Physics 203 – Spring 2006 –Bjoern Seipel
Waves
What is a wave?
A disturbance or variation that transfers energy
progressively from point to point in a medium and that
may take the form of an elastic deformation or of a
variation of pressure, electric or magnetic intensity,
electric potential, or temperature.
In short:
1. A disturbance or variation which travels through a
medium
2. Must transfer energy from one location to another
Types of waves:
Mechanical waves
A wave which needs a medium in order to propagates
itself. Sound waves, waves in a Slinky, and water waves
are all examples of this. Sound waves need air molecules
in order to exist; the Slinky waves need the Slinky, and
the waves in the ocean need the water
It follows, then, that mechanical waves cannot exist in a
vacuum. This is the factor that distinguishes them from
electromagnetic waves
Electromagnetic Waves
Radio signals, light rays, x-rays, and cosmic rays
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Physics 203 – Spring 2006 –Bjoern Seipel
Mechanical Waves
Two basic types of mechanical waves
Longitudinal Waves:
In a longitudinal wave the
particle displacement is
parallel to the direction of
wave propagation. The figure
below shows a onedimensional longitudinal
plane wave propagating
through air. The particles do
not move down the tube with
the wave; they simply
oscillate back and forth about
their individual equilibrium positions
Transverse Waves:
In a transverse wave the particle
displacement is perpendicular to
the direction of wave
propagation. The animation
below shows a one-dimensional
transverse plane wave
propagating from left to right.
The particles do not move along
with the wave; they simply
oscillate up and down about their individual equilibrium positions as the
wave passes by. Pick a single particle and watch its motion.
http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html
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Physics 203 – Spring 2006 –Bjoern Seipel
Quantitative description of waves
Periodic waves – normal understanding of waves
- The disturbance repeats over and over
- Can be generated by simple harmonic resulting in SHM of
the elements of the medium
The wave is characterized by
Amplitude A: The maximum displacement of the elements of a
medium
Frequency f: Number of crests or dips that pass by a point per
unit of time
Phase φ: Where is the wave at time = 0 ?
Propagation speed v
From there we can derive:
Period T: How long it takes for an entire cycle to pass by T =
1/f
Wavelength λ : The distance from crest to crest or the length of
one complete cycle v = f λ
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Physics 203 – Spring 2006 –Bjoern Seipel
Mathematical description
A complete mathematical description will give the displacement at any position x
with any instant time t
y = A sin(2πft ± 2πx / λ )
+/- for direction of motion
substitute : λ = v / f
y = A sin(2πft ± 2πxf / v)
Sinusoidal Wavefunction
y = A sin(2πf (t ± x / v))
Examples
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Physics 203 – Spring 2006 –Bjoern Seipel
Velocity of the waves
Velocity is not arbitrary!
Depends on…
…characteristics of the medium through which waves move
…in case of EM radiation in vacuum it is constant
So: The frequency and the wavelength are not independent of
each other. Usually the frequency is fixed by the source.
Waves on a string (transverse wave)
Depends on Tension & mass of string
Higher tension (F) leads to a higher velocity Æ wave will travel
more rapidly
More mass or better mass per length = m/L leads to a higher
inertia and to lower velocity
Speed of a wave on a string
v=
F
(m / L)
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Physics 203 – Spring 2006 –Bjoern Seipel
Sound (longitudinal wave)
A sound wave is similar in nature to
a slinky wave for a variety of
reasons. First, there is a medium
which carries the disturbance from
one location to another. Typically, this
medium is air; though it could be any
material such as water or steel.
Speed of sound in Air
v ≈ 343 m/s ≈ 770 mi/h
at atmospheric pressure and room temperature
Material
Speed (m/s)
Aluminum
6420
Granite
6000
Steel
5960
Pyrex glass
5640
Copper
5010
Plastic
2680
Fresh water
(20 ºC)
1482
Fresh water (0 ºC)
1402
Hydrogen (0 ºC)
1284
Helium (0 ºC)
965
Air (20 ºC)
343
Air (0 ºC)
331
Frequency human ears can detect/hear are
in the range
20 – 20000 Hz
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Physics 203 – Spring 2006 –Bjoern Seipel
Sound Intensity
Notice that sound waves carry
energy. We define the intensity I
as the rate at which energy E
flows through a unit area A
perpendicular to the direction of
travel of the wave.
Intensity = Power / Area
I = P / A = E / (At)
For a point source, energy spreads out in all directions
Æ Area of a sphere A = 4 π r
2
Intensity with distance from
a point source
P
I=
4πr 2
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Physics 203 – Spring 2006 –Bjoern Seipel
Human Perception of Sound
The loudness of a sound depends on its intensity. A sound
perceived as roughly twice as loud as another actually has an
intensity that is 10 times greater.
