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Acoustics Crash Course 1 - Modes

A crash course in acoustics in Modes. This is by no means complete, but it should serve as a good starting point for someone looking to learn about acoustics. Great for someone building a home or professional studio, or looking to correct problems for their home theater system.


The Fun House Metaphor

Imagine a room whose surfaces reflected light to varying degrees. These surfaces don't reflect light perfectly, but they all reflect light. Some of them reflect more blue than red, others are sort of greenish*. You go in with protective glasses so you can watch light rays fly around the room without worrying about damaging your sight.

Light travels much too swiftly for us to see it go by. If someone took your picture in this room, the camera's flash would send out light in all directions, much like a speaker, and these would bounce off, travel through, and be reflected by various objects in the room - the walls, you, the camera, etc. until every frequency had been absorbed by one of the surfaces. It would be too quick for you to see every reflection, but you might get a brief impression of "this room seems sort of blue" and by using colored light, or sophisticated measuring equipment, you could verify that indeed it is more blue than other rooms you've been in.

This metaphor, while not totally accurate, can help you learn about the problems affecting rooms.

Standing Waves, Room Modes, and Eigentones

All of those phrases mean more or less the same thing. If you take a flashlight and shine it straight at one of the walls, it will bounce off of one wall and hit the opposite wall. Minor movements by your hand will cause radical movements of the beam of light as it bounces several times between the two walls.

In audio these are known as Standing Waves - a wave of sound that bounces between two or more surfaces emphasizing one frequency over others. The area between two parallel walls resonate at certain frequencies.

Light waves are tiny. Audio waves are gigantic by comparison. 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 around every which way.

Treble waves are small and fast like those created by a rock dropping into a pond. Bass waves are long like the ones that sweep you off your feet when you go to the beach.

If you've ever seen a leaf floating on wavy water, you'll know that the waves don't actually move the water anywhere. The leaf rises and falls, but doesn't move in any direction. Sound waves are very much the same, increasing and decreasing in pressure without really moving anywhere. Otherwise sound would be accompanied by a wind, and you would feel a breeze every time John Bonham hit his kick drum.

When a speaker moves forwards it compresses the air a bit, when it moves back it creates a bit of a vacuum. These are known as Compression and Rarefaction. If one speaker moves forward while the other moves backwards, they cancel each other out. If you can wire your speakers out of phase you'll hear this - all the low end will drop out and it will sound tinny.

Standing Waves are sounds that reinforce each other, like two speakers wired in phase, as they bounce back and forth between walls, the build up areas of high and low pressure that are consistently in the same places in the room.

If your walls are 14 feet apart and the wave is 13 feet long, it won't get the reinforcement it needs and it'll die down quickly. If the wave is 28 feet long, or 14 feet long, 7 feet long, etc., it will keep bouncing back and forth between the walls until it encounters a corner and dies off. Because of this, unless you have a very square room, the modes will terminate in the edges and corners. Once that 14 foot wave encounters your 8 foot 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).

LENGTH OF A SOUND WAVE CALCULATION

speed of sound (distance per second)
------------------------------------ = length of wave
frequency (cycles per second)

The speed of sound is around 1130 feet per second, or 344 meters per second. Do not mix feet and meters in your calculations. To make these calculations work you must keep the units of measure consistent. Either feet or meters, cycles or distance must always be per second.

Axial Room Modes

Axial Room Modes involve two parallel walls. In the imaginary reflective room, take your flashlight and point it directly at the wall and it will bounce back and forth between the two walls. These are Axial Room Modes.

Three Primary Axial Modes

The red line is a one mode (length), the blue line is one mode (width) and the green line is one mode (height).

[Image]



This diagram is very much simplified. If you're standing in the room and shine the light at one of the walls it will bounce back and forth between the walls like the blue beam. Remember sound will zig-zag around the room, and that sound sources aren't directional like flashlights. A speaker is more like a bare lightbulb, or a light bulb in a box.

Because of this zigzag, room modes are actually a range of frequencies centered around the number given in our calculations.

