Section 5 : Room Acoustics

Content

• Sound Propagation 5.1

5.2 • Room Modes and calculation: Axial : Tangential: Oblique

• Modal Behavior 5.3

• Geometrical Acoustics 5.4

5.5 • Absorption and Reverberation

• RT 60 5.6

GrassCA © 1 5.1: Room Acoustics: Introduction

“the acoustical consultant has to examine the objective acoustical properties of the design by calculation, by geometric ray considerations, by model investigations or even by computer simulation. As a general rule there will have to be some compromise in order to obtain a reasonable result.”

Heinrich Kuttruff

GrassCA © 2

5.1: Room Acoustics: Introduction

As the result of its size and geometry, a room excessively amplifies sound at certain frequencies. This is the result of standing waves (acoustic resonances/modes) of the room. These are waves whose original oscillation is continuously reinforced by their own reflections.

Rooms have many resonances, but only the low-frequency ones are discrete, distinct, unaffected by the sound absorbing material in the room, and accommodate most of the acoustic energy build up in the room. In a typical room similar in size to a home-theater, a listening room or studio control room the resonant frequencies of these standing waves fall in the bass frequency region. Provoking these low-frequency resonances result in boomy noise and frequency colouration in a room.

GrassCA © 3 5.1:Sound Propagation: Standing waves

Example: walls 4m apart, wave is 3.5m long: so will dissipate quickly as its not reinforced. If wave is 8m, 4m or 2m etc., it will reflect back and forth between the walls until it encounters a corner and will reduce in amplitude.

Because of this, unless you have a very square room, the modes will terminate in the edges and corners. When 3.5m wave encounters a 3m ceiling, it will die off.

You can calculate the length of a sound wave fairly easily. You just take the speed of sound in feet-per- second, and divide it by the frequency (waves per second).

Speed of sound ( m/second) = length of wave Frequency(cycles per second)

GrassCA © 4 5.1: Wavelength, Frequency and Standing waves

c f

f λ c 2d

GrassCA © 5 5.1:Sound Propagation: Standing waves

The wave energy bounces between two or more surfaces emphasizing one frequency over others.

The area between two parallel walls resonate at certain frequencies

LINK

A bass wave can be several feet long. This is important, especially below 300hz or so. Above 300hz, the waves become small enough that they aren't affected by the room size as much. They bounce/reflect around in all directions within the room..

GrassCA © 6 5.1:Sound Propagation: Standing waves

The formula for determining the fundamental frequency of a standing wave for a particular room dimension is:

f = c / 2d f = Fundamental frequency of the standing wave c = Velocity of sound (343m/sec(20ºC) (1130 ft/sec)) d = Room dimension being considered in meters or feet (length, width, or height)

Other standing waves occur at harmonics of the fundamental frequency - that is 2, 3, and 4 times the fundamental.

LINK 1 LINK 2

GrassCA © 7 5.1:Sound Propagation: Resonant Frequencies

Ineffective low frequency absorption of furniture, floor/ceiling covering, and wall treatments make the resonant modes highly noticeable and a major issue at low frequencies.

Standing waves can cause certain resonant frequencies to either be unduly enhanced (antinodes) or completely disappear(nodes) so it is always a good idea to listen to your work from a few different spots in the studio. AXIAL MODE.

GrassCA © 8 5.1:Sound Propagation: Standing waves

343ms ÷ (λ)7m = 49Hz 343ms ÷ (½ 3.5m= 98Hz ? m (d) λ) 343ms ÷ ( 2 λ ) 14m= 24.5Hz

343ms ÷ (λ) 3m = 114.3Hz 343ms ÷ (2λ) 6m = 57.2Hz 3m (d) 343ms ÷ (3 λ )9m = 38.1Hz 7m (d)

A room may be viewed as a complex resonator, having distinct acoustic modes at low frequencies. For acoustic resonance of sound at lower frequencies: the modal density (the number of modes in a frequency interval) at these frequencies is by far lower than that at high frequencies.

