Today’s Lecture
Linear Combinations of Waves
Standing Waves
Sound
What about wave shapes?
Are all periodic waves (not pulses) harmonic?
Harmonic waves are simplest kind of waves.
Other periodic waves (rectangular, triangular etc.) and even wave
pulses can be represented as sums of harmonic waves with
frequencies, which are multiples of the basic one (higher harmonics),
and different amplitudes.
Fourier decomposition.
Fourier decomposition.
A harmonic wave is the simplest kind of wave.
Real waves may have ugly profiles.
It is often convenient to present those complex waves as sums of
simple harmonic waves with frequencies, which are multiples of the
same basic frequency. Fourier decomposition.
http://www.kettering.edu/~drussell/Demos/Fourier/Fourier.html
Dispersion
If there are big waves and ripples on the water surface, which are going to
run faster?
For some kinds of waves the wave speed of a simple harmonic wave
depends on the wave length.
This phenomenon is called dispersion.
Example – for large waves on the surface of deep water
λg
v=
2π
If you Fourier decompose a pulse, it is made up of harmonic waves of many
different wavelengths. Hence a large water wave undergoes DISPERSION!
Limits of Applicability of the Superposition Principle.
Superposition principle suggests that disturbance of the composite wave
is an algebraic sum of the disturbances produced by the individual
waves.
+
=
This is NOT true, when the amplitudes of the individual waves are so
large that they change the properties of the medium through which
they propagate.
One can say that the second wave propagates through a different medium
due to the perturbation made by the first wave. Then the interference
does not result in algebraic addition.
v =
F
μ
As the amplitude grows, the string gets more stretched,
the tension force increases and so does the wave
speed. A wave with a higher amplitude goes faster!
Waves breaking at a sea shore…
Waves break when their amplitude becomes comparable with the sea depth.
Bigger waves break further away from the shore.
Two waves may interfere constructively and add algebraically far from the
shore 1.
The composite wave has a larger amplitude, which becomes
comparable with the depth further away from the shore 2-4.
Therefore the composite wave would break sooner than any of the two
constituent waves.
The breaking wave is no longer more a sum of the two constituent waves
– the superposition principle is violated, when the amplitude becomes
large compared with the depth.
Summary of Waves
A periodic continuous wave is
characterized by its wavelength
and period.
A simple harmonic wave can
be written as
Newton’s laws show that for small
amplitude waves on a string
Waves carry energy characterized
by intensity or flux that falls off as 1/r2
Many waves of small amplitude satisfy
the linear wave equation and those that
do obey the superposition principle.
Two waves of the same frequencies and wavelengths
propagating in opposite directions interfere to make a
composite standing wave.
Unlike the two interfering propagating waves, the standing wave
does not seem to go anywhere, just like an oscillation…
But it has all the essential properties of the waves:
spatial structure, wavelength, wave number.
It also has amplitude and frequency just as the oscillations do.
How do you create a standing wave?
Make a traveling wave interfere with its reflection off some boundary!
There are some points, nodes, N,
where the string does not move at
all.
There are other points, antinodes,
A, where the oscillations have the
maximum amplitude.
How far apart are those
nodes and antinodes?
We normally consider standing waves in a confined space,
like on string clamped on both sides.
It imposes some boundary conditions that have to be
matched by all standing waves, which are allowed.
f =
v
λ
A propagating wave on a string can have any frequency/length.
In contrast, stable standing waves on a string can only have some
discrete, well defined wavelengths and frequencies.
How many different wavelengths are allowed? What is the
smallest/largest one?
Largest wavelength / Lowest frequency
λ
2
λ = 2 L,
=L
v
f =
2L
Higher harmonics, wavelengths:
λm
2L
= L , λm =
m⋅
2
m
Higher harmonics, frequencies:
Some math:
largest wavelength / lowest frequency
v
f1 =
2L
f m = mf1
Fundamental mode
Higher harmonics
What About Open End Boundary Conditions?
Largest wavelength / Lowest frequency
Higher harmonics, wavelengths:
Higher harmonics, frequencies:
Only odd harmonics!
What if both ends are open?
More Math: Interference
y1 = A cos( kx − ωt )
y2 = − A cos( kx + ωt )
Composite wave:
amplitude oscillations
in time
depends
on position
More Math: Interference
For closed boundary conditions:
For
For open end boundary conditions:
What about open boundary conditions at both ends?
Resonance - a vibration of large amplitude in a mechanical or
electrical system caused by a relatively small periodic stimulus
of the same or nearly the same period as the natural vibration
period of the system.
Requires:
• a system capable to vibrate with a reasonable small
damping;
• an external driving force with a frequency close to that of the
system.
Question:
why are we discussing resonance now and have not
discussed it while talking about propagating waves?