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 by two - usually we divide the speed of sound by two because it's an easier calculation to remember.

AXIAL MODE CALCULATION
Speed of sound / 2
------------------------- = standing wave frequency
distance between surfaces

My living room is 18 feet long, so it will enforce a frequency of (1130/2) / (18) = 565 / 18 = 31.39. Multiples of this will be a problem too. You can reverse this calculation easily enough - 1130 / 31.39 = 36 = twice the length of my room.

It will also enforce frequencies based on it's width and height (11 feet and 8 feet). The other walls are modes of 51.36 and 70.625. Since I know multiples will be a problem, I know at a glance that frequencies around 150 and 210 will be a problem. How do I know? Because both 30 and 50 go into 150, and both 30 and 70 go into 210. Calculating the exact frequencies using the calculator I can confirm this. The following are several modes I found by multiplying the original mode two or more times: 156.94, 154.09, 219.72, 211.875, 282.5, and 282.5. Since these modes are close to each other I know they will be problems. I'm not concerned with modes above 300hz.

Tangential Modes

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

Sound, like a rubber ball, bounces off a surface at about the same angle it arrives at. In tennis the angle is normally very shallow, maybe 15 degrees. On the other hand, throwing a ball straight at the wall, it bounces off at roughly 90 degrees.

Three Primary Tangential Modes

Starting at in the middle of the right wall and aim the flashlight back at the rear wall. The light would follow the blue line and bounce off until it hit the left wall. It would then bounce off of the left wall and hit the front wall, and then bounce back to the right wall.

[Image]



Again, this diagram is an over simplifications, but should help you visualize what the problems are. Based on these diagrams you would always place acoustical treatment in the center of each surfaces, but these are not necessarily the best places for room treatment.

Now imagine you're standing in the middle of one wall and you aim the flashlight (the blue line) at an angle towards the center of the wall to your side. No matter what the dimensions of the room it will bounce off of that wall and hit the center of the wall opposite you. Then it will bounce off that wall and hit the wall to your other side. It will bounce off of that wall and come back towards you. If you weren't there to block the light, it would do the same thing again. Aim it up slightly and it will spiral up the room.

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 circuit. So if one frequency that's supported is 100hz, another one would be 200hz.

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

If you remember your geometry you'll remember that the calculation for a right angle triangle is a^2 + b^2 = c^2. (^2 is the symbol for squared) We know the values for a and b - these are parts of the lengths of the walls at the point where the sound wave bounces off of them, so we're solving the equation for c.

Althought we're only calculating for two surfaces, it's important you do this calculation for every variation of Length, Width and Height. In other words, do it for L & H, L & W, and W & H. The calculation is:


TANGENTIAL MODE CALCULATION

Frequency = c/2 * sqrt(p^2/L^2 + q^2/W^2)

     c = speed of sound
  sqrt = Square Root Of
    ^2 = squared

     L = Length of Room
     W = Width of Room
     H = Height of Room

p & q represent the mode we're solving for
p = 1, q = 1 for the first mode
p = 2, q = 1 or p = 2, q = 2 etc. for higher modes

So let's figure this out for one of the dimensions of my living room.

TANGENTIAL MODE CALCULATION: EXAMPLE

Length = 18
 Width = 11
     p = 1
     q = 1

Frequency = c/2 * sqrt(1^2/18^2 + 1^2/11^2)
Frequency = 565 * sqrt(1/324 + 1/121)
Frequency = 565 * sqrt(121/39204 + 324/39204)
Frequency = 565 * sqrt(445/39204)
Frequency = 565 * sqrt(0.011350882563003775124987246199367)
Frequency = 565 * 0.10654052075620700252419556208632
Frequency = 60.195394227256956426170492578774

Now you see why we have automatic calculators to do this for us! Imagine doing that calculation for the two other dimensions using higher modes.

Oblique Room Modes

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 just moving around on a flat plane, it bounces off of the ceiling and floor on it's way around.

