Post on 31-Oct-2021
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P773 AcousticsJune, 2008
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Physicalprinciple
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Denotes anExample
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Measure pressure inside sealed box
7Acoustic impulse response inside the box
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Air spring only
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For Plane Waves
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Mathematicalprinciple
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Along the wavedirection, longitudinal
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= p v
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Motivate thiswith energyconservation
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Out of phase,reactive flow
In-phase,power flow
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Measure pressure outside sealed box at dust cap
23Acoustic impulse response at cone surface
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25Relative acoustic frequency response: [at cone]/[inside box]
Slope of 40dB/decade
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Showsp=f(r,t)/r
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However, for a 3-dimensional spherically-spreadingwave from a point source, the amplitude of theharmonic solution to the wave equation for thepressure [1,2] can be written as
p(r,ω) = ρ jω U exp(−jkr)/(4πr),
where U is the volume velocity [m3/s] of the pointsource. The jω factor represents a time derivative,and we can thus write the solution in the time domainas
p(r,t) = ρ A(t-r/c)/(4πr),
where A(..) is the volume acceleration [m3/s2] of thesource. Note that the solution for the pressure doesnot change shape as it propagates, but the amplitudefalls off as 1/r.
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The particle velocity, v, relates to the pressure by theNewtonian equation of motion for the air
∇p = −ρ ∂v/∂t.
Mathematically, for a spherically-symmetric wavesolution, ∇p=∂p/∂r, and thus
∇[exp(−jkr)/r]=−jk exp(−jkr)/r−exp(−jkr)/r2.
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As a result, the Newtonian equation relates thepressure to the particle velocity of the air, at radius r,as
(1 + 1/jkr) p = ρ c v,
which can also be written as
(jkr + 1) p = jkr ρ c v = ρ j ω r v = ρ a r,
where a is the acceleration of the air particles.
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Plot of Z = R+jωM for a sphere.
For a sphere, the acoustic impedance iseasy to work out. For kr<<1, the size ofthe sphere is much less than awavelength, and the shape of the sourceis not very important.
Thus we expect all acoustic impedancesof monopole acoustic sources to actsimilarly.
The rising low-frequency imaginary part represents the reasonablyconstant inertial air load.
Note the curve is very smooth with no resonances or interference. That istrue since diffraction from edges does not occur for a sphere.
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Figure 1. Showing the real and imaginary parts of theacoustic surface impedance for a piston in an infinitebaffle, versus ka, where a is the piston radius. AfterBeranek.
The LF imaginary part ofthe acoustic impedance isproportional to frequency,representing a constantinertial air load.
The real part is proportionalto frequency^2, which isexpected for a monopolesource.
The oscillations relate tointerference from edgereflections.
Piston in Infinite Baffle
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At bass frequencies
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Listen to baffle…
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From Olson Acoustics
Conclusion: we need a huge baffle to give decent bass
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Horn design from Olsen
Horns are the most efficient speakers,but often give a coloured response.
40The new “robot” lookfor the 800 series
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Direct signalDiffraction from an edge
Emitting an impulse
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Point source tweeter1st-order diffraction
Point-source tweeter diffraction
Tweeter finite size diffraction
Approximate effect of boxincluding midrange
Diffraction calculations using a simple model
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Question: if equalized,could we hear this effect?
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For paper,f ~ 1.7kHz
Movie screen
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Directivity relatesto diffraction
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ka = 1 means550 Hz for a = 10cm
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This shows that asmall source will
be omnidirectionalif size << wavelength
ka = 1 means550 Hz for a = 10cm
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Ultrasonic frequenciesdecay rapidly
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This sloperepresents amass of air
of 0.85 x radius
The “air load”
Radiation impedance= pressure/velocity
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Low sensitivity
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Acceleration is proportional to force
Conclusions: (1) Displacement is opposite direction to force ! (2) For constant displacement, acceleration is proportional to freq^2 (3) For constant force, displacement is proportional to 1/freq^2
Kinematics and dynamics of shaking a mass
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The 1925 Rice-Kellogloudspeaker model
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B&W philosophy is to:absorb the rear wave!
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The End