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Acoustic Wave Equation

Acoustic Wave Equation

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Acoustic Wave Equation

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  1. Acoustic Wave Equation

  2. Acoustic Variables • Pressure • Density – Condensation • Velocity (particle) • Temperature

  3. Sound Speed Please Memorize!!!

  4. Necessary Differential Equations to Obtain a Wave Equation • Mass Continuity • Equation of State • Force Equation – N2L Assumptions: homogeneous, isotropic, ideal fluid

  5. Equations of State Ideal Gasses: Real Fluids:

  6. Continuity Equation

  7. Force Equation

  8. Fluid Acceleration

  9. Lagrangian and Eulerian Variables • Eulerian – Fixed Moorings • Lagrangian – Drifting Buoys Convective Term Material, substantial or Lagrangian Derivative Eulerian Derivative

  10. Newton’s Second Law

  11. Linear Continuity Equation

  12. Linear Force Equation

  13. Linear Wave Equation

  14. Velocity Potential

  15. Variation of sound speed with temperature

  16. Speed of sound in water-temperature, pressure, and salinity

  17. Class Sound Speed Data

  18. Harmonic 1-D Plane Waves

  19. Condensation and Velocity Potential

  20. Specific Acoustic Impedance Mechanical Impedance For a plane wave: In general:

  21. Sound Speed Analogous to E-M wave impedance

  22. Plane wave in an arbitrary direction

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

  24. k in x-y plane

  25. Energy

  26. Energy Density

  27. Average Power and Intensity A cdt For plane waves

  28. Effective Average - RMS

  29. 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.

  30. Sound Level - Decibel

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

  32. Reference Level Conventions

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 Historical Reference • 1 microbar • 1 bar = 1 x 105 Pa • 1 mbar = 1 x 105mPa • So to convert from intensity levels referenced to 1 mbar to intensity levels referenced to 1 mPa,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

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

  35. 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

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

  37. Spherical Waves For Us