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: 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.

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. 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

7. 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)

8. 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

9. 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

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 = 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)

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 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

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, L I L P ≈