1 mechanical waves ch 21-23. 2 waves a wave is a disturbance in a medium which carries energy from...
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1
Mechanical Waves
Ch 21-23
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Waves
A wave is a disturbance in a medium
which carries energy from one point to another
without the transport of matter.The medium allows the disturbance to
propagate.
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Transverse Wave
Particles oscillate at right angles to the direction of motion.
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Longitudinal Waves
Particles oscillate parallel to the direction of motion.
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Periodic Waves & Pulses
A wave pulse is a single disturbance.
A periodic wave is a series of disturbances or wave train.
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Transverse Wave Speed
Determined by the medium and its properties.
elasticity or restoring force inertia
v =Restoring force factor
Inertia Factor
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Wave on a mediumwith tension.
String, rope, wire, etc…T is the tension, & is the linear density,
= m/L = mass per unit length.
T
v
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Waves
Speed:
v = f =
T
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Wave Terminology
Frequency (f) - cycles per second. (Hz)Period (T) - Seconds per cycle.Amplitude (A) - Maximum displacement
from equilibrium.The distance that a wave travels in one
period is the wavelength ().
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Example 1
A wave travels along a string. The time for a particular point to move from a maximum displacement to zero is 0.170 s. The wavelength is 1.40 m. What are the period, frequency, and wave speed?
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Example 1 continued
It takes 0.680 s for one cycle, so T = 0.680 s
f = 1/T, so f = 1.47 Hz
fv m 40.1Hz 47.1s
m06.2
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Example 2
What is the speed of a transverse wave in a rope of length 2.00 m and mass 60.0 g under a tension of 500 N?
T
v L
mT
m
TL
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Example 2 continued
kg 060.0
m 00.2N 500v
m/s 129v
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Polarization
Most transverse waves are linearly polarizedThey either move just up and down
Vertically polarizedOr just side to side
Horizontally polarized
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Circular polarization
If we combine two perpendicular waves that have equal amplitude but are out of step by a quarter-cycle, the resulting wave is circularly polarized.
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Polarizing filters
Only let through waves that are polarized one way.
Like passing a rope through a slot in a board – only waves in the direction of the slot will get through.
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Longitudinal Wave Speed
Depends on the pressure change and the fractional volume change
Where is the density. B is the bulk modulus of a fluid. Y is young’s modulus for a solid. See tables 12-1 and 12-2. B = 1/k
B
v Y
v
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Longitudinal waves
Don’t have polarizationWhen the frequency is within the range
of human hearing, it is called sound.
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Sound waves in gases
Temperature doesn’t remain constant as sound waves move through air.
So, we use the equation
Where is the ratio of heat capacities (ch 18), R is the ideal gas constant (8.314 J/mol∙K), T is temperature in K, and M is the molecular mass (ch 17).
M
RTv
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Sound waves
Humans can hear from about 20 Hz to about 20 000 Hz.
Air is not continuous – it consists of molecules.
Like a swarm of bees.Also sort of like wave/particle duality.
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Mathematical wave description
y(x, t) = A sin(t – kx)
(Motion to right)
or y(x, t) = A sin(t + kx)
(Motion to left)
k =2
= 2f =2
T
v = f =
k
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Reflection
When a wave comes to a boundary, it is reflected.
Imagine a string that is tied to a wall at one end. If we send a single wave pulse down the string,when it reaches the wall, it exerts an upward
force on the wall.
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Reflection
By Newton’s third law, the wall exerts a downward force that is equal in
magnitude.
This force generates a pulse at the wall, which travels back along the string in the opposite direction.
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Reflection
In a ‘hard’ reflection like this,
there must be a node at the wall
because the string is tied there.
The reflected pulse is inverted from the incident wave.
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Reflection
Now imagine that instead of being tied to a wall
the string is fastened to a ring which is free to move along a rod.
When the wave pulse arrives at the rod, the ring moves up the rod
and pulls on the string.
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Reflection
This sort of ‘soft’ reflection
creates a reflected pulse
that is not inverted.
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Transmission
When a wave is incident on a boundary that separates two regions of different wave speedspart of the wave is reflectedand part is transmitted.
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Transmission
If the second medium is denser than the first the reflected wave is inverted.
If the second medium is less dense the reflected wave is not inverted.
In either case, the transmitted wave is not inverted.
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Transmission
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Transmission
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Interference
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Interference
The effect that waves have when they occupy the same part of the medium.
They can add together or cancel each other out.
After the waves pass each other, they continue on with no residual effects.
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Constructive Interference
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Constructive Interference
out of phase = 360° = 1 cycle = 2 rad
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Destructive Interference
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Destructive Interference
/2 out of phase = 180° = 1/2 cycle = rad
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Superposition of waves
If two waves travel simultaneously along the same string
the displacement of the string when the waves overlap
is the algebraic sum of the displacements from each individual wave.
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Standing Waves
Consider a string that is stretched between two clamps, like a guitar string.
If we send a continuous sinusoidal wave of a certain frequency along the string to the right
When the wave reaches the right end, it will reflect and travel back to the left.
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Standing waves
The left-going wave the overlaps with the wave that is still traveling to the right.
