Pressure waves in open pipe

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Pressure waves in open pipe
Pressure waves in pipe closed at one end
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Musical Sounds
Note pressure has a node but displaement an
anti-node

• Consider a hollow pipe open at both ends
• a wave reflects even if the end is open gtfree
  end gtanti-node

Fundamental or first harmonic f1
v/? v/2L
for L.4m, v343m/s, f1429Hz
In general, ?n2L/n n1,2,3, fn
v/ ?n nv/2L
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Musical Sounds

• Consider a pipe with one end closed
• waves reflect at both ends but there is a node at
  the closed end and an anti-node at the open end

Fundamental has ?/4 L f1 v/? v/4L
Lower than both open
Lower frequency as L increases
In general, ?n 4L/n but n is odd! fn v/
?n nv/4L n1,3,5,...
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Problem

• Organ pipe A has both ends open and a fundamental
  frequency of 300 Hz
• The 3rd harmonic of pipe B (one open end) has the
  same frequency as the second harmonic of pipe A
• How long is a) pipe A ? b) pipe B ? if the
  speed of sound is 343 m/s

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Problem

• fundamental of A has LA?/2v/2f
  (343m/s)/2(300Hz) .572 m
• 2nd harmonic has LA?.572m fv/? 343/.572600Hz

3rd harmonic of pipe B has n3 v ? f(4
LB/3)600 343 m/s LB 343/800 .429 m
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Musical Sounds

• Actual wave form produced by an instrument is a
  superposition of various harmonics

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Complex wave
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Fourier Analysis

• The principle of superposition can be used to
  understand an arbitrary wave form
• Jean Baptiste Fourier (1786-1830) showed that an
  arbitrary wave form can be written as a sum of a
  large number of sinusoidal waves with carefully
  chosen amplitudes and frequencies
• e.g. y(0,t) -(1/?) sin(?t)-(1/2?) sin(2?t)
  -(1/3?) sin(3?t)-(1/4?) sin(4?t)-...

y(x,t)ym sin(kx- ?t)
Decomposition into sinusoidal waves is analogous
to vector components r x i y j z
k
y(0,t)-ym sin(?t)
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-(1/2?) sin(2?t)
-(1/?) sin(?t)
T2?/?
T2?/2?
-(1/?) sin(?t)-(1/2?) sin(2?t)
-(1/?) sin(?t)-(1/2?) sin(2?t) -(1/3?) sin(3?t)
-(1/4?) sin(4?t)
-(1/5?) sin(5?t) -(1/6?) sin(6?t)