Acoustics

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Acoustics

• Week 1
• Fundamentals of Sound
• Sound Levels and the Decibel

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Fundamentals of Sound

• Sound can be defined as a wave motion in air or
  other elastic media (stimulus) or as that
  excitation of the hearing mechanism that results
  in the perception of sound (sensation).
• Frequency is a characteristic of periodic waves
  measured in hertz (cycles per second), readable
  on an oscilloscope or frequency counter. The ear
  perceives a different pitch for a quiet tone than
  a loud one. The pitch of a low-frequency tone
  goes down, while the pitch of a high-frequency
  tone goes up as intensity increases. We cannot
  equate frequency and pitch, but they are
  analogous.
• The same situation exists between intensity and
  loudness. The relationship between the two is not
  linear.
• Similarly, the relationship between waveform (or
  spectrum) and perceived quality (or timbre) is
  complicated by the functioning of the hearing
  mechanism.

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Sine Waves

• The sine wave is a basic waveform closely related
  to simple harmonic motion. Vibration or
  oscillation is possible because of the elasticity
  of the spring and the inertia of the weight.
  Elasticity and inertia are two things all media
  must possess to be capable of conducting sound.

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Sine Wave Language

• The easiest value to read is the peak-to-peak
  value (of voltage, current, sound pressure,
  etc.). If the wave is symmetrical, the
  peak-to-peak value is twice the peak value.
  Another common way to measure the sine wave is
  using RMS (Root Mean Square) values. The other
  two scales of measurement used are peak and
  average. The picture to the right shows four
  different ways that a sine waves amplitude can
  be measured and the equations used for
  conversions.

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Propagation of Sound

• If an air particle is displaced from its original
  position, elastic forces of the air tend to
  restore it to its original position. Because of
  the inertia of the particle, it overshoots the
  resting position.
• Sound is readily conducted in gases, liquids, and
  solids such as air, water, steel, etc., which are
  all elastic media.
• Without a medium, sound cannot be propagated.
  Outer space is an almost perfect vacuum no sound
  can be conducted due to the absence of air.
• Particles of air propagating a sound wave do not
  move far from their undisplaced positions.

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Forms of Particle Motion

• There is more than a million molecules in a cubic
  inch of air.
• The molecules crowded together represent areas of
  compression and the sparse areas represent
  rarefactions.

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Sound in Free Space

• The intensity of sound decreases as the distance
  to the source is increased.
• Doubling the distance reduces the intensity to
  1/4 the initial value, tripling the distance
  yields 1/9, increasing the distance four times
  yields 1/16 the initial intensity.
• The inverse square law states that the intensity
  of sound in a free field is inversely
  proportional to the square of the distance from
  the source.

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Wavelength, Period, Frequency

• The wavelength is the distance a wave travels in
  the time it takes to complete one cycle. The
  period is the time it takes a wave to complete
  one cycle. The frequency is the number of cycles
  per second (Hertz).

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Wavelength and Frequency
Formulas for calculating wavelength and
frequency. The speed of sound in air is about
1,130 feet per second at normal temperature and
pressure.
Two graphical approaches for an easy solution to
the above equations.
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Complex Waves

• The sine wave with the lowest frequency (f1) is
  called the fundamental, the one with twice the
  frequency (f2) is called the second harmonic, and
  the one three times the frequency (f3) is the
  third harmonic. Harmonics are whole number
  multiples of the fundamental frequency.

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Phase

• Phase is the time relationship between waveforms.
  Each waveform is lagging the previous by 90
  degrees.

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Combinations of Waveforms

• Combinations of waveforms that are not in phase.
  The difference in waveshapes is due entirely to
  the shifting of the phase of the harmonics with
  respect to the fundamental.

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Harmonics and Octaves
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Spectrum

• The audio or frequency spectrum of the human ear
  is about 20 Hz to 20 kHz. The spectrum tells how
  the energy of the signal is distributed in
  frequency. For the ideal sine wave, all the
  energy is concentrated at one frequency. All
  other types of waveforms have more than one
  frequency present.
• The diagram shows various types of waveforms and
  their harmonic content. The sine, triangle, and
  square waves are known as period waves due to
  their cyclic pattern.

