Nicholas J. Giordano

www.cengage.com/physics/giordano

Chapter 12 Waves

Wave Motion • A wave is a moving disturbance that transports

energy from one place to another without transporting matter

• Questions about waves • What is being disturbed? • How is it disturbed?

• The motion associated with a wave disturbance often has a repeating form, so wave motion has much in common with simple harmonic motion

Introduction

Waves, String Example

• One example of a wave is a disturbance on a string • Shaking the free end creates a disturbance that moves

horizontally along the string • A single shake creates a wave pulse • If the end of the string is shaken up and down in simple

harmonic motion, a periodic wave is produced Section 12.1

Waves, String Example cont. • The disturbances are examples of waves • Portions of the string are moving so there is kinetic

energy associated with the wave • There is elastic potential energy in the string as it

stretches • The wave carries this energy as it travels • The wave does not carry matter as it travels

• Pieces of the string do not move from one end of the string to the other

Section 12.1

Analysis of The Wave Pulse

• A single pulse propagates to the right

• The graph (part D in the figure shown) shows the displacement of point D on the string • It is perpendicular to

the direction of propagation

• The wave transports energy without transporting matter

Section 12.1

Wave Terminology • The “thing” being disturbed by the wave is its

medium • When the medium is a material substance, the wave

is a mechanical wave • In transverse waves the motion of the medium is

perpendicular to the direction of the propagation of the wave • The string was an example

• In longitudinal waves the motion of the medium is parallel to the direction of the propagation of the wave

Section 12.1

Example: Longitudinal Wave

• The spring is shaken back and forth in the horizontal direction

• At some places the coils are compressed

• At other places the coils are stretched

• This motion produces a longitudinal wave

Section 12.1

Describing Periodic Waves

• Assume a person is shaking the string so that the end is undergoing simple harmonic motion

• The crest is the maximum positive y displacement

• The trough is the maximum negative y displacement

Section 12.2

Periodic vs. Nonperiodic Waves • Nonperiodic waves

• The wave disturbance is limited to a small region of space

• Periodic waves • The wave extends over a very wide region of space • The displacement of the medium varies in a repeating

and often sinusoidal pattern • A periodic wave involves repeating motion as a

function of both space and time

Section 12.2

The Equation of a Wave • Assume the displacement generating the wave in the

string vibrates as a simple harmonic oscillator with yend = A sin (2 π ƒ t)

• The string’s displacement is given by • λ is the symbol for wavelength • This is a mathematical description of a periodic wave • It shows the transverse displacement y of a point on

the string as it varies with time and location

Section 12.2

More Wave Terminology • Periodic waves have a frequency

• The frequency is related to the “repeat time” • The period is the time that a point takes to go from a

crest to the next crest in its motion • Then ƒ = 1 / T

• Periodic waves have an amplitude • Wave crests have y = + A • Wave troughs have y = - A

Section 12.2

Wavelength

• The wavelength is the “repeat distance” of the wave

• Start at a given value of y

• Advance x by a distance equal to the wavelength and y will be at the same value again

Section 12.2

Periodic Wave, Summary • Periodic waves have both a repeat time and a repeat

distance • A periodic wave is a combination of two simple

harmonic motions • One is a function of time • The other is a function of space

Section 12.2

Speed of a Wave • The mathematical description of a wave contains

frequency, wavelength and amplitude • The speed of a wave is

• This is based on the definitions of period and

wavelength

Section 12.2

Direction of a Wave • To determine the direction of the wave, you can focus on

the motion of a crest • As x becomes larger, the wave has moved to the right

and the wave velocity is positive and its equation is

• The equation of a wave moving to the left and having a negative velocity is

Section 12.2

Displacement of the medium as a function of location (x) and time (t)

Amplitude

Frequency

Wavelength

Section 12.2

Interpreting the Equation of a Periodic Wave

Waves on a String • Waves on a string are mechanical waves • The medium that is disturbed is the string • For a transverse wave on a string, the speed of the

wave depends on the tension in the string and the string’s mass per unit length • Mass / length = μ • Tension will be denoted as FT to keep the tension

separate from the period • The speed of the wave is

Section 12.3

Waves on a String, cont. • The speed of the wave is independent of the

frequency of the wave • The frequency will be determined by how rapidly the

end of the string is shaken • The speed of transverse waves on a string is the

same for both periodic and nonperiodic waves

Section 12.3

Sound Waves

• Sound is a mechanical wave that can travel through almost any material • Travels in solids,

liquids, and gases • Assume a speaker is

used to generate the waves

Section 12.3

Sound Waves, cont. • The speaker moves back and forth in the horizontal

direction • As it moves, it collides with nearby air molecules • The x component of the velocity of the air molecules

is affected by the speaker • The displacement of the air molecules associated

with the sound wave is also along the x direction • The result is a longitudinal wave