GrassCA © 9 5.1:. Sound Propagation: Resonant Frequencies

24.5Hz 7m x 3m x 5m At certain frequencies the dimensions of a room are 34Hz integer multiples of the 42.5Hz 57.2Hz wavelength of the tones 49Hz corresponding to those frequencies. 68.6Hz If 34 and 68.6Hz what is height of room? 38Hz This causes the reflection 58Hz of the wave from the walls to reinforce each other MFP and establish standing waves in the room.

These resonant frequencies (characteristic frequencies) of the room and their corresponding standing wave pattern are called mode shapes of the room. The resonance frequencies and the corresponding mode shapes depend primarily on the shape and size of the room.

GrassCA © 10 5.2: Room Modes

Room Modes are the 3 different ways in which acoustic wave energy is reflected within a walled room space( 6 sides).

• Simplest and strongest Axial • Reflected from 2 walls

• Half the power of axial reflections Tangential • Reflected from 4 walls

• Quarter power of axial reflections Oblique • Reflected from 6 walls

GrassCA © 11 5.2: Room Modes: Axial

3 Primary Axial Modes

Red line = width mode Blue line = length mode Green line =height mode

Axial Room Modes involve two parallel walls. In the imaginary reflective room, think of how a laser would reflect of mirrored surfaces. It would bounce back and forth between the two walls. These are Axial Room Modes.

GrassCA © 12

5.2: Room Modes: Axial

3.4 m x 4m x 4.8m

Ratio 1: 1.18 : 1.41

Room ratios are an important part of the initial design process of a new studio.

Calculating the resonances that will be favored between two parallel surfaces is the inverse of the calculation to determine the length of a sound wave. Since a room can enforce a wave twice as long as it is, you can multiply the length of the room dimension by two. f = c / 2d

GrassCA © 13 5.2: Room Modes: Axial Mode Calculation

3.4m 4m 4.8m n n n Fundamental f= V/2d f= V/2d f= V/2d x y z f Type 1 0 0 35.9 Axial ( 2 λ) 50.44Hz 42.88Hz 35.73Hz 0 1 0 43.1 Axial ( λ) 100.88Hz 85.75Hz 71.46Hz 0 0 1 50.7 Axial ( ½ λ) 201.76Hz 171.5Hz 142.92Hz 2 0 0 71.8 Axial 0 2 0 86.1 Axial 3 0 0 107.6 Axial The basic calculations above, in general, match 0 3 0 129.2 Axial the axial mode grid produced by a MODE 0 0 2 101.3 Axial calculator. The calculator required dimensions 4 0 0 143.5 Axial in inches. Therefore some approximations were 0 4 0 172.2 Axial taken by the spreadsheet.(i.e. no decimal point) 5 0 0 179.4 Axial It can be seen, however, that the mode 0 0 3 152.0 Axial calculator gives more detail about mode numbers and other frequencies that may not be 0 5 0 215.3 Axial obvious by basic calculation. 0 0 4 202.7 Axial

GrassCA © 14 5.2: Room Modes: Tangential

Red line = 0.1.1. vertical mode

Blue line = 1.1.0. horizontal mode

Green line = 1.0.1. lateral mode

Since we know the dimensions of the room, and assuming the walls are perpendicular to each other, we're calculating the triangle formed by two adjacent surfaces and the sound ray.(Ray tracing)

GrassCA © 15 5.2: Room Modes: Tangential

Tangential Modes involve four surfaces (two sets of parallel walls) and have about half the energy of Axial modes.

The distance between bounces is not arbitrary. Since sound waves that are out of phase cancel each other out, they must be multiples of each other in order to support the wave throughout the path/circuit. So if one frequency that's supported is 100hz, another one would be 200hz.