Propagating waves do not
have any special natural
lengths and frequencies.
Both lengths and frequencies
can vary continuously.
Standing waves have a
particular set of allowed
frequencies, which are
characteristic for the system.
v
f1 =
2L
Fundamental mode
f m = mf1
Higher harmonics
A longitudinal standing wave:
Sound wave – longitudinal wave of compressions and rarefactions.
What are those
black circles?
Speed of Sound
v=
F
•
Tension of the string, F, provides the restoring force
μ
•
Mass per unit length, μ = m/L, measures inertia of
the string, that slows down any wave propagation
Speed of sound ?
v=
B
ρ
•
•
ΔP
B=−
ΔV / V
Bulk modulus of elasticity, B, defines the restoring
force
Density of the medium, ρ, measures inertia of the
medium that slows down any wave propagation
ΔV/V is the fractional change of volume (can be
measured in %)
Large bulk modulus of elasticity corresponds to a
large change of pressure – vigorous restoring
force – at small fractional change of volume
v=
B
ρ
ΔP
B=−
ΔV / V
• A hard material with a low compressibility and low density Aluminum - high speed of sound.
• A soft material with a high density – Lead – low speed of sound.
Speed of Sound in a Gas
v=
B
ρ
Can we calculate the bulk modulus of elasticity, B,
of a gas?
What process should we assume?
Sound waves are propagating quickly.
No time for heat exchange!
⇒ Adiabatic process.
For an adiabatic expansion:
Hence:
With the result:
PV γ = c = const
Speed of Sound in a Gas
γP
v=
ρ
γ = Cp/Cv – the constant for an adiabatic process
in the gas
P - pressure of the gas
ρ - density of the gas
What happens to the speed
of sound if you compress a
gas isothermally?
Change the gas temperature?
Speed of Sound in Air
γP
v=
ρ
Air is essentially 79% nitrogen (N2) and 21% oxygen (O2).
The masses of nitrogen and oxygen molecules are 28au
32au respectively.
At one atmosphere of pressure and room temperature, 20oC, one
mole of gas under these conditions occupies V=22.4x293/273L=24L,
and the density is:
For a diatomic molecule γ =1.4 and
In air the speed of sound must satisfy
Speeds of Sound for Different Gases at Normal
Conditions
γP
v=
ρ
γ = Cp/Cv – the constant for an adiabatic process
in the gas
P - pressure of the gas – atmospheric pressure
ρ - density of the gas
Ratio of sound speeds:
Find the sound speed in hydrogen, H2, at one atmosphere and 20oC.
Example – “Donald Duck talk”
You may have heard someone inhale He gas after which, for
a short time, they sounded like Donald Duck. Why??
Consider a sound wave generated by your vocal cords with a
wavelength of λ=64cm. The velocity of sound in He is found from
Since the wavelengths are identical, the two frequencies are
Pressure
Speeds:
(i) of molecular motion;
(ii) of the collective
motion of the air.
Normal conversation –
speed of collective
motion 45 μm/s.
Amplitude – 10 nm.
displacement
Intensity of Sound
Intensity of sound = Average Power / unit area
Intensity of sound = ΔF x u / unit area = ΔP x
Where u is the displacement speed of the air
from equilibrium
u
so – the amplitude of displacement of the air
ΔP – the amplitude of pressure variations
From the last slide ΔP and displacement, s, are π/2 out of phase:
Averaging over time
Intensity of Sound
Using the results from our work in fluids we can write
these expressions to make a bit more sense:
From Newton’s third law in a fluid medium we have
Using this result:
Intensity is quadratic in the amplitude of pressure variations (ΔP) or
the speed of the spatial fluctuations (ωsο) of the gas molecules. For
example:
Sensitivity of Human Ear
Most sensitive at about 4000Hz.
343 m/s
λ =v/ f =
=
4000 Hz
= 8.6 cm
Eardrum - about 2cm
from opening
Measuring Sound Intensity in Decibels, dB
Number of dB, decibel,
dimensionless parameter
I 0 = 10
−12
W/m
2
⎛I ⎞
β = 10 log ⎜⎜ ⎟⎟
⎝ I0 ⎠
- reference value, corresponding to
threshold of sensitivity of a normal
ear at about optimal frequency
Number of dB, decibel,
dimensionless parameter
I 0 = 10
−12
W/m
I = I 0 ⋅ 10
2
⎛I ⎞
β = 10 log ⎜⎜ ⎟⎟
⎝ I0 ⎠
- reference value the threshold of sensitivity of a
normal ear at about optimal frequency
β / 10
An example: TV is turned down from 75dB to 60dB.
How does its sound intensity change?
Goes down by a factor of
In general: sound level is going up/down by x dB sound intensity is multiplied by
10
± x / 10