Oblique Room Mode

Similar to the Tangential Room Mode - notice that it hits the front, back, left, and right in the same places, but also hits the floor and ceiling in between every time. Of course, you'd need to rotate this diagram for all six dimensions to fully appreciate ever tangential room mode you had.

[Image]



One last time, this diagram is an over simplification. Oblique room modes are the most difficult to visualize. The crosses are there to help indicate which wall the sound ray is hitting at any given time.

Again, the calculation is similar to the previous ones, but you're solving for a 3 dimensional shape, a pyramid of sorts. In the diagram above all lengths are the same, though they don't look it because of foreshortening in the 3D diagram.

OBLIQUE MODE CALCULATION

Frequency = c/2 * sqrt(p^2/L^2 + q^2/W^2 + r^2/H^2)

     c = speed of sound
  sqrt = Square Root Of
    ^2 = squared

     L = Length of Room
     W = Width of Room
     H = Height of Room

p, q and r represent the mode we're solving for

This is actually a sort of master equation, and you'll often see Room Modes in the format "1 0 0" which symbolizes the p, q, and r values for the equation. Setting any of the values to 0 essentially removes that value from the equation (0^2/x = 0). This means that two positive numbers and one zero is the calculation for tangential, three positive numbers is the calculation for oblique, and one positive number and two zeros is the calculation for axial. There are no negative numbers as there are no negative room dimensions.

AXIAL MODE CALCULATION REVISITED

Axial Mode Calculation (p = 1, q = 0, r = 0)

Frequency = c/2 * sqrt(p^2/L^2 + q^2/W^2 + r^2/H^2)
Frequency = c/2 * sqrt(1^2/L^2 + 0^2/W^2 + 0^2/H^2)
Frequency = c/2 * sqrt(1^2/L^2 + 0 + 0)
Frequency = c/2 * sqrt(1^2/L^2)
Frequency = c/2 * 1/L
Frequency = c/2L
Frequency = (c/2) / L
Frequency = (The Speed of Sound / 2)
            ------------------------
            Length of Room

Some Final Notes

So Axial Modes are the easiest to compute, and they're the most important, which is very nice. Tangential Modes are about half as loud, and Oblique about a quarter as loud, but if an oblique room mode occurs near another mode, that frequency may still be a problem. It's best to calculate all room modes, Axial, Tangential and Oblique to see where any overlap may be.

I've added a Room Mode Calculator to my site to prevent you from having to do all this number crunching by hand.

These modes actually hit the whole wall surfaces, so don't think you can treat just the center of every wall as depicted in the diagrams. For example, there's nothing that says a Tangential Mode has to happen halfway up the wall, and it and does spiral up the wall until it hits a corner where the angle change and the mode dies out, unless it becomes an oblique. Axial Modes can and do happen at any and every part of the wall.

Also, since there are sleight angles involved - nothing is exactly like this simplified mathematical model - the numbers you get from these equations are the center of a band of frequencies that are affected. Which is why modes that happen close to each other are a problem. It's like boosting 140 and 150 Hz on two EQ's, the "Q" is likely to overlap causing a larger buildup of sound in the area of 140-150hz.

Since waves above 300hz are so small (282hz is a four foot wave) there is a much more even spread and much less of a build up in certain areas. A four foot wave can get into lots of places, while a 20 foot wave doesn't fit in may places. Above 300hz, sound is more influenced by what the room is made of and things that are in the room than the shape of the room.

* More on material reflectivity in a later article

Sound Wave quick reference. These are the frequencies of the lowest notes on a piano. A = 27.5, A# = 29.135, B = 30.868, C = 32.703, C#, 34.048, D = 36.708, D# = 38.891, E = 41.203, F = 43.654, F# = 46.249. G = 48.999, G# = 51.913.

These will allow you to quickly test the room modes without sophisticated equipment. Simply play a note at one of these frequencies or their octaves to see whether or not it's emphasized in your room. Using your ears isn't as precise as other measuring equipment, but it's a step in the right direction.

Things are much more complicated than this, please don't do anything stupid or spend any money just because you "read it online."


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page first created on Saturday, June 08, 2002


© Mark Wieczorek