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Addition of waves and wave speed
If conditions are just right, you will get stationary patterns of waves.
We can have the ends either fixed (tied down and not moving) or free (not tied down).
To start with, we will consider fixed ends.
From the demo, what are the first three patterns of standing waves? Sketch them
below.
_____________________________________________
_____________________________________________
_____________________________________________
Now for the patterns, the sine function is zero at x=0 and at λ/2, λ, 3 λ/2 and so on.
Using L for the length of the string above, see if you can develop an expression for the
wave pattern of the form
L = stuff with n and λ.
The table below might help.
Mode, n
Wavelengths λ
/string length L
1
2
3
You will use these patterns in the vibrating strings lab.
1
For one fixed and one free end, the patterns are:
n=1
_____________________________________________ free
n=2
_____________________________________________ free
n=3
_____________________________________________ free
What is the relation between length (L), wavelength (λ) and n for this case?
Mode, n
Wavelengths λ
/string length L
1
2
3
You will work with these patterns in the sound resonance lab.
Prediction: Can you predict the first 3 patterns and the relationship for two free ends?
Do so on the next page and then we will see if it checks out.
2
For two free ends, the patterns would be:
n=1
_____________________________________________ free
n=2
_____________________________________________ free
n=3
_____________________________________________ free
The relation between length (L), wavelength (λ) and n for this case would be:
Mode, n
Wavelengths λ
/string length L
1
2
3
Check: (v = 340 m/s)
Tube Length
λ1
λ2
f1
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_______
_______
__________
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_______
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Animations of these and the Doppler Shift can be found at
http://web.clark.edu/ggrey/PS101web/waves/waves.htm
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f2
________
________
Additional results of superposition of waves.
In addition to standing waves, there are other phenomena which are results of wave
addition. One very noticeable effect is the phenomenon of beats. When 2 sources
have nearly the same frequency, the amplitude of the sum of the two waves “wavers”
with a frequency equal to the difference in frequency between the two sources:
A sin[(ω+δ)t-kx] + A sin[(ω-δ)t-kx] = 2A cos[δ(t-x/v)] sin[ω(t-x/v)] (details in
appendix) where the new amplitude is 2A cos[δ(t-x/v)]. This is an envelope, just like
the decaying exponential of a damped oscillator. A good animation is at
http://www.mta.ca/faculty/science/physics/suren/Beats/Beats.html
Below is a graph of the sum of two sine waves of frequencies 10 and 12 Hz. The
low frequency curve is the cosine amplitude function.
3
0
-3
0
0.2
0.4
0.6
0.8
1
1.2
Notice that amplitude of the wave sum appears to oscillate with twice the frequency
of the envelope, so the audible frequency of the amplitude variation is twice that of
the envelope; here that would be 2 Hz, or the difference in frequency between the
two original waves. The frequency of the sum is 11 Hz, the average of the two
original frequencies. Audio applet:
http://qbx6.ltu.edu/s_schneider/physlets/main/beats.shtml
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Doppler shift.
When a source of sound or an observer is moving, there is an apparent shift in
frequency of the waves, like a siren or the whine of the tires of an approaching car.
The Physlet we use in class gives a good basis for understanding this phenomenon.
Hecht does a great job of explaining the resulting formula simply.
Here, the source is moving to the left.
Left observer:
Right observer:
Emitted wave
wave crests
crests closer,
farther apart,
higher frequency
lower frequency
d = vt-vst = nλo
d = vt + vst = nλo
For both cases, n=fst and λo = v/fo Substitute these in and solve for fo/fs :
The combined formula for moving sources and observers is
fo = fs (v+vs)/(v+vo)
fun animation: http://www.colorado.edu/physics/2000/applets/doppler.html
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Intensity
The intensity of a wave (loudness or brightness) is the power per unit area. Power is
energy/time and the energy in a wave is proportional to the square of the amplitude.
We use a log scale for intensity of sound, because your ear responds logarithmically.
The decibel scale is defined as
I (db) = 10 log(I/Io) where Io is 1.0 x 10-12 W/m2, the threshold of hearing.
This means that a 30db drop in intensity or volume is a thouandfold drop in
power per unit area.
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Addition of 2 waves:
We will make use of the double angle formula for sines:
sin(a±b) = sin(a)cos(b) ± cos(a)sin(b)
Then, using k = 2π/λ and λ = v/f so k = 2πf/v = ω/v :
A sin[(ω+δ)t-kx] + A sin[(ω-δ)t-kx] = A sin[(ω+δ)(t-x/v)] + A sin[(ω-δ)(t-x/v)]
Now, write (t-x/v) = φ to give
A sin[(ω+δ) φ] + A sin[(ω-δ) φ ] = A sin[ ωφ + δφ ] + A sin[ ωφ - δφ ]
= A [sin(ωφ) cos(δφ) + cos(ωφ)sin(δφ)] + A [sin(ωφ) cos(δφ) - cos(ωφ)sin(δφ)]
= 2A sin(ωφ) cos(δφ) = [2A cos(δφ) ] sin(ωφ)
Good general website for wave addition:
http://www.rwc.uc.edu/koehler/biophys.2ed/stand.html
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