GrassCA © 16 5.2: Room Modes: Tangential Mode Calculation

Tangential (ƒxy) = c ÷ 2 √ (x ÷ L)² + (y ÷ W)²

ƒxy(tangential) = tangential modal frequencies (Hz) x = the number of half wavelengths between one set of two surfaces y = the number of half wavelengths between the other set of surfaces L, W = the distance between the reflecting surfaces (m)

H a² + b² = c² 3.4m 11.56+ 16 = 5.25m

W 4m a² + b² = c² H 11.56+ 23.04 = 5.88m 3.4m L 4.8m a² + b² = c² W 11.56+ 16 = 6.25m 4m

GrassCA © 17 L 4.8m 5.2: Room Modes: Tangential Mode Calculation

1 Mode Dimensions nx ny nz f Type Fun f= ½c x (x/a²) +(y/b²) x y z 1 1 0 56.0 Tang 1 0 1 62.1 Tang ( λ) 55.97Hz 1.1.0. 4m x 4.8m 0 1 1 66.5 Tang ( λ) 61.99Hz 1.0.1. 3.4m x 4.8m 2 1 0 83.7 Tang ( λ) 66.39Hz 0.1.1. 4m x 3.4m 1 2 0 93.3 Tang 2 0 1 87.8 Tang ( λ) 83.5Hz 2.1.0. 4m x 3.4m 0 2 1 99.9 Tang ( λ) 87.73Hz 2.0.1. 4.8m x 3.4m 2 2 0 112.1 Tang 3 1 0 115.9 Tang 3 0 1 119.0 Tang 1 3 0 134.0 Tang Use a good room mode 1 0 2 107.5 Tang 0 1 2 110.1 Tang calculator. 3 2 0 137.8 Tang 0 3 1 138.7 Tang

GrassCA © 18 5.2: Room Modes: Oblique

Oblique: Having a slanting or sloping direction, course, or position; inclined

Geometric lines or planes that are neither parallel nor perpendicular.

Similar to Tangential: the ray hits the front, back, left, and right in the same places, but also hits floor and ceiling in between every time. To appreciate every tangential room mode we would need to rotate this diagram for all six dimensions.

GrassCA © 19 5.2: Room Modes: Oblique

Oblique Modes are the most difficult to describe.

They involve all six surfaces, and have about half the energy of Tangential Modes, one quarter of the energy of Axial modes.

This mode looks something like the Tangential Mode, except instead of only reflecting around on a flat plane, it also reflects off of the ceiling and floor

GrassCA © 20 5.2: Room Modes: Oblique Mode Calculation

3.4m 4m 4.8m nx ny nz f Type Fundamental 1 1 1 75.6 Obl 2 1 1 97.8 Obl ( 2 λ) Way Too Dull 1 2 1 106.2 Obl ( λ) Cant Be Bothered 2 2 1 123.0 Obl ( ½ λ) Joking Aren’t You 3 1 1 126.5 Obl 1 1 2 115.8 Obl ƒxyz(oblique) 1 3 1 143.3 Obl = c ÷ 2√(x ÷ L)² + (y ÷ W)² + (z ÷ H)² 2 1 2 131.4 Obl x, y, z = half wavelengths between 3 2 1 146.9 Obl surfaces. 1 2 2 137.7 Obl 2 3 1 156.2 Obl L, W, H= distance of reflecting surfaces(m) 4 1 1 158.2 Obl 2 2 2 151.1 Obl This equation in fact works as a universal 3 1 2 154.0 Obl mode equation, as for axial modes two of the factors will drop out.

GrassCA © 21 5.2:. Mode Calculation; The room 7m x 3m x 5m Odd numbered 38Hz 45.7Hz harmonic multiples 3 λ produce a drop in 9m 3/2 λ 65.3Hz 7.5m 76Hz sound pressure at 3/4 λ that frequency. 5.25m 3/2 λ 4.5m The response graph 22.8Hz can be easily read 3 λ 60 and understood by 15m 32.6Hz using: 3/2 λ 73.5Hz 3 λ , and 3/2 λ and 10.5m 53 2/3 λ 2/3 λ and 3/4 λ for our 4.67m room dimensions.