Section 12.3

Speed of Sound Waves • The speed of sound depends on the properties of

the medium • At room temperature, the speed of sound in air is

approximately 343 m/s • The speed is independent of the frequency • The speed applies to both periodic and nonperiodic

waves • Sound waves in a liquid or solid are also longitudinal • The speed of sound is generally smallest for gases

and highest for solids

Section 12.3

Waves in a Solid

• Solids can support both longitudinal and transverse waves

• The longitudinal waves are considered sound waves

• The speed of the sound depends on the solid’s elastic properties

Section 12.3

Speed of Sound in a Solid • For a thin bar of material, the speed of sound is

given by

• The speed of a transverse wave is more complicated and depends on the shear modulus and other elastic constants

• In general, the speed of the transverse wave is slower than the speed of longitudinal waves

Section 12.3

Transverse Waves • Transverse waves can travel through solids • They cannot travel through liquids or gases

• The displacements in transverse waves involve a shearing motion

• Liquids and gases flow and there is no restoring force to produce the oscillations necessary for a transverse wave

Section 12.3

Electromagnetic Waves • Electromagnetic (em) waves are not mechanical

waves • They are electric and magnetic disturbances that can

propagate even in a vacuum • No mechanical medium is required

• The electric and magnetic fields are always perpendicular to the direction of propagation • So they are transverse waves

• EM waves are classified according to their frequency • The speed of an em wave in a vacuum is 3.00 x 108

m/s • It is independent of the frequency of the wave

Section 12.3

Speed of Waves, Summary • The speed of a wave depends on the properties of

the medium through which it travels • The speed varies widely

• From slow waves on a string • To very fast em waves

• Generally, the wave speed is independent of both frequency and amplitude • There are cases in light and optics where the speed

does depend on the frequency • The speed is the same for periodic and nonperiodic

waves

Section 12.3

Water Waves

• A water wave can be generated by dropping a rock onto the surface

• The waves propagate outward

Section 12.3

Water Waves, cont.

• The motion of the water’s surface is both transverse and longitudinal

• A bug on the surface moves up and down as well as backward and forward

Section 12.3

Wave Fronts: Spherical Waves

• A spherical wave travels away from its source in a three-dimensional fashion

• The wave crests form concentric spheres centered on the source • The crests are also

called wave fronts

Section 12.4

Spherical Waves, cont. • The direction of the wave propagation is always

perpendicular to the surface of a wave front • The direction is indicated by rays • Each wave carries energy as it travels away from the

source • Power measures the energy emitted by the source per

unit time • Units of power are J/s = W • W for Watt

Intensity • Intensity is the power carried by the wave over a unit

area of the wave front • SI units of intensity is W/m2 • Once a wave front is emitted, its energy remains the

same • The intensity falls as the wave moves farther from

the source • The area is becoming larger

Section 12.4

Intensity, cont. • At a distance r from the source, the surface area of

the sphere is 4πr2 • The intensity is

• The intensity falls with distance as

Section 12.4

Plane Waves

• Wave fronts are not always spherical • Another type is a plane wave • In a perfect plane wave, each crest and trough extend over an infinite

plane in space • The intensity is approximately constant over long propagation

distances • Intensity is ideally independent of distance

Section 12.4

Intensity and Amplitude • The intensity of a wave is related to its amplitude

• Spring example

• The potential energy is ½ k x2 • For a wave on a spring, the displacement is

proportional to the amplitude • Therefore, the energy and intensity are proportional to

the square of the amplitude

Section 12.4

Superposition • Waves generally propagate independently of one

another • A wave can travel though a particular region of

space without affecting the motion of another wave traveling though the same region

• This is due to the Principle of Superposition • When two (or more) waves are present, the

displacement of the medium is equal to the sum of the displacements of the individual waves

• The presence of one wave does not affect the frequency, amplitude, or velocity of the other wave

Section 12.5

Constructive Interference

• Two wave pulses are traveling toward each other

• They have equal and positive amplitudes

• At C, the two waves completely overlap and the amplitude is twice the amplitude of the individual waves

• The emerging pulses are unchanged

• This is an example of constructive interference

Section 12.5

Destructive Interference

• Two pulses are traveling toward each other

• They have equal and opposite amplitudes

• At C, the two waves completely overlap, total displacement is zero

• The emerging pulses are unchanged

• This is an example of destructive interference

Section 12.5

Interference • Constructive interference causes the waves to

produce a displacement that is larger than the displacements of either of the individual waves

