chapters 16, 17 waves. types of waves mechanical – governed by newton’s laws and exist in a...

Chapters 16, 17

Waves

Types of waves

• Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.)

• Electromagnetic – governed by electricity and magnetism equations, may exist without any medium

• Matter – governed by quantum mechanical equations

Types of waves

Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as:

• Transverse – if the direction of displacement is perpendicular to the direction of propagation

• Longitudinal – if the direction of displacement is parallel to the direction of propagation

The wave equation

• Let us consider transverse waves propagating without change in shape and with a constant wave

velocity v

• We will describe waves via vertical displacement

y(x,t)

• For an observer moving with the wave

the wave shape doesn’t depend on time y(x’) = f(x’)

The wave equation

For an observer at rest:

• the wave shape depends on time y(x,t)

• the reference frame linked to the wave is moving

with the velocity of the wave v

vtxx ' vtxx '

)()'( vtxfxf )(),( vtxftxy

The wave equation

• We considered a wave propagating with velocity v

• For a medium with isotropic (symmetric) properties, the wave equation should have a symmetric solution

for a wave propagating with velocity –v

)(),(1 vtxftxy

))((),(2 tvxftxy

)( vtxf

The wave equation

• Therefore, solutions of the wave equation should have a form

• Considering partial derivatives

)(),( vtxftxy

x

vtxf

x

txy

)(),(

x

vtx

vtx

vtxf

)(

)(

)()(' vtxf

t

vtxf

t

txy

)(),(

t

vtx

vtx

vtxf

)(

)(

)()()(' vvtxf

The wave equation



)(),( vtxftxy

x

vtxf

xx

txy )(),(2

2

)(' vtxfx

)('' vtxf

t

vtxf

tt

txy )(),(2

2

)()(' vvtxft

2)('' vvtxf

The wave equation



)(),( vtxftxy

)(''),(

2

2

vtxfx

txy

22

2

)(''),(

vvtxft

txy

2

22 ),(

x

txyv

2

22

2

2 ),(),(

x

txyv

t

txy

The wave equation

• The wave equation (not the only one having solutions of the form y(x,t) = f(x ± vt)):

• It works for longitudinal waves as well

• v is a constant and is determined by the properties of the medium. E.g., for a stretched string with linear

density μ = m/l under tension τ

v

2

22

2

2 ),(),(

x

txyv

t

txy

Superposition of waves

• Let us consider two different solutions of the wave equation

• Superposition principle – a sum of two solutions to the wave equation is a solution to the wave equation

21

22

21

2

x

yv

t

y

22

22

22

2

x

yv

t

y

22

22

21

22

22

2

21

2

x

yv

x

yv

t

y

t

y

221

22

221

2 )()(

x

yyv

t

yy

+

Superposition of waves

• Overlapping solutions of the wave equation algebraically add to produce a resultant (net) wave

• Overlapping solutions of the wave equation do not in any way alter the travel of each other

Chapter 16Problem 27

Reflection of waves at boundaries

• Within media with boundaries, solutions to the wave equation should satisfy boundary conditions. As a results, waves may be reflected from boundaries

• Hard reflection – a fixed zero value of deformation at the boundary – a reflected wave is inverted

• Soft reflection – a free value of deformation at the boundary – a reflected wave is not inverted

Sinusoidal waves

• One of the most characteristic solutions of the wave equation is a sinusoidal wave:

• ym - amplitude, φ - phase constant

)2/)(cos(

))(sin()(

vtxky

vtxkyvtxy

m

m

Wavelength

• “Freezing” the solution at t = 0 we obtain a

sinusoidal function of x:

• Wavelength λ – smallest distance (parallel to the direction of wave’s travel) between repetitions of the wave shape

))(cos(),( vtxkytxy m

)cos()0,( kxyxy m

Wave number

• On the other hand:

• Angular wave number: k = 2π / λ

)cos()0,( kxyxy m ))(cos( xkym

)cos( kkxym

)2cos()cos( kxkx /2k

Angular frequency

• Considering motion of the point at x = 0 we observe a simple harmonic motion (oscillation) :

• For simple harmonic motion (Chapter 15):