We struggle to find a dimension that relates to the 53Hz and 63Hz anti-nodes. We know it is not an even function of one of the obvious dimensions.

UWS 2009 GrassCACG:ASD © 22 5.2: Mode Calculation: Tangential

59 68.6 Pythagoras

5.8m 60.2 a² + b² = c² 78.8 5m 25 + 9 = 34 = 5.8

53 a² + b² = c² 7.6m 49 + 9 = 58 = 7.6 3m 60.2Hz = 3/4 λ of 7.6m:ANTINODE 7m 78.8Hz = 3/4 λ of 5.8m: ANTINODE a² + b² = c² 59Hz = 2/3 λ of 8.6m: NODE 5m 25 + 49 = 74= 8.6 53Hz = 3/4 λ of 8.6m: ANTINODE 8.6m 68.6Hz = 2/3 λ of 7.6m: NODE GrassCA © 7m 23 5.2: Mode Calculation: Calculators = .xls spreadsheet

All Modes nx ny nz f Type 7m x 3m x 5m 1 0 0 24.7 Axial 0 1 0 57.5 Axial 1 1 0 62.5 Tangential 0 0 1 34.4 Axial 1 0 1 42.3 Tangential 2 0 0 49.3 Axial 0 1 1 67.0 Tangential 1 1 1 71.4 Oblique 2 1 0 75.7 Tangential 2 0 1 60.1 Tangential The mode calculator will only 2 1 1 83.2 Oblique calculate NODES (+ ve ) 3 0 0 74.0 Axial (even multiples). 0 0 2 68.8 Axial It is only important to measure 3 0 1 81.6 Tangential for positive interference and 1 0 2 73.1 Tangential room resonances, not antinodes.

GrassCA © 24 5.3: Modal Behaviour: Bonello Criteria

The Bonello criteria is a method for assessing the modal behaviour in a room and allows for an easy comparison between different rooms. ( Density of Modes )

These criteria try to ascertain how significant the modal behaviour of a room is in perceptual terms, of whether a room is more likely to sound 'smoother' in the low frequency range.

It is done by dividing the audio frequency spectrum into third octave bands, as an approximation of critical bands, and then counting the number of modes per band.

GrassCA © 25 5.3: Modal behaviour: Relative mode strength

3 Modes

2 Modal Density can be seen to be far greater at 1 high frequencies Relative Mode Strength Mode Relative than at lower

0 frequencies 0 50 100 150 200 250

Frequency (Hz)

Modes that coincide (2 or more on the graph at the modal frequency) are only considered problematic if the 1/3-octave band in which the frequency falls has a total modal density equal to or less than 5.

GrassCA © 26 5.3: Modal Behaviour: Relative mode strength

There will be room modes across the audio spectrum. Their colouration level will depend on the following 5 factors. (Gilford)

Frequency Degree of bandwidth of excitation of the the mode. mode.

Separation Position of Frequency from strong source and mic content of neighboring to standing source. modes. wave system.

GrassCA © 27 5.3: Modal Behavior: Room Size Ratios

Height Width Length 1.00 (2.7m) 1.14 (3.1m) 1.39 (3.8m) 1.00 (3.1m) 1.28 (3.9m) 1.54 (4.8m) 1.00 (2.7m) 1.60 (4.2m) 2.33 (6.3m)

A room with a suspended tile In a basement with exposed ceiling has a real height, joists the true height is to the concerning LF’s, to the solid bottom of the floor above, surface above the tiles not the bottom of the joists

There are several good ratios to use but those shown The ideal room has a ratio of above are the ones most height, width, and length often referenced similar to one of these.

GrassCA © 28 5.3: Modal Behavior: Room Size Ratios

The importance of room modes can be overstated. Avoid similar dimensions for width and height or even multiples(i.e. 3m and 6m). modes just describe where the resonances will be worst

Regardless of the room's size and shape, standing waves and acoustic interference happen at all low frequencies. Bass traps need handle the entire range, not just the frequencies determined by the room modes.

