converting dBcmMHz to Nepers

1
converting dB/cm/MHz to Nepers

• another common attenuation value for tissue is
  0.5 dB/cm/MHz
• Q For this attenuation value, what is the
  attenuation in Nepers for a plane wave that
  propagates a distance of 5cm at 500kHz?
• .5 dB/cm/MHz 5cm .5MHz 1.25dB
• A 1.25dB / (8.7dB/Np) 0.1437 Np
• verify 20 log10(exp(- 0.1437 Np))
• -1.25dB

2
converting dB/cm/MHz to Nepers

• Q Using the same attenuation as before, what is
  the intensity of a plane wave after propagating a
  distance of 5cm at 500kHz if the initial
  intensity is 5W/cm2?
• A1 5W/cm2 exp (- 2 0.1437 Np) 3.75
  W/cm2
• A2 5W/cm2 10(-1.25/10) 3.75 W/cm2

3
converting dB/cm/MHz to Nepers

• Exercise verify that an attenuation of
  5.75x10-8Np/cm/Hz is the same as 0.5dB/cm/MHz

4
Resonance in pipes

• recall that for plane wave, planar boundary,
  normal incidence, (6.2.7 in book)
• These equations are the same as 10.2.2

5
Radiation Impedance

• impedance seen by wave emanating from a pipe is
  the radiation impedance (think optics where a
  plane wave incident on a slit - plane wave on the
  left, diffraction pattern on the right)
• for a circular pipe, this is roughly the same as
  that for a circular piston (use the low frequency
  limit)

6
Radiation impedance

• recall also that the radiation impedance is
  defined as the ratio between the force and the
  particle velocity
• use this expression for the impedance not only
  outside of the pipe, but inside as well, so
  and

7
Input impedance

• similar approach to derivation in sections 6.2
  and 6.3 (continuity of specific normal impedance)
• combine expressions and obtain

8
Radiation impedance -flanged opening

• flanged (baffled piston in the low frequency
  limit)
• see derivation on p. 274
• resonance frequency closed pipe w/ rigid cap is
  f(n) (n c) / (2 L)

effective length is therefore L L .85a
9
Radiation impedance -unflanged opening

• no longer represented by baffled piston
• resonance frequency closed pipe w/ rigid cap is
  f(n) (n c) / (2 L)

effective length is therefore L L .6a
10
Power transmission from pipes -flanged vs.
unflanged

• details on p. 275 (section 10.3)
• low frequency limit
• flanged
• unflanged
• flange doubles the output power - result of
  better impedance match

11
Pipes as filters (mufflers)

• sections 10.10 and 10.11
• junctions
• branches
• design filter based on equivalent lumped models
  for L and C

12
Exponential Horn (loudspeaker)

• acoustic transformer
• provides an even better impedance match than a
  flanged pipe
• some details on p. 414 (section 14.7)

13
Helmholtz Resonator (10.8)

• open end radiates - need to include radiation
  resistance (acts as a mass)
• fluid in neck - provides additional mass
• cavity - compression of fluid in here provides
  stiffness
• square root of stiffness divided by mass -
  natural frequency

14
Helmholtz Resonator - Assumptions

• wavelength is much greater than the cube root of
  the volume (V) of the cavity
• wavelength is much greater than the square root
  of the area (S) of the opening
• wavelength is much greater than the length (L) of
  the neck
• more details provided on p.284, section 10.8

15
Effective length of the neck (includes radiation
resistance), where a is the radius of the
neck

• L L (.85 .85)a L 1.7a,
• when the outer end is flanged (inside, where
  neck attaches to cavity is always assumed to be
  flanged)
• L L (.6 .85)a L 1.45a,
• when the outer end is unflanged (inside, where
  neck attaches to cavity is again assumed to be
  flanged)

16
Helmholtz Resonator - Assumptions

• the total mass (the mass in the plug plus the
  radiation mass) is equal to the density times the
  volume, so
• m r0 S L
• stiffness is (see derivation in book)
• s r0 c2 S2 / V
• resonant frequency is then

17
Helmholtz Resonator - Example

• 2 liter bottle
• f 343 / 2 / pi
• sqrt(pi / 4 (2.1e-2)2 / (4e-2 1.45
  1.05e-2)/ (2e3 (1e-2)3 ))
• f 96.67Hz

18
Helmholtz Resonator - 2 liter
19
Helmholtz Resonator - 500ml bottle

• f 343 / 2 / pi
• sqrt(pi / 4 (2.1e-2)2 / (2.8e-2 1.45
  1.05e-2)/ (500 (1e-2)3 ))
• f 218 Hz

20
Helmholtz Resonator - 739ml bottle

• f 343 / 2 / pi
• sqrt(pi / 4 (2.1e-2)2 / (2.8e-2 1.45
  1.05e-2)/ (739 (1e-2)3 ))
• f 180 Hz

21
f 110 exp((i/12) ln(2))

• A - 110.0000 Hz
• A - 116.5409
• B - 123.4708
• C - 130.8128
• C - 138.5913
• D - 146.8324
• D - 155.5635
• E - 164.8138
• F - 174.6141
• F - 184.9972
• G - 195.9977
• G - 207.6523

22
f 220 2(i/12)

• A - 220 hz
• A - 233.0818808
• B - 246.9416506
• C - 261.6255653
• C - 277.182631
• D - 293.6647679
• D - 311.1269837
• E - 329.6275569
• F - 349.2282314
• F - 369.9944227
• G - 391.995436
• G - 415.3046976