page 480. problem 33 the amplitude of an oscillator ...€¦ · the amplitude of an oscillator...
TRANSCRIPT
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Review question chapter 14
Page 480. problem 33
The amplitude of an oscillator decreases to 36.8%...
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Traveling Waves and Sound
• Mechanical and Field waves
• Transverse and longitudinal waves
• Wave speed
• Sinusoidal waves
• Wave intensity
• Sound waves
• Doppler effect and shock waves
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Traveling Waves
Sound waves
Doppler effect
Shock waves
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Mechanical and Field waves
• Mechanical waves travel through a well defined medium, e.g. sound through air, ocean waves across water.
• Field waves can travel through a vacuum –such as EM waves (radio, light, X-rays), and quantum waves, gravitational waves.
• We are going to cover mechanical waves.
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Two types of waves
Transverse waves – medium moves perpendicular to the direction of the wave.
Longitudinal waves – medium moves parallel.
Transverse – ropes, water, earthquakes (S), (light)
Longitudinal – sound, compression springs, earthquakes (P)
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Traveling Waves
• Traveling waves transfer energy from one point to another.
• They travel through a medium which does not get transferred from the source to destination.
Pulse Wave
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Wave speed
• The speed of the wave is a property of the medium – tension of the rope or spring, pressure of the air.
• For perfect systems, the speed does not depend on the frequency or amplitude of the wave.
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Wave speed in a string
The speed of a wave increases with the tension on the string, Ts, measured in Newtons.
The speed decreases with the linear density of the string, μ. where
Linear density, μ , has units kg/m, and is the mass per unit length.
L
m
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Wave speed in a string
It can be shown that the speed of a wave along a string is
Where Ts is the tension (in N) on the string, and μ is the mass per unit length (m/L in kgm-1) of the string.
String tension and linear density
s
string
Tv
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Sound Waves
Sound waves are longitudinal waves in a medium, where the molecules or atoms are compressed and rarefied
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Speed of sound in a gas
The speed of sound in a gas is related to the average speed of individual molecules in that gas (Chapter 11)
T is temperature (in Kelvin), kB is Boltzmann’s constant and m the mass of the molecules.
m
Tkv B
rms
3
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Speed of sound in a gas
It can be shown, that the speed of sound in a gas is
T is temperature (in Kelvin), R is the ideal gas constant (8.31JK/mol) and M is the molar mass of the gas. The adiabatic index, γ, depends on the gas – 1.4 for Air, 1.7 for Helium.
M
RT
m
Tkv B
sound
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Speed of sound in solids
In general, the speed of sound waves is related to the coefficient of stiffness and the density
C is the coefficient of stiffness, ρ the density (Mass/Volume).
Speed increases with the strength of the molecular bonds, and decreases with the mass of the molecules
Cv
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Describing the wave
The wave travels through space (x) and time (t)– we want to describe the displacement (y) as a function of f(x,t).
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Describing the wave
The wave travels through space (x) and time (t)– we want to describe the displacement (y) as a function of f(x,t).
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Mapping a Sinusoidal Wave
From Ch. 14 we found that the displacement, y(t) is
T
tAty 2cos)(
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Mapping a Sinusoidal Wave
If we look along x, we see that y(x) as well
Where λ is the wavelength.
xAxy 2cos)(
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Mapping a Sinusoidal Wave
Combining these we get
For a wave traveling to the right.
T
txAtxy
2cos),(
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Mapping a Sinusoidal Wave
For a wave traveling to the left, the sign on the time gets flipped:
For a wave traveling to the left.
T
txAtxy
2cos),(
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Wavelength, λ, Frequency, f, and Speed, v
• If we wait at a point, we will see f waves pass by per second.
• This distance is fλ per second.
• So the speed of the wave, v=fλ.
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Energy in Sound waves
The intensity, I, of a wave, is defined as the power per unit area:
Units of J s-1 m-2.
A
PI
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Energy in Sound waves
For a point source – the area A, is the area of a sphere:
Units of J s-1 m-2.
2
source
4 r
PI
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Energy in Sound waves
The ratio of intensities for a point source goes as the 1/r2 law
Units of J s-1 m-2.
2
1
2
2
2
1
r
r
I
I
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Measuring sound intensity
• Human perception of sound is logarithmic
• Increasing the sound intensity by x10 increases the perceived loudness by x2
• The ratio of the quietest to the loudest sound corresponds to an intensity ratio of 1012
• So we use a logarithmic scale, called deciBels, or dB (after Alexander Graham Bell)
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Measuring sound intensity
The decibel is defined as
Where I0 = 1.0x10-12 W/m2, which is the quietest sound that the “average” person can hear 0dB
0
10log)10(I
IdB
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Measuring sound intensity
Relationship between intensity (W/m2) and dB
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Decibels and calculations
• Decibels are logarithmic. To calculate the effects of sound amplifiers or sound absorbers on the intensity of a sound, we use addition and subtraction.
• If a sound is attenuated by 3dB, a sound intensity of 10dB will be 7dB.
• If the same sound is amplified by 3dB, it would be 13dB.
• Inew=Iold + Ichange
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The Doppler Effect
A stationary listener will experience a change in pitch or frequency from a moving sound source.
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The Doppler Effect
• The human ear detects the frequency or sound – the rate of pressure pulses.
• The speed of sound does not change – it’s a property of the medium, and that’s not moving.
• The frequency of the pressure pulses will increase.
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Doppler Effect
To calculate the change in frequency, we can look at the change in the wavelength:
Where λ+ is the new wavelength, v the speed of sound, vs is the speed of the noise source
v
vs00
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Doppler Effect
The change in frequency for a source moving towards the listener:
Where f+ is the new frequency, v the speed of sound, vs is the speed of the noise source
vv
ff
s
1
0
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Doppler Effect
The change in frequency for a source moving away from the listener:
Where f- is the new frequency, v the speed of sound, vs is the speed of the noise source
vv
ff
s
1
0
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The Doppler Effect
• Similarly a moving observer experiences a change in the frequency of the pressure pulses
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Doppler Effect
The change in frequency for a listener when the listener is moving:
Where f+, f- are the new frequencies, v the speed of sound, vobs is the speed of the observer.
0
0
)/1(
)/1(
fvvf
fvvf
obs
obs
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Uses of Doppler shifts
• Blood speed monitors sometimes part of an ultrasound monitor
• Galaxy red shifts –the Doppler shift can be seen in EM radiation
• Radar guns
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Shock Waves
Shock waves happen when objects travel near or faster than the speed of sound in that medium.
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Shock Waves
Shock waves or sonic booms happen when objects travel near or faster than the speed of sound in that medium.
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Shock waves for EM radiation
• How can something travel faster than light ?
• Inside a medium, such as clear plastic or water, high energy particles can travel faster than the speed of light in that medium
• Called Čerenkov (Cherenkov) radiation
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Summary
• Mechanical and Field waves
• Transverse and longitudinal waves
• Wave speed
• Sinusoidal waves
• Wave intensity
• Sound waves
• Doppler effect and shock waves
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Homework problems
Chapter 15 Problems
42, 43, 52, 53, 65, 66, 68, 69.