As far as acoustic interference is concerned, the only thing that changes with different room dimensions is where in the room the peaks and dips at each low frequency occur.

For small studios, calculation shows that no frequency is likely to become prominent unless it has a high early intensity and a long decay. This occurs only with axial modes, which are therefore likely to become individually significant.

GrassCA © 29 5.4: Room Acoustics: Geometrical Acoustics

A sound ray strikes a solid surface and is usually reflected from it. This process takes place according to the reflection law well known in optics.

Incidence Angle = Reflection Angle

Reflection Law: the ray during reflection remains in the plane defined by the incident ray and the normal to the surface, and that the angle between the incident ray and reflected ray is halved by the normal to the wall.

GrassCA © 30 5.4: Room Acoustics: Geometrical Acoustics

Any sound ray which undergoes a double reflection in an edge (corner) formed by two (three) perpendicular surfaces will travel back, Parallel to the source wave in the same direction as shown below, no matter from which direction the incident ray arrives.

r i

GrassCA © 31 5.4: Room Acoustics: Geometrical Acoustics

If the angle of the edge deviates from a right angle by δ, the direction of the reflected ray will differ by 2δ from that of the incident ray as shown below

2 

‘whispering gallery’. Speaker’s head is parallel to the wall, sound rays hit the wall at grazing incidence and are repeatedly reflected from it. The rays remain confined within an annular region; wall conducts the sound along its perimeter.

Source St. Paul’s Cathedral in London

which has a narrow annular

platform above the floor.

The acoustical effects caused Pickup by it are detrimental to music.

Reverberation plays a major role in every aspect of room acoustics and is a lesser criterion for the judgement of the acoustical qualities of every kind of room

Another important subject to look at is the diffuse sound field. Both reverberation and diffusion are closely related to each other:

The laws of reverberation can be formulated in a simple way only for sound fields where all directions of sound propagation contribute equal sound intensities.

Reverb: the sum total of all sound reflections arriving at a certain point in the room after the room was excited by an impulsive sound signal.

Our brain blends together all of the sounds reaching our ears within 5-30 ms of the original. Our ear will attend to their direction.

Reflections arriving approximately 30-50 ms or more after the original will be perceived as separate sounds. This phenomenon is known as the Haas effect.

The Haas(precedence) effect. It is important to ensure the first reflections arrive at listener before 30ms to avoid echo perception.

It is these initial reflections that are most important to the brain in determining the apparent size of the listening room. Consider the importance for critical listening.

WC Sabine interested in room volume and a Mean Free Path of reflection. A rough guide to the average delay-line length is the ``mean free path'' in the desired reverberant environment.

The mean free path is defined as the average distance a ray of sound travels before it encounters an obstacle and reflects.

This approximation requires the diffuse field assumption that plane waves are travelling in random directions. Late reverberation satisfies this assumption, provided the room is not too dead.

Calculating the MFP allows judgement on the first reflection distance and time average within a room. Wave energy in small rooms reflects between 4- 6 times before dissipating.

5m MFP = 4V S 3m V= Volume m³ S = Surface Area m² 7m (7 x 3 x 5) x 4 . T = d/c (time .distance .speed) [(3 x 5) x 2] + [ (7 x 3) x 2 ] + [(7 x 5) x2] = 2.9577/ 343 105 x 4 . 420 = 0.0086 30 + 42 + 70 = 142 = 8.6 miliseconds

= 2.9577m (57.9Hz …115Hz) ( average wave reflection distance) 4 reflections = 34.4ms 6 reflections = 51.6ms GrassCA © 37 5.5: Room Acoustics: (Absorption: The Sabine)

The Sabine equation (calculations) is used to measure material absorbency and provide the Sabine unit. The modern version of Sabine's technique involves a special room and sophisticated measuring equipment.

Sabine unit : a number between 0 and 1. Zero is completely reflective, and One is completely absorbent. An open window ,provided there's nothing outside to reflect sound back, would measure 1.

