Standing Waves
•  A standing wave is one which is reflected
back and forth between fixes ends of a
string or pipe.
•  Reflection may be fixed or open-ended.
•  Superposition of the wave upon itself
results in a pattern of constructive and
destructive interference and an enhanced
wave. Let’s see a simulation
•  http://www.5min.com/Video/The-Rubens-TubeFrequency-of-Fire-1858291
Fixed-end standing waves - guitar
or violin string
•
•
•
Fundamental
1st harmonic
λ = 2L
•
First overtone
•
•
2nd harmonic
λ=L
•
Second Overtone
•
•
3rd harmonic
λ = 2L/3
http://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html
Problem
•  A string of length 12 m that’s fixed at both ends
supports a standing wave with a total of 5 nodes.
What are the harmonic number and wavelength
of this standing wave?
•  L = 4(1/2 λ ) à λ = 2L/4 4th harmonic since it
matches λn = 2L/n for n = 4
•  wavelength: λ4 = 2(12m) / 4 = 6 m
Open-ended standing waves flute & clarinet
λ = 4L
λ = 2L
λ = (4/3)L
λ=L
λ = (4/5L
λ = (2/3)L
physics.indiana.edu/~p105_f02/standing_waves_...
http://upload.wikimedia.org/
wikibooks/en/3/32/
Fhsst_waves40.png
•
open ends one end
–
closed
both ends
closed
Sample Problem
•  12-30. a) Determine the length of an organ pipe that
emits middle C (262Hz). The air temp. is 21oC.
•  A) v = 331m/s + 0.6 m/soC(21oC) = 344m/s
•  A) λ = 2L v = fλ = 2λf
L = v/2f = 344m/s/{2(262/s)]
•
L = 0.656m
•  B) What are the wavelength and frequency of the 1st
harmonic?
•  Frequency is 262 Hz
•  Wavelength is twice the length of the pipe, 1.31 m.
•  C) What is the wavelength and frequency in the traveling
sound wave produced in the outside air?
•  They are the same because it is air that is resonating in
the organ pipe: 262Hz and 1.31 m
Superposition of 2 sound waves
http://www.ece.utexas.edu/~nodog/me379m/superposition.html
Resonance and Beats
•  Resonance occurs when a vibration from
one oscillator occurs at a natural
frequency for another oscillator.
•  The first oscillator will cause the second to
vibrate.
•  See next slide.
Resonance
•
http://www.isd-dc.org/ISD-Wash/GIFS%20Pictures%20&%20Whatnots/
tuningforkresonance.jpg
Beats
•  The word physicists use to describe the
characteristic loud/soft pattern that
characterizes two nearly matched
frequencies.
•  Musicians call this “being out of tune”.
Beats
•  When two sound waves whose
frequencies are close but not exactly the
same, the resulting sound modulates in
amplitude changing from loud to soft to
loud. This is called beat frequency and is
shown by:
•
fbeat =
f1-f2
Diffraction
•  Bending of a wave around a barrier
•  Diffraction of waves combined with
interference of the diffracte waves causes
“diffraction patterns”.
•  Here is an example using a “ripple tank”.
•  http://www.falstad.com/ripple/
Double-slit or multi-slit diffraction
•
•
micro.magnet.fsu.edu/.../doubleslit/
Remove frame
Mathematical Description of a
Traveling Wave
•  Y = A sin (ϖ t + κ x )
•  Y dependent of x and t; y(x,t) or “y of x &
t”
•  If the - sign is used, wave is traveling in +x
direction à
•  A is amplitude of the wave
•  µ (omega) is angular frequency (µ = 2πf)
•  κ angular wave # (κ = 2 πk, k = 1/λ)
Other forms
•  Important features of the wave: amplitude,
frequency f (through ϖ), period T (which is
1/f = 2π/ϖ), wavelength (λ = 2π/κ) and wave
speed v (which is λf = ϖ/k)
•  y = Asin2π[ft + (1/λ)x] or
•  y = Asin(2π/λ)(vt + x)
Sample Problem:
The vertical position y of any point on a rope that supports a transverse wave traveling
horizontally is given by the equation
y = 0.1 sin (6 π t + 8 π x)
Find:
amplitude: 0.1
angular frequency: ϖ = 6 π s -1
frequency: f = ϖ / (2 π) = (6 π s -1 )/ (2 π ) = 3 Hz
angular wave number: k = κ / (2 π ) = 8 π m -1) / ((2 π) =
4 m-1
wavelength: λ = 1/ k = 1/ (4 m-1) = 0.25 m
period: T = 1/f = 1 / 3 Hz = 0.33s
wave speed: v = f λ = 0.25m (3 Hz) = 0.75 m/s
Sound Level
•  Intensity: Rate at which sound waves
transmit energy is measured in energy per
unit area: watt/m2 or watt/cm 2
•  Intensity level or loudness level, Β
•  B = 10log I/Io where Io = 1x10-12 w/m2
•
or 1x10-16 w/cm2
More math…
β = 10 log I
Io
-16
2
10
w/cm
Intensity level = 10 log Intensity / threshhold of
hearing
We all don’t hear the same, so this is a
comparative measurement in decibels
Flow chart for β problem
•  If I = 4.7 x 10^-10 w/cm^2:
•  10xlog(4.7 2nd EE -10 / 1 2nd EE -16) =
66.7 dB
•  If I = 2.9 x 10^-3 w/cm^2:
•  10xlog(2.9 2nd EE -3 / 1 2nd EE -16) =
135 dB
Problem
•  Now we are going backwards from intensity level
(dB) to intensity (w/cm2)
•  If the intensity level is 83 dB, convert that
to intensity in w/cm2.
•  B = 10 log I / Io
get to a working eqtn:
•  B /10 = log I / Io
•  Log-1(B/10) = Log-1(log I/Io)
•  Log-1(B/10) = I/Io
•  Let’s say that the intensity level of a
sound is 25.3 dB. What is the intensity of
the sound in w/cm2?
•  B = 10 log I / Io
•  25.3 dB = 10 log (I/10-16 w/cm2)
•  2.53 = log I – log 10-16
•  2.50 + log 10-16 = log I
•  2.50 – 16 = log I