When the left-going wave reaches the left end
it reflects again and overlaps both the original right-going wave and the reflected left-going wave.
Very soon, we have many overlapping waves which interfere with each other.
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Standing waves
For certain frequenciesthe interference produces a standing wave
patternwith nodes and large antinodes.This is called resonanceand those certain frequencies are called
resonant frequencies.
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Standing waves
A standing wave looks like a stationary vibration pattern,
but is the result of waves moving back and forth on a medium.
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Standing waves
Superposition of reflected waves which have a maximum amplitude and appear to be a stationary vibration pattern.
y1 + y2 = -2Acos(t)sin(kx)
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Standing Waves
If the string is fixed at both endsthere must be nodes there.The simplest pattern of resonance that can
occur is one antinode at the center of the string.
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Standing Waves on Strings
Nodes form at a fixed or closed end.Antinodes form at a free or open end.
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Standing waves
For this pattern, half a wavelength spans the distance L.
This is called the 1st harmonic. It is also called the fundamental mode of vibration.
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Standing waves
For the next possible pattern, a whole wavelength spans the distance L.
This is called the 2nd harmonic, or the 1st overtone.
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Standing Waves
For the next possible pattern, one and a half wavelengths span the distance L.
This is called the 3rd harmonic, or the 2nd overtone.
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Standing waves
,.......3,2,1for ,2
nn
L
,.......3,2,1for ,2
nL
vn
vf
In general, we can write
frequency.resonant 1st theis where 11 fnff
or
number harmonic
theis where n
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Standing Waves
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Standing Waves on a String
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Overtones
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String fixed at ONE end
Note: Only the odd harmonics exist!
=4L
n
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Example
The A-string of a violin has a linear density of 0.6 g/m and an effective length of 330 mm.
(a) Find the tension required for its fundamental frequency to be 440 Hz.
(b) If the string is under this tension, how far from one end should it be pressed against the fingerboard in order to have it vibrate at a fundamental frequency of 495 Hz, which corresponds to the note B?
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Example
= 0.6 g/m = 6 x 10–4 kg/mL = 330 mm = 0.33 ma) Ft = ?
b) 0.33 m – L2 = ?
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Example - A)
v Ft
v2
Ft
Ft v2 v f
Ft 2f 2 2L
Ft 4L2f 2 51N
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Example - B)
v1 = v2
f11 = f22
2 f11
f2
0.59m
L2 0.29m
0.330m – 0.293m 0.037m
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Wave Example 1
The stainless steel forestay of a racing sailboat is 20 m long, and its mass is 12 kg. To find its tension, it is struck by a hammer at the lower end and the return of the pulse is timed. If the time interval is 0.20 s, what is the tension in the stay?
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Example 1
L = 20 m, t = 0.20 s, m = 12 kgFind: F
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Example 1
v xt
2Lt
mL
v F
F mv2
L
mL
(2Lt
)2 4mL
t2
F (4)(12kg)(20m)
(0.20s)2 = 2.4 x 104 N
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Note:
Wave speed is determined by the medium.
Wave frequency is determined by the source.
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Sound Waves
p = BkAcos(t - kx)If y is written as a sine function, P is
written as a cosine function because the displacement and the pressure are/2 rad out of phase.
pmax = BkA
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Waves in 3 Dimensions
x
A 0 1 4 9 16
For Spherical Wavefronts: A = 4r2
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Intensity
Power per unit areaW/m2
B
p
v
pkABI
222
1 2max
2max2
A
PI
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Loudness of Sound
Also called intensity levelDetermined by the intensitywhich is a function of the sound's
amplitude.The human ear does not have a linear
response to the intensity of sound.The response is nearly logarithmic.
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Decibel Scale (dB)
= 10 logI
Io
Where: Io = 1 x 10-12 W/m2
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Common decibel levels
Threshold of hearing0 dB = 1 x 10-12 W/m2
Whisper20 dB = 1 x 10-10 W/m2
Conversation65 dB = 3.2 x 10-6 W/m2
Threshold of pain120 dB = 1 W/m2
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EXAMPLE
How many times more intense is an 80-dB sound than a 40-dB sound?
= 10logI
IologI
Io=
10
logI – logIo =
10
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EXAMPLE
logI =
10+ logIo
logI1 = 8 +– 12 = –4 I1 = 10
–4
logI2 = 4 +– 12 = –8 I2 = 10
-8
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EXAMPLE
Number of times greater = I1/I2
I1I2=10–4
10 -8= 104 = 10, 000
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Beats
When two sound waves that are at nearly the same frequency interfere with each other, they form a beat pattern.
It is an amplitude variation.The beat frequency
21 fffbeat
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The Doppler effect
When a source of sound is moving towards you, it sounds higher pitched (higher frequency).
When it moves away, it sounds lower pitched.
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The Doppler Effect
S
S
L
L
vv
f
vv
f
The S’s stand for the source of the sound. The L’s stand for the listener. v by itself stands for the speed of sound. Be careful with the signs on your velocities!!
The direction from listener toward source is positive The direction from source toward listener is
negative