• Angular frequency ω

))(cos(),( vtxkytxy m

)cos(),0( kvtyty m )cos( kvtym

)cos()( tyty m

/2 vkv

Frequency, period

• Definitions of frequency and period are the same as for the case of rotational motion or simple harmonic motion:

• Therefore, for the wave velocity

2//1 Tf /2T

fTkv //

)cos(),( tkxytxy m

Chapter 16Problem 7

Interference of waves

• Interference – a phenomenon of combining waves, which follows from the superposition principle

• Considering two sinusoidal waves of the same amplitude, wavelength, and direction of propagation

• The resultant wave:

)cos(),(2 tkxytxy m)cos(),(1 tkxytxy m

),(),(),( 21 txytxytxy

)cos()cos( tkxytkxy mm

2

cos2

cos2coscos

)2/cos()2/cos(2 tkxym


• If φ = 0 (Fully constructive)

• If φ = π (Fully destructive)

• If φ = 2π/3 (Intermediate)

)2/cos()2/cos(2),( tkxytxy m

)cos(2),( tkxytxy m

0),( txy

)3/cos(

)3/cos(2),(

tkx

ytxy m

)3/cos( tkxym


• Considering two sinusoidal waves of the same amplitude, wavelength, but running in opposite directions

• The resultant wave:

)cos(),(2 tkxytxy m)cos(),(1 tkxytxy m

),(),(),( 21 txytxytxy

)cos()cos( tkxytkxy mm

2

cos2

cos2coscos

)2/cos()2/cos(2 tkxym


• If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions, their interference with each other produces a standing wave

)sin()sin(2),( kxtytxy m

...2,1,0

)21(

n

nkx

22

1

nx

Antinodes

1|sin| kx

tyy m sin2

...2,1,0

n

nkx

0sin kx

0y

2

nx

Nodes


cm 8.1H

Standing waves and resonance

• For a medium with fixed boundaries (hard reflection) standing waves can be generated because of the reflection from both boundaries: resonance

• Depending on the number of antinodes, different resonances can occur

Standing waves and resonance

• Resonance wavelengths

• Resonance frequencies

L2

2

2L

3

2L

...3,2,1,2

nn

L

v

f ...3,2,1,2

nL

nv

Harmonic series

• Harmonic series – collection of all possible modes - resonant oscillations (n – harmonic number)

• First harmonic (fundamental mode):

...3,2,1,2

nL

vnfn

L

vf

21

More about standing waves

• Longitudinal standing waves can also be produced

• Standing waves can be produced in 2 and 3 dimensions as well

Phasors

• For superposition of waves it is convenient to use phasors – vectors that have magnitude equal to the amplitude of the wave and rotating around the origin

• Two phase-shifted waves with the same frequency can be represented by phasors separated by a fixed angle

Phasors

• To obtain a resultant wave (add waves) one has to add phasors as vectors

• Using phasors one can add waves of different amplitudes

Rate of energy transmission

• As the wave travels it transports energy, even though the particles of the medium don’t propagate with the wave

• The average power of energy transmission for the sinusoidal solution of the wave equation

• Exact expression depends on the medium or the system through which the wave is propagating

vyP mavg22

Sound waves

• Sound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies detectable by human ears (between ~ 20 Hz and ~ 20 KHz)

• Ultrasound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies higher than detectable by human ears (> 20 KHz)

• Infrasound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies lower than detectable by human ears (< 20 Hz)

Speed of sound

• Speed of sound:

ρ – density of a medium, B – bulk modulus of a medium

• Traveling sound waves

B

v

V

VBP

)cos(

))(cos(),(

tkxs

vtxkstxs

m

m

Intensity of sound

• Intensity of sound – average rate of sound energy transmission per unit area

• For a sinusoidal traveling wave:

• Decibel scale

β – sound level; I0 = 10-12 W/m2 – lower limit of human

hearing

A

PI

22

2

1 mvsI

0

log)10(I

IdB

Sources of musical sound

• Music produced by musical instruments is a combination of sound waves with frequencies corresponding to a superposition of harmonics (resonances) of those musical instruments