• Destructive interference causes the waves to produce a displacement that is smaller than the displacements of either of the individual waves

• In either case, the energy of each wave is contained in the kinetic energy of the medium

• The waves can interfere, even destructively, and still carry energy independently

Section 12.5

Interference of Periodic Waves

• The crests of the waves travel away from the initial source • There is constructive interference where the wave crests

overlap • There is destructive interference where a crest and trough

overlap • The result shows an interference pattern with regions of

constructive and destructive interference Section 12.5

Reflection

• Reflection changes the propagation direction of the wave

• Rays can be used to indicate the direction of energy flow

• The rays change direction when a wave reflects from the boundary of the medium

• The wave is inverted as it reflects from a fixed end

Section 12.6

Example: Reflection of Light

• The light wave from a laser reflects from a mirror

Section 12.6

Reflection – Light Ray Details

• The rays make an initial angle of θi with a line drawn perpendicular to the surface

• The perpendicular component of the wave’s velocity reverses direction

• The parallel component of the wave’s velocity is not affected by the reflection

• The angle of incidence will equal the angle of reflection: θi = θr

Section 12.6

Reflection – Free Surface

• The end of the string is attached to a ring that is free to move up and down

• When the wave is reflected, it is not inverted

• The properties of the medium at the boundary will determine if the reflected wave will be inverted or not

Section 12.6

Radar

• An application of wave reflection is radar

• A radio wave pulse is sent from a transmitting antenna and reflects from some distant object

• A portion of the reflected wave will arrive back at the original transmitter, where it is detected

Section 12.6

Radar, cont. • Radar determines the distance to the object by

measuring the time delay between the original and reflected signals

• By using a rotating antenna, the direction of the object can also be detected

• The amplitude of the reflected rays gives information about the size of the object • A larger object reflects more of the wave energy and

gives a larger signal at the detecting antenna

Section 12.6

Refraction

• If the rays follow bent paths in a medium, they are said to be refracted

• The frequency of the wave stays the same • It is determined by the source

• The change in direction of the wave is due to a change in its speed Section 12.7

Standing Waves • Waves may travel back and forth along a string of

length L • If the string has both ends held in fixed positions, the

displacement at both ends must be zero • These conditions can be satisfied by a periodic wave

only for certain wavelengths • For these wavelengths, a standing wave can be

produced • It is called a standing wave because the outline of the

wave appears stationary

Section 12.8

Standing Waves, cont.

• The standing wave is obtained by the interference of two waves traveling in opposite directions

• The waves travel along the string and are reflected from the ends

Section 12.8

Standing Waves, final

• Points where the string displacement is zero are called nodes

• Points where the displacement is largest are called antinodes

• Many standing waves may “fit” into the length of the string

Section 12.8

Harmonics • The longest possible wavelength corresponds to the

smallest possible frequency • This frequency is called the fundamental

frequency, ƒ1 • The next longest wavelength is called the second

harmonic • The pattern of wavelengths and frequencies is

Section 12.8

Harmonics, cont • Combining the frequency and wavelength equations

gives other expressions for the frequency: • This is for standing waves on a string with fixed ends

• The allowed standing wave frequencies are integer multiples of the fundamental frequency

Section 12.8

Musical Tones • Many musical instruments use strings as a vibrating

element • Your fingers press down on the string and changes its

length • The string vibrates with all the possible standing wave

pattern frequencies • The pitch of note is determined by its fundamental

frequency • Two notes whose fundamental frequencies differ by

a factor of 2 are said to be separated by an octave

Section 12.8

Seismic Waves

• Seismic waves propagate through the Earth • Their source can be any large mechanical

disturbance such as an earthquake • There are three types of seismic waves

Section 12.9

Types of Seismic Waves • S waves

• S for shear • Transverse waves • The displacement of the solid Earth is perpendicular to

the direction of propagation • P waves

• P for pressure • Longitudinal sound waves

• Surface waves • Similar to water waves but travel through the surface

of the Earth • Seismic waves can be detected by a seismograph

Section 12.9

Structure of the Earth

• Seismic waves can help determine the interior structure of the Earth

• S waves do not propagate through the core • So the core contains a

liquid • Both S and P waves are

refracted

Section 12.9

Structure of the Earth, cont. • Analysis of the waves led to the following structure:

• Inner core • Outer core • Mantle • Crust

• Many characteristics of these sections also were obtained from the study of seismic waves

Section 12.9