Chapter 14 Lecture

Slide 14-1

Essential University Physics Richard Wolfson

2nd Edition

© 2012 Pearson Education, Inc.

Wave Motion

Slide 14-2 © 2012 Pearson Education, Inc.

In this lecture you’ll learn

• To explain waves as traveling

disturbances that transport

energy but not matter

• To describe waves

quantitatively in terms of

frequency, period, wavelength,

and amplitude

• To describe specific types of

waves

– Waves on strings

– Sound waves

• To describe interference,

reflection, and standing waves

• To describe the Doppler effect

and shock waves

Slide 14-3 © 2012 Pearson Education, Inc.

What’s a Wave?

• A wave is a traveling disturbance that transports

energy but not matter.

– Mechanical waves are disturbances of a material

medium.

• The medium moves briefly as the wave goes by, but the

medium itself isn’t transported any distance.

• The wave propagates as the disturbance of the medium is

communicated to adjacent parts of the medium.

– Electromagnetic waves, including light, do not require a

medium.

• Nevertheless, they share many of the properties of

mechanical waves.

Slide 14-4 © 2012 Pearson Education, Inc.

Longitudinal and Transverse Waves

• In a longitudinal wave, the

disturbance is parallel to the

wave motion.

• In a transverse wave,

the disturbance is

perpendicular to the wave

motion.

• Some waves, like surface

waves on water, involve both

longitudinal and transverse

motions.

Longitudinal wave on a mass-spring system:

Transverse wave on a mass-spring system:

Slide 14-5 © 2012 Pearson Education, Inc.

Properties of Waves

• Wavelength is the distance over which a wave repeats in space.

• Period T is the time for a complete cycle of the wave at a fixed position:

– Frequency f = 1/T

• Amplitude A is the peak value of the wave disturbance.

• Wave speed is the rate at which the wave propagates:

v = /T = f

• Wave equation: 2 2

2 2 2

1y y

x v t

Slide 14-6 © 2012 Pearson Education, Inc.

Simple Harmonic Waves

• A simple harmonic wave has a sinusoidal shape.

– A simple harmonic wave is described by a sinusoidal

function of space and time:

• y measures the wave disturbance at position x and time t.

• k = 2p / is the wave number, a measure of the rate at

which the wave varies in space.

• = 2pf = 2p /T is the angular frequency, a measure of the

rate at which the wave varies in time.

• The wave speed is v = f = /k.

y x t A kx t

These two waves have the same

speed. How do their wavenumbers,

angular frequencies, and periods

compare?

Slide 14-7 © 2012 Pearson Education, Inc.

Waves on Strings

• On strings, fibers, long springs, cables, wires, etc.,

tension provides the restoring force that helps

transverse waves propagate.

– Newton’s second law gives

where F is the tension and

is the mass per unit length.

– The speed of such waves is

v =F

m

2 222

2 2mv R v

F vR R

Slide 14-8 © 2012 Pearson Education, Inc.

Wave Power and Intensity

• The power carried by a wave is proportional to the wave

speed and to the square of the wave amplitude.

– Details depend on the type of wave; for waves on a string, the

average power is P = 1

2mw

2A

2v.

• Wave intensity is the power per unit

area.

– In a plane wave, the intensity remains

constant.

• The plane wave is a good

approximation to real waves far

from their source.

– A spherical wave spreads in three

dimensions, so its intensity drops as

the inverse square of the distance

from its source:

I =

P

A=

P

4pr2

Slide 14-9 © 2012 Pearson Education, Inc.

Sound

• Sound waves are longitudinal mechanical waves that

propagate through gases, liquids, and solids.

– Sound waves in air involve small

changes in air pressure and density,

associated with back-and-forth motion

of the air as the wave passes.

– Sound intensity levels are measured

in decibels, a logarithmic unit based

on comparison with a reference

intensity I0= 10–12 W/m2:

– The human ear

responds to a

broad range of

sound intensities

and frequencies.

b = 10 log I I

0( ).

Slide 14-10 © 2012 Pearson Education, Inc.

Wave Interference

• Unlike particles, two waves can be in the same place at

the same time.

• When they are, they interfere.

– In most cases, the waves superpose, or simply add.

• When wave crests coincide, the interference is constructive.

• When crests coincide with troughs, the interference is

destructive.

Slide 14-11 © 2012 Pearson Education, Inc.

Interference Phenomena

• When waves of slightly different

frequencies interfere, they

alternate between constructive

and destructive interference.

– This results in beats at the difference

of their frequencies.

• Interference from two closely

spaced sources results in patterns

of high- and low-amplitude waves.

– The photo shows such an

interference pattern with water waves.

Slide 14-12 © 2012 Pearson Education, Inc.

Wave Reflection

• Waves reflect at an interface

with a different medium.

– The outgoing wave interferes with

the incoming wave.

– The reflected wave is inverted,

depending on properties of the

second medium.

– The diagram shows waves on a

string reflecting at clamped and

free ends.

– More generally, a wave is partially

reflected and partially transmitted

at an interface between different

media.

Slide 14-13 © 2012 Pearson Education, Inc.

Partial Reflection and Refraction

• Partial reflection of light

occurs at an interface

between air and glass.

• Waves striking an interface

at an oblique angle undergo

refraction, a change in the

direction of propagation.

Slide 14-14 © 2012 Pearson Education, Inc.

Standing Waves

• Waves on a confined medium reflect at both ends.

– The result is standing waves that oscillate but don’t propagate.

– The length of the medium restricts the allowed wavelengths and

frequencies to discrete values.

On a string

clamped at one

end, the string

length must be an

odd integer

multiple of a half-

wavelength:

L = m/4,

with m odd.

On a string

clamped at both

ends, the string

length must be an

integer multiple of

a half-wavelength:

L = m/2,

with m an integer.

Slide 14-15 © 2012 Pearson Education, Inc.

Standing Waves in Musical Instruments

• Stringed instruments are analogous to the strings of the

previous slide: The string length determines the allowed

wavelengths and, together with the wave speed, the allowed

frequencies.

• Wind instruments are analogous, with sound waves in their

air columns.

– Wind instruments are typically open at one end or both.

Slide 14-16 © 2012 Pearson Education, Inc.

The Doppler effect

• When a wave source moves through the wave medium, a stationary observer experiences a shift in wavelength and frequency. – The frequency increases for an approaching source.

– The frequency decreases for a receding source.

– The shifted frequencies are given by • Here u is the source speed and v is the wave speed. • A similar effect occurs for a moving observer, but there’s no wavelength

shift. • The Doppler effect for light is similar but slightly different because light

has no medium. The formula above applies only for speeds much less than light.

1 .f f u v

Slide 14-17 © 2012 Pearson Education, Inc.

Shock Waves

• Shock waves occur when a wave source moves through

the medium at a speed u greater than the wave speed v.

– The ratio u/v is called the Mach number.

– Mach angle is defined by sin = u/v.

– Examples include sonic booms from aircraft, wakes of boats,

and astrophysical bodies moving through interplanetary and

interstellar gas.

Slide 14-18 © 2012 Pearson Education, Inc.

Summary

• A wave is a traveling disturbance that carries energy but not matter. – Mechanical waves involve the disturbance of a material medium.

• These include sound waves.

– Electromagnetic waves, including light, have no medium.

– Simple harmonic waves are sinusoidal in shape.

– The speed of a wave follows from its frequency and wavelength or from its angular frequency and wavenumber: v = f = /k.

– Important wave phenomena include

• Reflection and refraction • Interference • Standing waves • The Doppler effect • Shock waves

y x t A kx t