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Published byStewart Arnold Modified over 4 years ago

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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 AirWater Steel Bulk Modulus 1.4(1.01 x 10 5) Pa 2.2 x 10 9 Pa ~2.5 x 10 11 Pa Density1.21 kg/m 3 1000 kg/m 3 ~10 4 kg/m 3 Speed 343 m/s1500 m/s 5000 m/s 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 Eulerian – Fixed Moorings Lagrangian – Drifting Buoys Material, substantial or Lagrangian Derivative Eulerian Derivative Convective Term

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Newton’s 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 AirWaterSteel Bulk Modulus 1.4(1.01 x 10 5) Pa 2.2 x 10 9 Pa~2.5 x 10 11 Pa Density1.21 kg/m 3 1000 kg/m 3 ~10 4 kg/m 3 Speed 343 m/s1500 m/s5000 m/s Spec. Ac. Imp.415 Pa-s/m1.5 x 10 6 Pa-s/m 5 x 10 7 Pa-s/m Analogous to E-M wave impedance

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Plane wave in an arbitrary direction

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Shorthand x y z Direction Cosines Surfaces (planes) of constant phase Propagation Vector

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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. SourceIntensity (W/m 2 )Sound Level Jet Plane100140 Pain Threshold1120 Siren1x10 -2 100 Busy Traffic1x10 -5 70 Conversation3x10 -6 65 Whisper1x10 -10 20 Rustle of leaves1x10 -11 10 Hearing Threshold1x10 -12 1

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Sound Level - Decibel

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Ears judge loudness on a logarithmic vice linear scale Alexander Graham Bell deci = 1 bel = 10 decibel Why the decibel?

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Reference Level Conventions Location Reference Intensity Reference Pressure Air1 x 10 -12 W/m 2 20 Pa Water6.67 x 10 -19 W/m 2 1 uPa

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Historical Reference 1 microbar 1 bar = 1 x 10 5 Pa 1 bar = 1 x 10 5 Pa So to convert from intensity levels referenced to 1 bar to intensity levels referenced to 1 Pa, simply add 100 dB

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Sound Pressure Level Mean Squared Quantities: Power, Energy, Intensity Root Mean Squared Quantities: Voltage, Current, Pressure “Intensity Level” “Sound Pressure Level”

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Example Tube with a piston driver –a=2.5 cm –f = 1 kHz –154 dB in air What are the –rms piston displacement –intensity –power

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Spherical Waves Standing wave n=0,1,2,3,… m=-n,…,+n Traveling wave

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Spherical Waves For Us

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