Acoustic Wave Equation

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Acoustic Wave Equation
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Acoustic Variables

• Pressure
• Density Condensation
• Velocity (particle)
• Temperature

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Sound Speed
Please Memorize!!!
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Necessary Differential Equations to Obtain a Wave
Equation

• Mass Continuity
• Equation of State
• Force Equation N2L

Assumptions homogeneous, isotropic, ideal fluid
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Equations of State
Ideal Gasses
Real Fluids
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Continuity Equation
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Force Equation
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Fluid Acceleration
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Lagrangian and Eulerian Variables
Convective Term
Material, substantial or Lagrangian Derivative
Eulerian Derivative

• Eulerian Fixed Moorings
• Lagrangian Drifting Buoys

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Newtons Second Law
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Linear Continuity Equation
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Linear Force Equation
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Linear Wave Equation
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Velocity Potential
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Variation of sound speed with temperature
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Speed of sound in water-temperature, pressure,
and salinity
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Class Sound Speed Data
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Harmonic 1-D Plane Waves
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Condensation and Velocity Potential
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Specific Acoustic Impedance
Mechanical Impedance
For a plane wave
In general
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Sound Speed
Analogous to E-M wave impedance
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Plane wave in an arbitrary direction
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Shorthand
z
Direction Cosines
y
x
Propagation Vector
Surfaces (planes) of constant phase
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k in x-y plane
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Energy
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Energy Density
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Average Power and Intensity
A
cdt
For plane waves
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Effective Average - RMS
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Intensity of sound

• Loudness intensity of the wave. Energy
  transported by a wave per unit time across a unit
  area perpendicular to the energy flow.

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Sound Level - Decibel
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Why the decibel?

• Ears judge loudness on a logarithmic vice linear
  scale
• Alexander Graham Bell
• deci
• 1 bel 10 decibel

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Reference Level Conventions
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Historical Reference

• 1 microbar
• 1 bar 1 x 105 Pa
• 1 mbar 1 x 105 mPa
• So to convert from intensity levels referenced to
  1 mbar to intensity levels referenced to 1 mPa,
  simply add 100 dB

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Sound Pressure Level
Mean Squared Quantities Power, Energy, Intensity
Intensity Level
Root Mean Squared Quantities Voltage, Current,
Pressure
Sound Pressure Level
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Example

• Tube with a piston driver
• a2.5 cm
• f 1 kHz
• 154 dB in air
• What are the
• rms piston displacement
• intensity
• power

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Spherical Waves
Standing wave n0,1,2,3, m-n,,n
Traveling wave
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Spherical Waves For Us
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