Two port analysis and boundary conditions

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Two port analysis and boundary conditions

• Musical Acoustics
• http//www.salford.ac.uk/saee
• student_area - undergraduate - musical acoustics
• username students password module

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Two port analysis

• Consider an oscillating system such as a
• pipe
• string
• electrical circuit
• System has
• input and output
• energy flow
• driving or restricting forces at boundaries
• simple transfer properties

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Electrical Circuit
String
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Pipe
Pressure P and volume velocity u
Representing as matrix
A, B, C, D are functions of frequency,
independent of boundaries
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What are A, B, C, D?
We can determine A, etc by considering boundary
conditions
Consider Rigid Termination
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We know how pipes oscillate
Dalemberts Solution
This will be handy later
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Consider total pressures at either end of pipe
Can you see why given
Using
Using trig identity
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Similarly we can find B, C, and D to give
Multiplying out
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Input impedance for Rigid Termination
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Input impedance for Pressure Release Termination
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Input Impedance of a Resistive Termination
ZL is some multiple of acoustic impedance of
wave-guide, Where multiplying factor ? is a
real, positive number
Then input impedance
Given
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No modes

• Peaks
• not infinite
• broad (low Q value)

Closed/Close Pipe
Close/Open Pipe
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Implications for Brass Instruments
Mouthpiece is pressure control value Brass
player uses embouchure to vary m, K and R of
mouthpiece Hence match impedance peaks of
pipe Hence maximum energy transfer
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Room for Expression
Broad resonant peaks give brass player options
to -lip up and down for intonation -add
expressive slides and vibrato
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Missing Harmonics Problem
Closed / open pipe
Gives frequencies F, 3F, 5F, 7F , 9F where
F1/4f
This is not a harmonic series and will seriously
restrict options for building scales Solution
Conical Sections and Flared Ends (Bells, Horns)
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Modelling Conical Sections and Flared Ends
S is cross sectional area of the pipe
First order case
Multiple elements using computers
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Real Trumpet
Broad Impedance Peaks Decreasing With
Frequency High Frequency Cut Off
Almost harmonic series 0.7f, 2f, 3f, 4f, 5f.
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About Flare

• Plane waves become more spherical
• Driver frequency constant
• Distance between wave fronts increases
• Gives increase in phase speed
• Hence shortening of effective length
• Hence frequency modes shift up
• Hence nearer closed/closed pipe

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About Bends

• both flare and negative flare
• reducing and increasing the effective length
• radius of curvature of the inside greater
• net smaller effective length
• Other effects
• Large cross-section- lower characteristic
  impedance
• friction along the sides - dulls sound (more at
  high f)

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My Noddy Matlab Model
Zin for ZLltlt1 (1000 Matrix Multiplication Steps)
Without Flare
With Flare