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Acoustics of Concert Halls and Rooms
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Acoustics of Concert Halls and Rooms

Acoustics of Concert Halls and Rooms

SOUND WAVES AND SOUND FIELDS

Principles of Sound and Vibration, Chapter 6

Science of Sound, Chapter 6

Slide 2

The acoustic wave equation

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.

Slide 3

PLANE SOUND WAVES

PLANE SOUND WAVES

Slide 4

Plane Sound Waves

Plane Sound Waves

Slide 5

SPHERICAL WAVES

SPHERICAL WAVES

We can simplify matters even further by writing p = ψ/r, giving

(a one dimensional wave equation)

Slide 6

Spherical waves:

Spherical waves:

Particle (acoustic) velocity:

Impedance:

The solution is an outgoing plus an incoming wave

ρc at kr >> 1

Similar to:

ρ ∂2ξ/∂t2 = -∂p/∂x

outgoing incoming

Slide 7

SOUND PRESSURE, POWER AND LOUDNESS

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/W0 W0 = 10-12 W

(or PWL)

Sound Intensity Level: LI = 10 log I/I0 I0 = 10-12 W/m2

(or SIL)

Slide 8

FREE FIELD

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

Slide 9

HEMISPHERICAL FIELD

HEMISPHERICAL FIELD

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

Slide 10

Sound pressure level

Sound pressure level

Our ears respond to extremely small pressure

fluctuations p

Intensity of a sound wave is proportional to the sound

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