Since different materials absorb different frequencies better than others, they do this for six frequencies designed to give a general indication of the sound absorbency of that material.

If you're trying to calculate the absorbency/reflectivity of a room, you take the total Sabines for all of the objects in the room. The Sabine(S)is an absolute measure of absorption, independent of surface area, and it can be used to compare any two absorbing devices directly and on equal terms . Absorbtion Co-efficient.

Large objects are measured per square foot. Six square feet of carpet with a Sabine at a given frequency of 0.5 will have a total Sabine for the room of (0.5 x 6) = 3 Sabines. 15 square feet of door with a Sabine at a given frequency of 0.2 will have a total sabine of (15 x 0.2) = 3.

Manufactures of sound absorbent materials( acoustic publish sabine numbers at various frequencies. With these numbers, some mathematical equations, and a knowledge of the size and composition of your room you can begin to calculate the reverberation times of your room.

The accepted Reverb Time is the amount of time it takes for a sound to fall 60 decibels, RT60. Decibel (dB): logarithmic measurement. That is, 20 isn't twice as loud as 10, it's 10 times as loud. 60 decibels is 1,000,000 times as loud as 0 decibels.

The ambient level of sound in a living room is around 30-50 dB, in order to measure the RT60 time in a room we would need to activate a 110db sound impulse and see how long it takes to fade to 50. In practice this is not done.

GrassCA © 40 5.5: Room Acoustics: RT 60: The equation 0.161 V ------= RT60 Sa V = room volume (l x w x h) m³, S = surface area of the room(wall, floor, etc) m² a = average sabine of the material.

"Sa" is (Surface Area x average Sabines).

Add multiple Sa's together: Sa for the floor material + Sa for the walls material + Sa for ceiling material

.To only use “a = average Sabine of the material” is not very accurate due to the fact that absorption co-efficients differ .for materials at different frequencies.

To give a better picture of how a room space will reverberate, calculations should be done for the different frequency ranges that data is supplied for. Co-efficients are given for frequencies of: 125Hz, 250Hz, 500Hz, 1kHz, 2kHz, 4kHz and (8kHz) a is now a specific co-efficient for a particular frequency.

0.161 V ------= RT60 @ 125Hz, 250 Hz, 500Hz,1K, 2K, 4K, 8 K) Sa(125Hz)

Rt-60 measurements are most useful in determining the acoustic properties of larger spaces such as churches, auditoria, Medium sized music venues large cinemas, etc. Consideration must be made of the effect of the human body as regards sound absorption. An empty space will yield different acoustic properties compared to a venue full of people.

In smaller environments the Rt-60 measurements become so short as to be useless. In these confined spaces, individual reflections from nearby surfaces dominate the sonic picture and are the primary focus for the audiophile.

Room mode diagrams: Retrieved 20/08/09 Heinrich Kuttruff http://www.marktaw.com Room Acoustics: Fourth edition

NWAA Labs, Inc Low Frequency Acoustics: Retrieved 20/08/09 http://www.deicon.com 25132 Rye Canyon Loop. Santa Clarita, CA 91355

The Bonello Criteria: Acoustics and psychoacoustics Glen Ballou by David Martin Howard, James A. S. Angus Handbook for sound engineers

The Critical Frequency: Acoustics and psychoacoustics F. Alton Everest David Martin Howard, James A. S. Angus Master Handbook of Acoustics: Fourth edition

Links http://www.sengpielaudio.com/calculator-RT60.htm http://www.realtraps.com/art_monitor.htm

Frequency W = 3.5m L = 3.8m H = 4.5m λ f1 2λ 49Hz 45.1Hz 38.1Hz

f2 λ 98.Hz 90.3Hz 76.2Hz 196Hz 180.5Hz 152.4Hz f3 ½λ f4 ²/3λ 147Hz 136Hz 114.6Hz f5 ¹/3λ 294Hz 271Hz 229Hz f6 ²/5λ 245Hz 226Hz 191Hz f7 ¼ λ 392Hz 361Hz 304.8Hz