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Sound Levels and the Decibel

• Levels in decibels make it easy to handle the
  extremely wide range of sensitivity in human
  hearing.
• The threshold of hearing matches the ultimate
  lower limit of perceptible sound in air.
• A level in decibels is a convenient way of
  handling the billion fold range of sound
  pressures to which the ear is sensitive without
  getting bogged down in a long string of zeros.

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Ratios vs. Difference

• Ratios of pressure seem to describe loudness
  changes better than difference in pressure.
• Ernst Weber (1834), Gustaf Fechner (1860),
  Hermann von Helmholtz (1873), and other
  researchers pointed out the importance of ratios.
• Ratios of stimuli come closer to matching up with
  human perception than do differences of stimuli.
• Ratios of powers, intensities, sound pressure,
  voltage, or anything else are unitless. This is
  important because logarithms can be taken only of
  unitless numbers.

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Handling Numbers

• Here are three different ways that numbers can be
  expressed

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Handling Numbers, contd.
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Back to Basics Math

• The study of acoustics requires a knowledge of
  some basic algebra.

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More Math...
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More Math...
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Exponents A Review
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Exponents, contd.
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Exponents, contd.
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Exponents, contd.
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Exponents, contd.
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Exponents, contd.
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Exponents, contd.
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Exponents, contd.
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Practice Problems
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Logarithms
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Logarithms
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Logarithms
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Practice Problems
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Practice Problems
Theorem 1
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Decibels

• A level is a logarithm of a ratio of two
  like-power quantities.
• A level in decibels is ten times the logarithm to
  the base 10 of the ratio of two power quantities.

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Decibels

• Sound pressure is proportional to (sound power)2.

• The squaring of the sound power results in the
  equation SPL 20 log (p1/p2) instead of 10 log.

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Sound Pressure Level (SPL)

• Sound pressure is usually the most accessible
  parameter to measure in acoustics, as voltage is
  for electronic circuits. For this reason, the
  Equation (2-3) form is more often encountered in
  day-to-day technical work.

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Reference Levels

• A sound level meter is used to read sound
  pressure levels. A sound level meter reading is a
  certain sound pressure level, 20 log (p1/p2). For
  sound in air, the standard reference pressure is
  20 µPa (micropascal). The µPa is a very minute
  sound pressure and corresponds closely to the
  threshold of human hearing.
• The equations to the right show how to convert
  SPL (in dB) to µPa.

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Log-to-Exponent Conversion

• Heres another quick look at the math used to
  convert logs to exponents

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Acoustic Reference Levels
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Acoustic Reference Levels

• Greek prefixes used for the powers of 10

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Pascals vs. Decibels

• Doubling of acoustic power is an increase of 3 dB
  (10 log 2 3.01).
• Doubling of acoustic pressure is an increase of 6
  dB (20 log 2 6.02)

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Pascals vs. Decibels
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Decibel Level Examples

• Here are some examples of decibel level
  conversions

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Decibel Level Examples
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Examples, contd.
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Examples, contd.
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Reflected Sound

• The equations below show how to calculate
  Reflection Delay and Reflection Level.

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Examples, contd.
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Examples, contd.
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Examples, contd.
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Ratios and Octaves

• An octave is defined as a 21 ratio of two
  frequencies. The ratio 32 is a fifth and 43 is
  a fourth.

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Examples
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Examples, contd.
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Measuring Sound Pressure Level

• Sound level meters usually offer a selection of
  weighting networks designated A, B, and C.
  Network selection is based on the general level
  of sounds to be measured. For SPLs of 20 55 dB,
  use network A. For 55 85 dB use network B, and
  for 85 140 dB, use network C.

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Equal Loudness Contours
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The Precedence (Haas) Effect
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SPL Changes With Distance

• The Inverse Square Law says that the intensity of
  sound is inversely proportional to the square of
  the distance from the point source.

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Inverse Distance Law

• The Inverse Square Law of intensity becomes the
  Inverse Distance Law for sound pressure. Sound
  pressure level is reduced 6 dB for each doubling
  of distance.

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Critical Distance

• The Critical Distance is that distance at which
  the direct sound pressure is equal to the
  reverberant sound pressure.