Hearing by numbers

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Hearing by numbers

• Dr Max Graham
• Faculty of Advanced Technology
• University of Glamorgan

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First principles Generation of a sound wave from
a vibrating source
A simple plane progressive wave
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Relationship between velocity (v), frequency (f),
wavelength (?) and period (T)
V ?f m/s
T 1/f s
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Some velocities of sound
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Frequency range of human hearing

• Roughly between 20 Hz and 20 kHz
• Below 20 Hz is known as infrasound
• Above 20 kHz is known as ultrasound
• Perception of maximum loudness is around 4 kHz

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Equal loudness contours (humans)
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Frequency ranges of animals
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Noise measurement Single and third octave band
centre frequencies
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Noise spectrum based on single octave band
frequency analysis
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Wavelengths of frequency limits of human hearing

• V ?f m/s, so ? V/f m 340/f m
• At 20 Hz, ?340/20 17 m
• At 20 kHz, ?340/20 000 17 mm
• And beyond..
• At 20 MHz, ?340/20 000 000 0.017 mm
• (less than the thickness of a human hair)

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A wavelength of 100 mm

• Frequency V/? 340/0.1 3400 H z

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Measurement of sound levels?

• Decibels (dB)
• But what are they?

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Interpretation of the decibel scale
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Range of sound powers
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Range of sound pressures

• Threshold of hearing 2 x 10-5 N/m2 or Pa
• Threshold of pain 200 Pa
• Atmospheric pressure 101325 N/m2

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A linear scale?

• Adopting a linear scale for say sound power over
  the range from the threshold of hearing (10-12
  W) to the threshold of pain (100 W)
• A scale of 1014 increments
• If each increment was represented by an atom
  (typical diameter 10-10 m), the scale would
  stretch for 10 km!

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A logarithmic scale

• By definition
• if logab c,
• then ac b

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The Bel scale

• Sound power level (LW)
• Sound power of source / Threshold power
• W/Wo
• Wo 10-12 Watts
• Using logarithms, we create the Bel scale
• LW log (W/Wo) Bels
• But this gives us a range between
• Threshold of hearing log (10-12/10-12) log 1
  0 Bels
• Threshold of pain log (100 / 10-12) log 1014
  14 Bels
• A bit limited!

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The deci-Bel (dB) scale

• The Bel scale was too limited. It is opened up by
  a factor of 10 to provide the decibel scale
• LW 10 log (W/Wo) dB re Wo 10-12 W
• We now have a more realistic range of between 0
  dB (threshold of hearing) and 140 dB (threshold
  of pain)

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Alternative decibel scales

• Power (W) of a noise source is expressed as the
  sound power level (dB)
• Intensity (W/m2) at some distance is expressed as
  the sound intensity level (dB)
• Pressure (N/m2) at some distance is expressed as
  the sound pressure level (dB)

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Definition of decibel (dB) parameters

• Sound power level (LW)
• LW 10 log (W/Wo) dB re Wo 10-12 W
• Sound intensity level (LI)
• LI 10 log (I/Io) dB re Io 10-12 W/m2
• Sound pressure level (Lp)
• Lp 20 log (P/Po) dB re 2 x 10-5 N/m2 (Pa) or 20
  µPa

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Sound pressure level (Lp)

• Where does the 20 come from?
• P vIz N/m2 or P2 Iz
• Remember log xn n logx
• Z is the characteristic impedance of the medium
  through which the sound is travelling
• Z density of the medium (kg/m3) x velocity of
  sound through it (m/s)
• Zair 1.2 x 340 408 kg/m2s (Rayls), roughly
  400 Rayls
• Consider reference values
• Po vIoz v10-12 x 400 v4 x 10-10 2 x 10-5
  N/m2

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What are the limits?

• How quiet can we go?
• How loud can we go?

