Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a...
-
Upload
kristian-berry -
Category
Documents
-
view
225 -
download
0
Transcript of Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a...
![Page 1: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/1.jpg)
Chapters 16, 17
Waves
![Page 2: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/2.jpg)
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
![Page 3: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/3.jpg)
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
![Page 4: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/4.jpg)
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
![Page 5: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/5.jpg)
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’)
![Page 6: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/6.jpg)
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
![Page 7: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/7.jpg)
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
![Page 8: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/8.jpg)
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
![Page 9: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/9.jpg)
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
![Page 10: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/10.jpg)
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
![Page 11: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/11.jpg)
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
![Page 12: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/12.jpg)
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
+
![Page 13: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/13.jpg)
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
![Page 14: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/14.jpg)
Chapter 16Problem 27
![Page 15: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/15.jpg)
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
![Page 16: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/16.jpg)
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
![Page 17: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/17.jpg)
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
![Page 18: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/18.jpg)
Wave number
• On the other hand:
• Angular wave number: k = 2π / λ
)cos()0,( kxyxy m ))(cos( xkym
)cos( kkxym
)2cos()cos( kxkx /2k
![Page 19: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/19.jpg)
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
![Page 20: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/20.jpg)
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
![Page 21: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/21.jpg)
Chapter 16Problem 7
![Page 22: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/22.jpg)
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
![Page 23: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/23.jpg)
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
![Page 24: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/24.jpg)
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
![Page 25: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/25.jpg)
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
![Page 26: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/26.jpg)
Chapter 16Problem 54
cm 8.1H
![Page 27: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/27.jpg)
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
![Page 28: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/28.jpg)
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
![Page 29: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/29.jpg)
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
![Page 30: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/30.jpg)
More about standing waves
• Longitudinal standing waves can also be produced
• Standing waves can be produced in 2 and 3 dimensions as well
![Page 31: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/31.jpg)
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
![Page 32: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/32.jpg)
Phasors
• To obtain a resultant wave (add waves) one has to add phasors as vectors
• Using phasors one can add waves of different amplitudes
![Page 33: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/33.jpg)
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
![Page 34: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/34.jpg)
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)
![Page 35: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/35.jpg)
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
![Page 36: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/36.jpg)
Chapter 17Problem 12
![Page 37: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/37.jpg)
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
![Page 38: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/38.jpg)
Chapter 17Problem 18
![Page 39: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/39.jpg)
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
![Page 40: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/40.jpg)
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
![Page 41: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/41.jpg)
Organ pipes
![Page 42: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/42.jpg)
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
![Page 43: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/43.jpg)
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
![Page 44: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/44.jpg)
Musical instruments
• Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument
• Guitar resonances (exaggerated) at low frequencies:
![Page 45: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/45.jpg)
Musical instruments
• Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument
• Guitar resonances at medium frequencies:
![Page 46: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/46.jpg)
Musical instruments
• Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument
• Guitar resonances at high frequencies:
![Page 47: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/47.jpg)
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
![Page 48: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/48.jpg)
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
![Page 49: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/49.jpg)
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
![Page 50: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/50.jpg)
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
![Page 51: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/51.jpg)
Chapter 17Problem 29
![Page 52: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/52.jpg)
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)
![Page 53: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/53.jpg)
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
![Page 54: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/54.jpg)
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
'
![Page 55: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/55.jpg)
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
![Page 56: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/56.jpg)
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
'
![Page 57: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/57.jpg)
Doppler effect
• For a moving detector and a moving source
• Doppler radar:
S
D
vv
vvff
'
![Page 58: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/58.jpg)
Chapter 17Problem 52
![Page 59: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/59.jpg)
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
![Page 60: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/60.jpg)
Supersonic speeds
• The Mach cone produces a sonic boom
![Page 61: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/61.jpg)
Answers to the even-numbered problems
Chapter 16:
Problem 2
(a) 3.49 m−1; (b) 31.5 m/s
![Page 62: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/62.jpg)
Answers to the even-numbered problems
Chapter 16:
Problem 24
198 Hz
![Page 63: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/63.jpg)
Answers to the even-numbered problems
Chapter 16:
Problem 26
1.75 m/s
![Page 64: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/64.jpg)
Answers to the even-numbered problems
Chapter 16:
Problem 30
(a) 82.8º; (b) 1.45 rad; (c) 0.23 wavelength
![Page 65: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/65.jpg)
Answers to the even-numbered problems
Chapter 16:
Problem 46
260 Hz
![Page 66: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/66.jpg)
Answers to the even-numbered problems
Chapter 17:
Problem 6
44 m
![Page 67: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/67.jpg)
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
![Page 68: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/68.jpg)
Answers to the even-numbered problems
Chapter 17:
Problem 14
4.12 rad
![Page 69: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/69.jpg)
Answers to the even-numbered problems
Chapter 17:
Problem 36
(a) 57.2 cm; (b) 42.9 cm
![Page 70: Chapters 16, 17 Waves. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic –](https://reader035.fdocuments.in/reader035/viewer/2022062409/5697bfd01a28abf838caad3f/html5/thumbnails/70.jpg)
Answers to the even-numbered problems
Chapter 17:
Problem 50
zero