16.107 L01 Jan 23/02
Pressure waves in open pipe
Pressure waves in pipe
closed at one end
Musical Sounds
• Consider a hollow pipe open at
both ends
• a wave reflects even if the end is
open =>free ‘end’ =>anti-node
Note: pressure has a
node but displaement
an anti-node
Fundamental or first harmonic
f1 = v/λ = v/2L
for L=.4m, v=343m/s, f1=429Hz
In general, λn=2L/n n=1,2,3,…
fn = v/ λn =nv/2L
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16.107 L01 Jan 23/02
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 λ/4 = L
f1 = v/λ = v/4L
Lower than both open
In general, λn = 4L/n but n is odd!
fn = v/ λn = nv/4L
Lower frequency
as L increases
n=1,3,5,...
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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16.107 L01 Jan 23/02
Problem
• fundamental of A has
LA=λ/2=v/2f
=(343m/s)/2(300Hz) =.572 m
• 2nd harmonic has LA=λ=.572m
f=v/λ =343/.572=600Hz
3rd harmonic of pipe B has n=3
v= λ f=(4 LB/3)600 =343 m/s
LB = 343/800 = .429 m
Musical Sounds
• Actual wave form produced by an instrument is a
superposition of various harmonics
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16.107 L01 Jan 23/02
Complex
wave
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16.107 L01 Jan 23/02
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/π) sin(ωt)-(1/2π) sin(2ωt)
-(1/3π) sin(3ωt)-(1/4π) sin(4ωt)-...
y(x,t)=ym sin(kx- ωt)
y(0,t)=-ym sin(ωt)
-(1/π) sin(ωt)
T=2π/ω
Decomposition into sinusoidal waves
is analogous to vector components
r=xi+yj+zk
-(1/2π) sin(2ωt)
T=2π/2ω
-(1/π) sin(ωt)-(1/2π) sin(2ωt)
-(1/π) sin(ωt)-(1/2π) sin(2ωt) -(1/3π) sin(3ωt) -(1/4π) sin(4ωt)
-(1/5π) sin(5ωt) -(1/6π) sin(6ωt)
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