LECTURE 7
STANDING WAVES
Instructor: Kazumi Tolich
Lecture 7
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Reading chapter 16-2
¤ Standing
waves
Phase difference due to path difference
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A phase difference between two waves is often the result of a
difference in path length, Δr.
Constructive interference occurs when δ = Nπ, where N = 0, 2, 4, …
Destructive interference occurs when δ = Nπ, where N = 1, 3, 5, …
4λ
5λ
4λ
4.5λ
Demo 1
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Two Speaker Interference
¤ Demonstration
of constructive and destructive interference due to path
length differences.
Quiz: 1
5
Example 1
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Two audio speakers facing in the same
direction oscillate in phase at the same
frequency. They are separated by a
distance equal to one-third of a
wavelength. Point P is in front of both
speakers, on the line that passes
through their centers. The amplitude of
the sound at P due to either speaker
acting alone is A. What is the
amplitude (in terms of A) of the
resultant wave at point P?
Standing waves & resonant frequencies
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If waves are confined in space, reflections at both
ends cause the waves to travel in in opposite
directions and interfere.
For a given string or pipe, there are certain
frequencies for which superposition results in a
stationary vibration pattern called a standing wave.
Frequencies that produce standing waves are called
resonant frequencies of the string system.
The lowest resonant frequency is called fundamental,
or first harmonic. Then the higher ones are 2nd
harmonic, 3rd harmonic, etc.
Nodes and antinodes
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Nodes are spaced a distance λ/2 apart, and
they include the point at which the string is
anchored to the wall. Nodes are points of
maximum destructive interference.
Antinodes are spaced λ/2 apart also. They
are points of maximum constructive
interference.
y or s
Standing waves on a string (fixed-fixed)
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The standing wave condition and resonant frequency for the nth harmonic for
a string with length L, and both ends fixed are
L
Standing wave on a string (fixed-driven)
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The standing wave condition for nth harmonic for a string with length L, and
one end fixed and the other end connected to a vibrator is
Demo 2
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Standing Waves in Rubber Tubing (Vary Frequency)
¤ When
the right frequencies are reached, the tubing vibrates in various
standing wave modes.
Standing wave on a string (fixed-free)
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The standing wave condition and resonant frequency for the nth harmonic for a string
with length L, and one end fixed and the other end free are
The free end must be at antinode so that the string meets the boundary condition.
Sound waves in a pipe (closed-closed)
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A closed end causes a node there for the displacement wave due to the
boundary conditions.
Physical representation of the n = 2
mode.
Sound waves in a pipe (open-open)
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node
Sound waves in a pipe (open-closed)
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Pipes and modes summary
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n = 1
n = 1
n = 2
n = 2
n = 3
n = 3
n = 3
n = 5
Open-Open or Closed-Closed
n = 1
Open-Closed
Example 2
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A shower stall is L = 2.45 m tall.
Assuming the shower stall is a
closed-closed pipe, for what
frequencies less than 500 Hz can
there be vertical standing sound
waves in the shower stall?
Assume the speed of sound is
v = 343 m/s.
Quiz: 2
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Demo 3 & 4
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Open and Closed End Pipes
¤ Various
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pipes with different resonant frequencies.
Beats (Singing Pipes)
¤ Demonstration
of resonance and beats due to difference in the resonant
frequencies of two pipes.
Pop music
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When air is blown across the open top of a pop bottle, the turbulent air
flow can cause an audible standing wave.
The standing wave will have an antinode at the top and a node at the
bottom.