Acoustic Wave Equation

Acoustic Variables Pressure Density – Condensation Velocity (particle) Temperature

Sound Speed AirWater Steel Bulk Modulus 1.4(1.01 x 10 5) Pa 2.2 x 10 9 Pa ~2.5 x Pa Density1.21 kg/m kg/m 3 ~10 4 kg/m 3 Speed 343 m/s1500 m/s 5000 m/s Please Memorize!!!

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

Equations of State Ideal Gasses: Real Fluids:

Continuity Equation

Force Equation

Fluid Acceleration

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

Newton’s Second Law

Linear Continuity Equation

Linear Force Equation

Linear Wave Equation

Velocity Potential

Variation of sound speed with temperature

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

Class Sound Speed Data

Harmonic 1-D Plane Waves

Condensation and Velocity Potential

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

Sound Speed AirWaterSteel Bulk Modulus 1.4(1.01 x 10 5) Pa 2.2 x 10 9 Pa~2.5 x Pa Density1.21 kg/m 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

Plane wave in an arbitrary direction

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

k in x-y plane

Energy

Energy Density

Average Power and Intensity A cdt For plane waves

Effective Average - RMS

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 Plane Pain Threshold1120 Siren1x Busy Traffic1x Conversation3x Whisper1x Rustle of leaves1x Hearing Threshold1x

Sound Level - Decibel

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

Reference Level Conventions Location Reference Intensity Reference Pressure Air1 x W/m 2 20  Pa Water6.67 x W/m 2 1 uPa

Historical Reference 1 microbar 1 bar = 1 x 10 5 Pa 1  bar = 1 x 10 5  Pa So to convert from intensity levels referenced to 1  bar to intensity levels referenced to 1  Pa,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

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

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

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

Spherical Waves For Us