Chapter 18 Superposition and standing waves

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Chapter 18 Superposition and standing
waves October 27 Standing waves 18.1
Superposition and interference Superposition
principle
If two or more waves are moving through a medium,
the resultant wave function at any point is the
algebraic sum of the individual waves y(x,t)
y1(x,t) y2(x,t)
Interference The combination of separate waves
in the same space to produce a resultant
wave. Constructive interference (displacements
are in the same direction), destructive
interference.
Quiz 18.1
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Superposition of sinusoidal waves Two waves
traveling in the same direction, with the same
frequency, wavelength and amplitude, but
different phase.
1) The sum wave is also a sinusoidal wave, with
the same frequency and wavelength. 2) The
amplitude of the sum wave is determined by the
phase difference f. 3) f 0, 2p, constructive
interfere f p, 3p, destructive interference
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18.2 Standing waves Superposition of two waves
with the same amplitude, same frequency and
wavelength, but traveling in opposite directions

• Standing wave an oscillation pattern with a
  stationary outline
• Every element oscillates in a simple harmonic
  motion.
• The amplitude of the simple harmonic motion
  depends on the location of the element.

Node point of zero amplitude, Antinode point
of maximum amplitude, The distance between
adjacent antinodes or adjacent nodes is l/2.
Quiz 18.2
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Read Ch181-2 Homework Ch18 (1-16)
5,13,16 Due November 7
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October 29 Standing waves in strings and air
columns
18.3 Standing waves in a string fixed at both
ends Boundary condition of waves The ends of the
strings must be nodes. Normal modes of vibration
A mode is an oscillation which has a
characteristic frequency and satisfies the
boundary condition. Quantization The situation
that only certain frequencies of oscillation are
allowed.
Fitting y sinkx into L while keeping y(0) 0
and y(L)0
Fundamental frequency
Harmonics
Example 18.3
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18.4 Resonance Resonance If a periodic force is
applied to a system with one or more normal
modes, the amplitude of the resulting motion is
the greatest when the frequency of the applied
force is equal to one of the natural frequencies
of the system.

• 18.5 Standing waves in air columns
• Features of a closed end and an open end of a
  pipe
• The closed end is a displacement node in the
  standing wave since the wall does not allow
  longitudinal motion in the air. It is a pressure
  antinode which has a maximum pressure variation.
• 2) The open end is approximately a displacement
  antinode in the standing wave. It is a pressure
  node since the open end corresponds to a point of
  no pressure variation.

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I) A pipe open at both ends 1) Both ends are
displacement antinodes. 2) The fundamental
frequency is
3) The higher harmonics are
II) A pipe closed at one end 1) The closed end
is a displacement node, the open end is a
displacement antinode. 2) The fundamental
frequency is
3) The higher harmonics are
Quiz 18.4, 18.5
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Read Ch183-6 Homework Ch18 (17-42)
19,21,37 Due November 7
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October 31 Beats and Fourier analysis 18.7
Beats Interference in time Spatial interference
The amplitude of the oscillation varies with the
position of the element, e.g., standing waves.
Temporal interference The amplitude of the
oscillation varies with time. It occurs when the
interfering waves have slightly different
frequencies so that they are periodically in and
out of phase at the point of superposition. Beatin
g The periodic variation in amplitude at a given
point due to the superposition of two waves
having slightly different frequencies.
Aresultant
The intensity of the resultant wave varies at a
frequency of .
Example 18.8
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18.8 Nonsinusoidal wave patterns Music The wave
patterns produced by a musical instrument are the
result of the superposition of various harmonics.
Fouriers theorem Any periodic function can be
synthesized by a series of sine and cosine terms.
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Read Ch187-8 Homework Ch18 (43-) 44,45 Due
November 7