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Decibel manipulation 1Combination of multiple
sources (equal noise levels)

• 1 noise source x dB
• 2 noise sources x 3 dB
• ?L 10 log(2I/Io) 10 log(I/Io) dB
• 10 log 2 3 dB
• 10 noise sources x 10 dB
• ?L 10 log(10I/Io) 10 log(I/Io) dB
• 10 log 10 10 dB
• N noise sources x 10 logN dB
• ?L 10logN dB

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Addition of decibels

• A general expression
• L 10 log(10L1/10 10 L2/10 . 10 Ln/10) dB

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Decibel manipulation 2Subtraction of decibels

• Subtract a noise level L2 from a noise level L1
• New level, L 10 log(10L!/10 10 L2/10) dB
• Example
• 80 dB 85 dB 10 log (108.0 108.5) dB 86.2
  dB
• 86.2 85 dB 10 log (108.62 108.5) dB 80
  dB

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Decibel manipulation 3(Averaging decibels of
equal time periods)

• If noise levels are sounded over equal time
  periods the average noise level over the total
  period they have been sounding for is given by
• LAVG 10 log((10L1/10 10L2/10 10Ln/10)/n)
  dB
• Example, the average of 80 dB and 100 dB is given
  by
• LAVG 10 log((108.0 1010)/2) dB 97 dB
• NOT (80 100) / 2 90 dB (the arithmetic
  average)

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Decibel manipulation 4(Time weighted averages)

• The average noise level (having the same acoustic
  energy) for several noise levels sounding over
  different time periods can be calculated and
  expressed over any time period (T)
• Leq,T 10 log ((t1 x 10L1/10 t2 x 10L2/10 .
  tn x 10Ln/10)/T) dB
• This can be applied to a single source or any
  number of sources

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Time weighted averagesAn example

• A noise level of 90 dB sounds continuously over a
  period of one hour.
• The noise level this is equivalent to over a
  period of 15 minutes is given by
• L 10 log ((t x 10L/10)/T dB
• 10 log (60 x 109/15) dB 96 dB
• Similarly, the noise level this is equivalent to
  over a period of 8 hours is given by
• L 10 log(1 x 109/8) dB 81 dB

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Dont forget your basics

• 90 dB for 1 hour 96 dB for 15 minutes
• Why?
• 4 x energy for a quarter of the time
• 90 dB for 1 hour 81 dB for 8 hours
• 1/8 of the energy for 8 x the time
• We are equating the energy in all cases.

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Similar principles apply to heat and light
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The analogy between sound and heat (also applies
to light)
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Distribution of energy from airborne sound
striking a partition
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Introducing the transmission coefficient (t)

• The transmission coefficient (t) is the fraction
  of the acoustic energy which is transmitted
  through a partition
• TL and t are related
• TL 10 log(1/t) dB

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Measures of sound insulation
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Implications for composite construction

• TL 10 log (ATOT /(A1t1 A2t2 An tn)) dB
• The presence of a relatively small area of
  construction with a high transmission coefficient
  will seriously reduce the overall sound
  insulation

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An example

• The incorporation of air gaps to the extent of 1
  of the area of a concrete block wall which
  should, if properly built, have an insulation of
  50 dB, will reduce this to 20 dB the equivalent
  of cheaply fitted single glazing!

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Sound transmission characteristics of a
partition The mass law
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Resonances

• Prediction of resonant frequencies
• F 0.45Vh((Nx/Lx)2 (Ny/Ly)2) Hz
• F resonant frequency (Hz)
• V velocity of sound (m/s)
• H thickness of element (m)
• Nx, Ny are integers 1,2,3 etc
• Lx width (m)
• Ly height (m)

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An example

• Calculate the lowest resonant frequency for 6 mm
  single glazing which is 2 m high and 1m wide
• F 0.45x5300x0.006x((1/1)2 (1/2)2) Hz
• 18 Hz
• Second and third resonant frequencies are 72 Hz
  and 161 Hz

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Critical frequency(the lowest frequency at which
coincidence occurs)

• Critical frequency VA2/1.8hVL Hz
• VA velocity of sound in air (340 m/s)
• H thickness of element (m)
• VL longitudinal velocity of sound through the
  element (m/s)

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An example

• Calculate the critical frequency for a 100 mm
  brick wall. The longitudinal velocity of sound
  through brickwork is 2350 m/s
• Critical frequency 3402 / (1.8 x 0.1 x 2350) Hz
• 273 Hz

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Thank you