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

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)

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

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

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

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).

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.