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PreClass Notes: Chapter 14, Sections 14.1-

14.4

• From Essential University Physics 3rd Edition

• by Richard Wolfson, Middlebury College

• ©2016 by Pearson Education, Inc.

• Narration and extra little notes by Jason Harlow,

University of Toronto

• This video is meant for University of Toronto

students taking PHY131.

Outline

A wave involves“a disturbance

that moves or propagates through

space. The disturbance carries

energy, but not matter. Air doesn’t

move from your mouth to a

listeners ear, but sound energy

does.”– R.Wolfson

• Wave Properties

• The Wave Equation

• Waves on a string

• Wave Power and Intensity

• Sound Waves

[Image of waves in a field of grass from https://threeriversdeep.wordpress.com/2014/07/ ]

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What is 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.

• Electromagnetic waves,including light, do not require a medium.

[image from https://webspace.utexas.edu/cokerwr/www/index.html/waves.html ©1999 by Daniel A. Russell ]

Transverse Waves

• In a transverse wave, the disturbance is perpendicular

to the wave motion.

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Longitudinal Waves

• In a longitudinal wave, the disturbance is parallel to the

wave motion.

Sound is an

longitudinal wave.

Properties of Continuous Waves

• Wavelength λ is the

distance over which

a wave repeats in

space.

• Period T is the time

for a complete

oscillation of the

wave at a fixed

position.

• Frequency f is the

number of wave

cycles per unit time:

f = 1/T

• Amplitude A is the

maximum value of

the wave

disturbance.

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Wave Speed

• Wave speed is the

rate at which the

wave propagates.

• Wave speed,

wavelength, period,

and frequency are

related:

Simple Harmonic Waves

• 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 = 2π/λ is the wave number, a measure of the rate at

which the wave varies in space.

• ω = 2π/T is the angular frequency, a measure of the rate

at which the wave varies in time.

• The ± is written so we can describe a wave going in the

+x direction (− sign) or the −x direction (+ sign).

• The wave speed is:

y x,t( ) = Acos kx ±wt( )

𝑣 = 𝜆𝑓 =𝜔

𝑘

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Got It? [1 of 5]

• These two waves have the same speed. Which

has the greater amplitude?

A

B

Got It? [2 of 5]

• These two waves have the same speed. Which

has the greater wavelength?

A

B

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Got It? [3 of 5]

• These two waves have the same speed. Which

has the greater period?

A

B

Got It? [4 of 5]

• These two waves have the same speed. Which

has the greater wave number?

A

B

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Got It? [5 of 5]

• These two waves have the same speed. Which

has the greater frequency?

A

B

The Wave Equation

• Many different types of media can support the

propagation of waves.

– Analysis of disturbances in different media often

result in the following equation (which is known as the

one-dimensional wave equation):

– This equation relates the space and time derivatives

of the disturbed quantity, y. Note that v is the wave

speed, where:

– It can be shown that any function of the form

satisfies the wave equation.

¶2 y

¶x2=

1

v2

¶2 y

¶t2

v =l

T= l f

y = f (x ± vt)

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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 μ

is the mass per unit length.

• The speed of such waves is

2Fq =mv2

R=

2qRmv2

R= 2qmv2

v =F

m

Wave Power

• The power carried by a wave is proportional to

the wave speed and to the square of the wave

amplitude.

– For waves on a string, the average

power is:

P = 1

2 mw2A

2v

• Other types of waves have similar equations for

average power, always dependent on the

square of the wave amplitude, 𝐴2.

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Wave Intensity

• Wave intensity is the

power crossing

a unit perpendicular area.

– In a plane wave, the

intensity remains

constant.

– 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

4p r2

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.

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The Speed of Sound

• The speed of sound in a gas depends on the

background pressure P, density ρ, and a factor γ

that is determined by the number of atoms that

form a typical gas molecule:

v =g P

r

• The speed of sound is determined by the air and

is not dependent upon the amplitude, frequency,

or wavelength of the sound itself.

• In dry air at atmospheric pressure and room

temperature, the speed of sound is v = 343 m/s.

Human Hearing and the Decibel

• The human ear responds to a broad range of sound

intensities and frequencies

– The audible range extends from about 20 Hz to 20

kHz in frequency and over 12 orders of magnitude in

intensity

– The sound intensity level β is measured in decibels:

– The decibel is a logarithmic unit based on a

comparison to the nominal threshold of hearing:

b = 10 log I I0( )

I0 = 10-12 W / m2

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Human Hearing and the Decibel