We measure loudness β by a
logarithmic scale of the intensity
level of a wave:
β = 10 log (I/I0),
where I0 = 10-12 W/m2
β is dimensionless but we give it
a name, decibels (dB).
3dB is a factor 2 change in
intensity
Every 10dB is a factor 10 change
in intensity
20 dB is a factor 100 change in
intensity
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Physics 203 – Spring 2006 –Bjoern Seipel
Doppler Effect
Heard an ambulance go by recently? Remember how the siren's
pitch changed as the vehicle raced towards, then away from you?
First the pitch became higher, then lower. This change in pitch
results from a shift in the frequency of the sound waves and is
known as the
Doppler Effect
+ means that the observer is moving
toward from the source
⎛ 1 ± uobserver
⎜
v
f '= ⎜
⎜ 1 m u source
v
⎝
⎞
⎟
⎟ f
⎟
⎠
+ means that the source is moving away from the observer
uobserver/source = speed of observer/source
As a rule: When the motion of detector and source is toward
the other, there is an upward shift in the frequency. When
the motion of pf detector and source is away from each other
there is a downward shift in frequency.
Three cases: Moving Observer, Moving Source or both observer
and source are in motion
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Physics 203 – Spring 2006 –Bjoern Seipel
Moving Observer
We know:
v=λf
If observer moves towards the source with speed u more
compressions move past the observer per time (higher f).
f' = v’/λ = (v+u)/ λ = (v+u)/ (v/f) = f (1+u/v)
If observer moves away less compressions move past the
observer:
v’ = (v - u) and f’ = f (1 - u/v)
Doppler Effect for moving observer: f’ = (1 ± u/v) f
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Physics 203 – Spring 2006 –Bjoern Seipel
Moving Source
Towards the observer:
λ' = vT – uT = (v - u)T
Away from the observer:
λ' = vT + uT = (v + u)T
And therefore with v = λ' f’
f' = v/λ’ = v/(v-u)T = v/(v-u) (1/f) = f (1/(1-u/v))
and
f' = v/λ’ = v/(v+u)T = v/(v+u) (1/f) = f (1/(1+u/v))
Doppler Effect for moving source
⎛
⎜
1
f '= ⎜
⎜ 1 m u source
v
⎝
⎞
⎟
⎟ f
⎟
⎠
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Physics 203 – Spring 2006 –Bjoern Seipel
Doppler-shifted frequency versus speed (400 Hz – source)
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Physics 203 – Spring 2006 –Bjoern Seipel
Superposition &Interference
If two or more waves are moving through a medium, the
resultant wave is found by adding together the displacements of
the individual waves, point by point.
As a result of superposition, waves can interfere. Interference
does not mean that waves are destroyed; they will pass through
each other.
Construcitve Interference
Destructive Interference
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Physics 203 – Spring 2006 –Bjoern Seipel
Standing Waves
A standing wave, also known as a stationary wave, is a wave that
remains in a constant position (and therefore v=0). This phenomenon can
occur because the medium is moving in the opposite direction to the
wave, or it can arise in a stationary medium as a result of interference
between two waves traveling in opposite directions.
Wire (Instrument)
Æ Transverse wave
Fundamental mode (lowest
frequency) or
First harmonic
Anti-node
Node
λ = 2L
f1 = v/λ = v/(2L)
Second harmonic
λ=L
f2 = v/λ = v/L = 2 f1
Ends are tight up so the wave stays in place
Third harmonic
λ = 3/2 L
f2 = v/λ = v/(2/3 L) = 3 v/(2L)= 3f1
nth harmonic: λn = 2L/n and fn = n v/(2L)
(n: 1,2,3,…..)
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Physics 203 – Spring 2006 –Bjoern Seipel
Vibrating Columns of Air (longitudinal waves)
a) λ = 4L
b) λ = 4/3 L
c) λ = 4/5 L
Standing wave in a bottle or
column closed at one end
f1 =
Anti-node
f2 =
f3 =
fn =
Important Example:
Node
Human ear canal
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Physics 203 – Spring 2006 –Bjoern Seipel
Standing wave in a pipe that is open
a) λ =
b) λ =
c) λ =
Standing wave in a bottle or
column closed at one end
f1 =
f2 =
f3 =
fn =
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Physics 203 – Spring 2006 –Bjoern Seipel
Beats
Overlapping of waves of same frequency leads to constructive and
destructive interference and the amplitude is constant. If the frequency of
the waves is slightly different we have the phenomenon of beats. The
amplitude is no longer constant and the sound level alternately rises and
falls
The number of times per second that the loudness rises and falls is the
beat frequency and is the difference between the two sound frequencies
f beat = f1 − f 2
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