one pulse alone would displace a particular point 3 cm upward, and the other alone would displace it 2 cm downward. (a) What is the displacement of that point when the pulses overlap? (b) After the pulses pass through one another, do they continue traveling? Explain using the superposition principle.

Exercise 26

How does destructive interference occur? What do you see in a destructive interference?

Exercise 27

Explain how noise-cancelling headphones work in terms of wave interference phenomena.

Exercise 28

A student says, “Nothing moves in a standing wave; that is why it is called standing.” Is this correct? Explain what remains in place and describe the motion of the string at a node and at an antinode.

Exercise 29

Explain how nodes and antinodes form on standing waves.

Exercise 30

Compare the three standing-wave patterns shown in Figure 5.6.6 and Figure 5.6.7. Assume the same string length and tension for all three. (a) Rank their wavelengths from longest to shortest. (b) Rank their frequencies from lowest to highest. Explain how the shapes of the patterns support your answers.

Exercise 31

A guitar string has a fundamental frequency of 150 Hz. (a) What are the frequencies of its second and third harmonics? (b) How many antinodes would appear in the standing-wave pattern for each of these harmonics?

Exercise 32

A guitarist tightens a string without changing its vibrating length. The speed of waves on the string increases. For its fundamental standing wave, what happens to (a) the wavelength and (b) the frequency? Explain why the fixed length matters.

Exercise 33

In the beat phenomenon, the loudness of the sound combined from two sound waves “wobbles” (it increases and decreases over time). Explain how the loudness of the combined sound becomes louder then quieter, even though individual sound waves remain at the same loudness, or amplitude.

5.7: Sound

Exercise 34

A loudspeaker produces sound in air. (a) What are compressions and rarefactions? (b) Do the air molecules near the speaker travel all the way to a listener’s ear? Explain how the disturbance reaches the listener.

5.8: Speed of Sound, Frequency, and Wavelength

Exercise 35

How do sound vibrations of atoms differ from thermal motion?

Exercise 36

You see a flash of lightning and hear the thunder 3.0 s later. Taking the speed of sound as 340 m/s, estimate how far away the lightning struck. Explain why the light’s travel time can be neglected in this estimate.

Exercise 37

When sound passes from one medium to another where its propagation speed is different, does its frequency or wavelength change? Explain your answer briefly.

Exercise 38

Using 20 Hz to 20,000 Hz as the approximate human audible range, classify sounds with frequencies of 10 Hz, 1,000 Hz, and 40,000 Hz as infrasound, audible sound, or ultrasound. If all three travel through the same air, which has the shortest wavelength? Explain.

Exercise 39

For a high-fidelity speaker containing a woofer and a tweeter (see Figure 5.8.4 for example), why is the woofer larger than the tweeter?

5.9: Doppler Effect and Sonic Booms

Exercise 40

Explain Doppler shift. How does the measured frequency increase or decrease?

Exercise 41

An aircraft travels at 680 m/s where the local speed of sound is 340 m/s. (a) What is its Mach number? (b) Is it subsonic or supersonic? (c) Does a given Mach number always correspond to the same speed in meters per second? Explain.

Exercise 42

Due to efficiency considerations related to its bow wake, the supersonic transport aircraft must maintain a cruising speed that is a constant ratio to the speed of sound (a constant Mach number). If the aircraft flies from warm air into colder air, should it increase or decrease its speed? Explain your answer.

Exercise 43

When you hear a sonic boom, you often cannot see the plane that made it. Why is that?

Source sections: Introduction to Oscillatory Motion and Waves, Period and Frequency in Oscillations, Simple Harmonic Motion: A Special Periodic Motion, Forced Oscillations and Resonance, Waves, Wave Interference: Standing Waves and Beats, Sound, Speed of Sound, Frequency, and Wavelength, Doppler Effect and Sonic Booms. Credits and licenses appear with each source section.