Post on 18-Jan-2016
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
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
• Therefore, solutions of the wave equation should have a form
• Considering partial derivatives
)(),( vtxftxy
x
vtxf
xx
txy )(),(2
2
)(' vtxfx
)('' vtxf
t
vtxf
tt
txy )(),(2
2
)()(' vvtxft
2)('' vvtxf
The wave equation
• Therefore, solutions of the wave equation should have a form
• Considering partial derivatives
)(),( 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
Interference of waves
• 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
Interference of waves
• 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
Interference of waves
• 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
Chapter 16Problem 54
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
Chapter 17Problem 12
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
Chapter 17Problem 18
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
• Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument
• Guitar resonances at medium frequencies:
Musical instruments
• Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument
• 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
Chapter 17Problem 29
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
'
Chapter 17Problem 52
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
Answers to the even-numbered problems
Chapter 16:
Problem 24
198 Hz
Answers to the even-numbered problems
Chapter 16:
Problem 26
1.75 m/s
Answers to the even-numbered problems
Chapter 16:
Problem 30
(a) 82.8º; (b) 1.45 rad; (c) 0.23 wavelength
Answers to the even-numbered problems
Chapter 16:
Problem 46
260 Hz
Answers to the even-numbered problems
Chapter 17:
Problem 6
44 m
Answers to the even-numbered problems
Chapter 17:
Problem 8
(a) 1.50 Pa; (b) 158 Hz; (c) 2.22 m;(d) 350 m/s
Answers to the even-numbered problems
Chapter 17:
Problem 14
4.12 rad
Answers to the even-numbered problems
Chapter 17:
Problem 36
(a) 57.2 cm; (b) 42.9 cm
Answers to the even-numbered problems
Chapter 17:
Problem 50
zero