normal modes and coupled rooms acoustics of concert halls and rooms principles of vibration and...

NORMAL MODES AND

COUPLED ROOMS

ACOUSTICS OF CONCERT HALLS AND ROOMS

Principles of Vibration and SoundChapters 6 and 11

NORMAL MODES IN CAVITIES

THE WAVE EQUATION IN THREE DIMENSIONS:

IN RECTANGULAR COORDINATES, THIS BECOMES:

WHOSE SOLUTIONS ARE:

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)

a) (2,0,0)axial mode

b) (3,2,0)tangential mode

CONTOURS OF EQUAL SOUND PRESSURE IN ARECTANGULARROOM

DISTRIBUTION OF MODE FREQUENCIES FOR 2 ROOMS

l : w : h = 2 : 2 : 2

l : w : h = 3 : 2 : 1

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’ 1.618 : 1 : 0.618 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

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)

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

TRANSMISSION LOSS (dB) WITHOUT A HOLE

COUPLED ROOMS

TWO ROOMS COUPLED BY AN OPENING WITH AREA S

ENERGY DENSITY IN ROOM 1 CONTAINING THE SOURCE

ENERGY DENSITY IN TWO ROOMS TREATED AS A SINGLE SPACE

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

DERIVATION APPEARS IN PRINCIPLES OF VIBRATION AND SOUND 2nd ed., CHAPTER 11 (ROSSING AND FLETCHER, 2004).

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.

DECAY OF REVERBERANT SOUND IN A ROOM WITH DIFFERENT REVERBERATION TIMES IN TWO

COUPLED SUBSPACES

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.