### Goal

Understand what a standing wave is and how to deal with them, and ideally avoid them.

## Standing Waves

**Standing Waves** are waves that appear to be stationary in a room. Their low
pressure, high pressure, and nodes do not move, they are stationary, or "standing" still.
The waves fall into a category of resonances. **Resonances** are a frequency that is naturally
strengthen by a system because it lines up with some aspect of the system. For example, in
a room with parallel walls resonances will naturally form at frequencies who have the same
wavelength as the distance between the walls.

At low frequencies these resonances are problematic because there are very few frequencies
that fall into this criteria and so uneven boosting or cutting occurs, while at high
frequencies many more resonances occur causing the spectrum to be much smoother. In
acoustics the most basic set of resonances that can occur are referred to as **modes**.

## Mechanism Behind Modes

The wave a resonance or mode is formed has to do with reflections. When a wave strikes a wall it is reflected. These reflections "interfere" with the incoming wave. If they interfere in a regular kind of a pattern they can cause static patterns to appear in a room. Here is a simulation to give you an idea of what happens:

Frequencies that do not line up will not form standing waves. It is this mechanism that makes instruments sound they wave they do. In the case of rooms, rooms are just boxes of air, and because of this if an event happens in the room it will cause pressure to vary and reflect off the walls. This variation is what leads to standing waves.

Here is a demonstration with a tube:

## Resonance in a pipe

The formula for resonance in a pipe that is open on both ends:

To adjust for a pipe closed at one end we use the quarter wavelength rules so we would have 4L instead of 2L. It would also only produce odd harmonics.

## Design Question

You want to create the note C4 with a pipe. What length should the pipe be? C4 has a hertz value of 261.6 Hz.

## Room Modes

There are 3 types of room modes, they are Axial, Tangential, and Oblique. Each one refers to the number of surfaces required to set up the wave. Axis only need 2 parallel surfaces, tangential needs 2 sets of parallel surfaces (4 walls), and oblique needs 3 sets of parallel surfaces (6 walls).

Walls that are not treated for the frequencies that set up modes in the room can be especially problematic. Naturally these are generally low frequencies.

## Axial Room Modes

The largest issue is generally the axial modes. Because they only require 2 parallel surfaces they tend to keep more of their energy. For all 3 sets of modes it is not just the fundamental that is a problem but also the harmonics which will also line up with the walls. As the harmonics increase however they will be far less problematic.

## dB Differences

In general if an axial wave is 0dB, then tangential is -3dB and oblique is -6dB. This assumes an ideal room, in real life the materials the surfaces are made of will make a large difference. From this looking at a spectrum may yield clues as to why a peak may occur.

Each room will be different but in general the low end will have issues with room modes and your treatment will be focused on them, the mids will be influenced by room modes but treatable by diffusion and absorption while the high end is easily treatable with diffusion and absorption.

## Room Mode Calculation

The following equation holds for rectangular rooms:

Room modes can be combined to consider different cases of room modes. For each pair of walls involved we represent that with a 1 in the equation. For example if we want to know about the axial mode involved with the length of the room we would use p=1, q=0,r=0. P is above teh L which is why we make it one and to remove the other sets of walls we make their coefficients 0. In this way we can talk about how many walls are involved. Want a tangential mode? Use a room mode with two 1's over the dimensions we are interested in. For example (1,1,0) evaluates the tangential room mode between the length and width of the room. If we had 3 non zero terms then it would be an oblique room mode.

If the room mode has a 2 or other higher number in the code then this corresponds to a higher harmonic of that room mode. Room mode (0,2,0) has a frequency twice as high as the room mode (0,1,0).

### Room mode equation observation

Consider the case of (1,0,0). This is a basic axial mode. An axial mode is just like a 1 dimensional pipe. So we expect the pipe resonance equation to come out of this. Plugging in we have:

If we take that 1 to instead by n where n is the harmonic number we arrive at the exact same equation as expected! So our equation for a room is an extension of the resonance equation for a pipe.

### Question

We saw the length axial is (1,0,0). What are the height and width axials?

A room has a length twice that of its width. Why is this an issue? How will the equation reflect this?

Given a room of 12ft, by 11ft, by 9ft. Find the (2,1,0) room mode frequency.

Given a room of 12ft, by 11ft, by 9ft. Find the axial room modes. Also find the (1,1,0) mode and the (1,1,1) mode.

## Room Modes in Reverb Tails

Beats showing up in a reverb tail is evidence of a problematic room mode.

## Mode Bandwidth

Each mode has a bandwidth:

Note the reverb time is inversely proportional. So the more “dead” a room the bigger the bandwidth.

Reverb time vs mode bandwidth

Reverberation Time (Seconds) | Mode Bandwidth (Hz) |
---|---|

0.2 | 11 |

0.3 | 7 |

0.4 | 5.5 |

0.5 | 4.4 |

0.8 | 2.7 |

1.0 | 2.2 |

## Room Modes in Rooms

In rectangular rooms room modes form regular patterns:

In three dimensions:

Higher order room mode:

By adjusting the room to a non-rectangular shape and avoiding parallel walls the room modes become much more dispersed.

For design various authors recommend different dimensions for rectangular rooms to help disperse modes more evenly, each with their own reasoning. Here are just a few:

Height | Width | Length |
---|---|---|

1 | 1.14 | 1.39 |

1 | 1.28 | 1.54 |

1 | 1.60 | 2.33 |

1 | 1.4 | 2.9 |

1 | 1.26 | 1.59 |

## Mode Density

Mode density increases with frequency, more modes next to each other means the modes in that range will matters less and less as they average each other out.

## The Bonello Criterion

A criteria for designing rooms which states that each 1/3 octave should have more modes than the previous one.

## Room Mode Calculators

There are many online calculators that can help you determine qualities about your room from its dimensions:

## Schlieren Optics

Finally these demos are some of the coolest around when it comes to standing waves:

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