**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 or being temporarily blinded by flashing lights.

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.

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. Remember sound will zig-zag around
the room, and that sound sources aren't directional like flashlights. A speaker
is more like a bare light bulb, 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 - 585/feet, or 172/meters.

**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.

Now imagine you're standing in the middle of one wall and you aim the flashlight
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.

The calculation for a tangential mode is similar to a that of axial modes,
but you have to calculate the length of the ray of sound. 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.

The calculation is:

**TANGENTIAL MODE CALCULATION**

` Speed of sound / 2`

`------------------------------------ = frequency`

`square root of (x^2/l^2) + (y^2/w^2)`
*l* and *w* are the length of the two surfaces involved in the calculation
- *length and width* or *length and height* or *width and height*.
This is essentially the same as the Axial Room Mode calculation, where the bottom
is the equation to find out the length of the of the wave based on the triangle.
Note that "x" and "y" can be different numbers. Figuring out for x=1 and y=1
yields the first mode. x=2 and y=1 or x=1 and y=2 would give you different shaped
triangles that are also valid. x=2 and y=2 would give you an octave of x=1 and
y=1.

**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.

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**

` Speed of sound / 2`

`------------------------------------------------ = frequency`

`square root of (x^2/l^2) + (y^2/w^2) + (z^2/h^2) `
"l" "w" and "h" are Length, Width and Height. This is actually a sort of master
equation, and you'll often see Room Modes in the format "1 0 0" which symbolizes
the X, Y, and Z values for the equation. Setting any of the values to 0 essentially
removes that value from the equation (0/1 = 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 is the calculation for
axial. There are no negative numbers as there are no negative room dimensions.

**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.

I hope this answers some of your questions and gives you a starting point for some more research. Remember that real life is more complicated than mathematical models, please don't do anything stupid or spend any money just because you "read it online."