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SOUND WAVES AND SOUND FIELDS

SOUND WAVES AND SOUND FIELDS. Acoustics of Concert Halls and Rooms. Principles of Sound and Vibration, Chapter 6 Science of Sound, Chapter 6. THE ACOUSTIC WAVE EQUATION. The acoustic wave equation is generally derived by considering an ideal fluid (a mathematical fiction).

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SOUND WAVES AND SOUND FIELDS

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  1. SOUND WAVES AND SOUND FIELDS Acoustics of Concert Halls and Rooms • Principles of Sound and Vibration, Chapter 6 • Science of Sound, Chapter 6

  2. THE ACOUSTIC WAVE EQUATION The acoustic wave equation is generally derived by considering an ideal fluid (a mathematical fiction). Its motion is described by the Euler equation of motion. In a real fluid (with viscosity), the Euler equation is Replaced by the Navier-Stokes equation. • Two different notations are used to derive the Acoustic wave • equation: • The LaGrange description • We follow a “particle” of fluid as it is compressed as well as displaced by an acoustic wave.) • The Euler description • (Fixed coordinates; p and c are functions of x and t. • They describe different portions of the fluid as it streams past.

  3. PLANE SOUND WAVES

  4. Plane Sound Waves

  5. SPHERICAL WAVES We can simplify matters even further by writing p = ψ/r, giving (a one dimensional wave equation)

  6. The solution is an outgoing plus an incoming wave Spherical waves: Similar to: ρ ∂2ξ/∂t2 = -∂p/∂x outgoing incoming Particle (acoustic) velocity: Impedance: ρc at kr >> 1

  7. SOUND PRESSURE, POWER AND LOUDNESS Decibels Decibel difference between two power levels: ΔL = L2 – L1 = 10 log W2/W1 Sound Power Level:Lw = 10 log W/W0W0 = 10-12 W (or PWL) Sound Intensity Level:LI = 10 log I/I0I0 = 10-12 W/m2 (or SIL)

  8. FREE FIELD I = W/4πr2 at r = 1 m: LI = 10 log I/10-12 = 10 log W/10-12 – 10 log 4p = LW - 11

  9. HEMISPHERICALFIELD I = W/2pr2 at r = l m LI = LW - 8 Note that the intensity I 1/r2 for both free and hemispherical fields; therefore, LI decreases 6 dB for each doubling of distance

  10. SOUND PRESSURE LEVEL Our ears respond to extremely small pressure fluctuations p Intensity of a sound wave is proportional to the sound Pressure squared: ρc ≈ 400 I = p2 /ρcρ = density c = speed of sound We define sound pressure level: Lp = 20 log p/p0 p0 = 2 x 10-5 Pa (or N/m2) (or SPL)

  11. TYPICAL SOUND LEVELS

  12. MULTIPLE SOURCES Example:Two uncorrelated sources of 80 dB each will produce a sound level of 83dB (Not 160 dB)

  13. MULTIPLE SOURCES What we really want to add are mean-square average pressures (average values of p2) This is equivalent to adding intensities Example: 3 sources of 50 dB each Lp = 10 log [(P12+P22+P32)/P02] = 10 log (I1 + I2 + I3)/ I0) = 10 log I1/I0 + 10 log 3 = 50 + 4.8 = 54.8 dB

  14. SOUND PRESSURE and INTENSITY Sound pressure level is measured with a sound level meter (SLM) Sound intensity level is more difficult to measure, and it requires more than one microphone In a free field, however, LI LP ≈

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