• In a musical instrument, energy of resonant oscillations is transferred to a resonator of a fixed or adjustable geometry

Open pipe resonance

• In an open pipe soft reflection of the waves at the ends of the pipe (less effective than form the closed ends) produces standing waves

• Fundamental mode (first harmonic): n = 1

• Higher harmonics:

...3,2,12

,2

nL

vnf

n

L

Organ pipes

Organ pipes

• Organ pipes are open on one end and closed on the other

• For such pipes the resonance condition is modified:

L

vnf

n

L

nnL

4,

4

...5,3,1;4

Musical instruments

• The size of the musical instrument reflects the range of frequencies over which the instrument is designed to function

• Smaller size implies higher frequencies, larger size implies lower frequencies

Musical instruments

• Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument

• Guitar resonances (exaggerated) at low frequencies:

Musical instruments


• Guitar resonances at medium frequencies:

Musical instruments


• Guitar resonances at high frequencies:

Beats

• Beats – interference of two waves with close frequencies

tss m 11 cos

tss m 22 cos+ tstssss mm 2121 coscos

ttsm 2cos

2cos2 2121

Sound from a point source

• Point source – source with size negligible compared to the wavelength

• Point sources produce spherical waves

• Wavefronts – surfaces over which oscillations have the same value

• Rays – lines perpendicular to wavefronts indicating direction of travel of wavefronts

Interference of sound waves

• Far from the point source wavefronts can be approximated as planes – planar waves

• Phase difference and path length difference are related:

• Fully constructive interference

• Fully destructive interference

2212 LLL

,...2,1,0L

,...2

5,

2

3,

2

1

L

Variation of intensity with distance

• A single point emits sound isotropically – with equal intensity in all directions (mechanical energy of the sound wave is conserved)

• All the energy emitted by the source must pass through the surface of imaginary sphere of radius r

• Sound intensity

(inverse square law)

A

PI

24 r

Ps

Doppler effect

• Doppler effect – change in the frequency due to relative motion of a source and an observer (detector)

Andreas Christian Johann Doppler

(1803 -1853)

Doppler effect

• For a moving detector (ear) and a stationary source

• In the source (stationary) reference frame:Speed of detector is –vD

Speed of sound waves is v

• In the detector (moving) reference frame:Speed of detector is 0

Speed of sound waves is v + vD

fv v

f

'

'v

f

Dvv

f

v

v

vvf D

Doppler effect

• For a moving detector (ear) and a stationary source

• If the detector is moving away from the source:

• For both cases:

v

vvff D

'

v

vvff D

'

v

vvff D

'

Doppler effect

• For a stationary detector (ear) and a moving source

• In the detector (stationary) reference frame:

• In the moving (source) frame:

*'

v

f

*Svv

f

f

vv S*

Svv

vf

Doppler effect

• For a stationary detector and a moving source

• If the source is moving away from the detector:

• For both cases:

Svv

vff

'

Svv

vff

'

Svv

vff

'

Doppler effect

• For a moving detector and a moving source

• Doppler radar:

S

D

vv

vvff

'

Supersonic speeds

• For a source moving faster than the speed of soundthe wavefronts form the Mach cone

• Mach number

Ernst Mach(1838-1916)

v

vs

vt

tvssin

1

Supersonic speeds

• The Mach cone produces a sonic boom

Answers to the even-numbered problems

Chapter 16:

Problem 2

(a) 3.49 m−1; (b) 31.5 m/s


Chapter 16:

Problem 24

198 Hz


Chapter 16:

Problem 26

1.75 m/s


Chapter 16:

Problem 30

(a) 82.8º; (b) 1.45 rad; (c) 0.23 wavelength


Chapter 16:

Problem 46

260 Hz


Chapter 17:

Problem 6

44 m


Chapter 17:

Problem 8

(a) 1.50 Pa; (b) 158 Hz; (c) 2.22 m;(d) 350 m/s


Chapter 17:

Problem 14

4.12 rad


Chapter 17:

Problem 36

(a) 57.2 cm; (b) 42.9 cm


Chapter 17:

Problem 50

zero

chapters 16, 17 waves. types of waves mechanical – governed by newton’s laws and exist in a...

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