1. NORMAL MODES AND COUPLED ROOMS
ACOUSTICS OF CONCERT HALLS AND ROOMS
NORMAL MODES AND COUPLED ROOMS
Principles of Vibration and Sound
Chapters 6 and 11

2. NORMAL MODES IN CAVITIES
THE WAVE EQUATION IN THREE DIMENSIONS:
IN RECTANGULAR COORDINATES, THIS BECOMES:
WHOSE SOLUTIONS ARE:

3. SOLUTIONS TO THE WAVE EQUATION IN RECTANGULAR COORDINATES
WHERE a,b, and c ARE THE ROOM DIMENSIONS and l, m, and n are three integers that denote the number of half-wavelengths in the 3 directions
CORRESPONDING MODE FREQUENCIES ARE:
(SEE CHAPTER 6 IN ROSSING AND FLETCHER)

4. CONTOURS OF
EQUAL SOUND
PRESSURE IN A
RECTANGULAR
ROOM
a) (2,0,0) axial mode
b) (3,2,0)
tangential mode

5. DISTRIBUTION OF MODE FREQUENCIES FOR 2 ROOMS
l : w : h = 2 : 2 : 2
l : w : h = 3 : 2 : 1

6. FREQUENCY DISTRIBUTION OF ROOM MODES
A CUBE HAS A VERY “PEAKY” RESPONSE; A RECTANGULAR ROOM WITH DIMENTIONS 3 : 2 : 1 HAS A MORE EVEN SPREAD. THE “GOLDEN RATIO’ : 1 : IS EVEN BETTER
NUMBER OF MODES WITH FREQUENCIES FROM 0 TO UPPER LIMIT f :
ABOVE THE SCHROEDER CUTOFF FREQUENCY fsc THE RESONANCE PEAKS BECOME A SMOOTHED OUT CONTINUUM, AND THE SUM OVER MODE INDICES CAN BE APPROXIMATED BY AN INTEGRAL

7. WALLS AND NOISE BARRIERS
WHEN A SOUND WAVE STRIKES A SOLID WALL, THE LARGEST PART IS REFLECTED WHEREAS SMALLER PORTIONS ARE ABSORBED AND TRANSMITTED THROUGH THE WALL
THE TRANSMISSION COEFFICIENT τ IS GIVEN BY τ = IT / I0
AND THE TRANSMISSION LOSS IN dB IS
WHERE M IS THE WALL MASS DENSITY AND f IS THE FREQUENCY.
TRANSMISSION LOSS MAY FALL BELOW THIS PREDICTED VALUE, HOWEVER, DUE TO WALL RESONANCES, LEAKS AND CRACKS, AND ESPECIALLY EXCITATION OF BENDING WAVES AT THE CRITICAL FREQUENCY (WHERE THEY TRAVEL AT THE SAME SPEED AS CERTAIN SOUND WAVES IN THE AIR)

8. TRANSMISSION LOSS (TL) OF A WALL AS A FUNCTI ON OF MASS AND FREQUENCY

9. TRANSMISSION LOSS (dB) WITHOUT A HOLE

10. TWO ROOMS COUPLED BY AN OPENING
COUPLED ROOMS
TWO ROOMS COUPLED BY AN OPENING
WITH AREA S

11. ENERGY DENSITY IN ROOM 1 CONTAINING THE SOURCE
SOUND POWER ABSORBED IN TWO ROOMS (ASSUMING DIFFUSE SOUND FIELDS ARE A10E1c/4 and A20E2 c/4.
POWER TRANSFERRED FROM ROOM 1 TO ROOM 2 IS SE1c/4 and POWER TRANSFERRED FROM ROOM 2 TO ROOM 1 IS SE2c/4
ENERGY DENSITY IN ROOM 1 CONTAINING THE SOURCE
ENERGY DENSITY IN TWO ROOMS TREATED AS A SINGLE SPACE
DERIVATION APPEARS IN PRINCIPLES OF VIBRATION AND SOUND 2nd ed., CHAPTER 11 (ROSSING AND FLETCHER, 2004).

12. REVERBERATION IN COUPLED ROOMS
IF THERE IS NO POWER SOURCE IN THE ROOMS, THE SOUND DECAY CAN BE WRITTEN
THE SOLUTION TO THESE TWO EQUATIONS LEADS TO COMPOUND REVERBERATION DECAY CURVE WITH TWO SLOPES.

13. DECAY OF REVERBERANT SOUND IN A ROOM WITH DIFFERENT REVERBERATION TIMES IN TWO COUPLED SUBSPACES

14. COMPOUND REVERBERATION DECAY CURVE
IN A ROOM WITH A COMPOUND DECAY CURVE, A LISTENER MIGHT CHARACTERIZE THE HALL AS “DRY” ON THE BASIS OF THE FASTER INITIAL DECAY EVEN THOUGH THE 60-dB DECAY IS SLOW.
THE EARLY DECAY TIME (EDT) IS AN IMPORTANT ROOM PARAMETER. SCHROEDER et al. (1974) FOUND THAT AUDIENCES PREFER EDT OF ABOUT 2 SECONDS.