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

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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: 1.The LaGrange description We follow a “particle” of fluid as it is compressed as well as displaced by an acoustic wave.) 2.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.

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PLANE SOUND WAVES

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Plane Sound Waves

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

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

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SOUND PRESSURE, POWER AND LOUDNESS Decibels Decibel difference between two power levels: ΔL = L 2 – L 1 = 10 log W 2 /W 1 Sound Power Level: L w = 10 log W/W 0 W 0 = 10 -12 W (or PWL) Sound Intensity Level: L I = 10 log I/I 0 I 0 = 10 -12 W/m 2 (or SIL)

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FREE FIELD I = W/4πr 2 at r = 1 m: L I = 10 log I/10 -12 = 10 log W/10 -12 – 10 log 4 = L W - 11

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HEMISPHERICAL FIELD I = W/2 r 2 at r = l m L I = L W - 8 Note that the intensity I 1/r 2 for both free and hemispherical fields; therefore, L I decreases 6 dB for each doubling of distance

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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 = p 2 /ρc ρ = density c = speed of sound We define sound pressure level: L p = 20 log p/p 0 p 0 = 2 x 10 -5 Pa (or N/m 2 ) (or SPL)

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TYPICAL SOUND LEVELS

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

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MULTIPLE SOURCES What we really want to add are mean-square average pressures (average values of p 2 ) This is equivalent to adding intensities Example: 3 sources of 50 dB each Lp = 10 log [(P 1 2 +P 2 2 +P 3 2 )/P 0 2 ] = 10 log ( I 1 + I 2 + I 3 )/ I 0 ) = 10 log I 1 / I 0 + 10 log 3 = 50 + 4.8 = 54.8 dB

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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, L I L P ≈

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