E-170 Loudspeaker Enclosure

Banff National Park AL, Canada (Photo by Yuichi)
This is to understand how standing wave behaves in a rectangular room.

A equation in the left is how to get standing wave frequencies. Since
n varies 0 to endless number, number of frequencies generated are also
Standing wave is indicated as (Nw, Nl, Nh). Assuming Width=6m, Length=7m and Height=3m, standing wave mode and frequencies are as follow.
(0, 1, 0) : 24.6Hz
(1, 0, 0) : 28.7Hz
(1, 1, 0) : 37.8Hz
(0, 2, 0) : 49.1Hz
(1, 2, 0) : 56.9Hz

(2, 0, 0) : 57.3Hz
(0, 0, 1) : 57.3Hz
(0, 1, 1) : 62.4Hz
(2, 1, 0) : 62.4Hz
(1, 0, 1) : 64.1Hz
(1, 2, 0) : 49.1Hz
(1, 1, 1) : 68.6Hz

(0, 3, 0) : 73.7Hz
(2, 2, 0) : 75.5Hz
(0, 2, 1) : 75.5Hz
(1, 3, 0) : 79.1Hz
(1, 2, 1) : 80.8Hz
(2, 0, 1) : 81.1Hz
(2, 1, 1) : 84.7Hz

(3, 0, 0) : 86.0Hz
(3, 1, 0) : 86.4Hz
up until
(n, n, n)

2D FEM (Finite Element Method) Simulation

To visualize standing wave in a room, I used FEM function in the Sound Easy simulation software. Left picture illustrates room model and how mesh is set for 2D visualization. In this model, a mesh (X=20cm, Y=22cm and Z=55cm) is assumed.

Frequency list in the left column is the standing waves under 100Hz. Lowest frequency is 21.44Hz. Then, 22.66Hz, 31.59Hz...and so on.

Center column shows strength of sound pressure. Red and deep blue indicate strong pressure. Red shows positive pressure and blue shows negative pressure.

Right column picture shows simulated pressure distribution in the room at 21.44Hz. When you point out frequency listed in the right most column, sound pressure distribution in the frequency is displayed. In this room, right side is now positive pressure. However, it will be blue in the next instance. Positive and negative pressure alternate.

This is the first standing wave
generated by right and left side
walls. Freq. is 21.44Hz.
This is second standing wave
by upper and lower walls.
Freq. is 22.88Hz.
Then, 31.39Hz. The wave behavior is not so simple as first two samples.
This is double of the first
standing wave which is 42.95 Hz.

Then, 63.08Hz
98.39Hz, now

3D Simulations are as follows

Frequency Response Sample
Assuming we have perfect speaker system which has 100% flat frequency response, we have to have this kind of frequency response curve if the speaker system is located into a rectangular room.

Standing wave effect is depending upon size of the room, sound absorption factor of the walls / floor / ceiling and listening position and other factors.

Therefore, room acoustic is very important as well as speaker system itself.

One of the advantages of FEM is that we can model all kind of room types. I tried to simulate a pentagon room and a round room which are often said that no standing wave is generated.

There are many standing wave exist.
48.19Hz mode is shown in a pentagon room.
Standing waves
below 100Hz
There are many standing wave exist.
52.34Hz mode is shown in a round room.
Standing waves
below 100Hz

Sound Easy software is used for all of simulations in this page.

To avoid standing wave effect in low frequency domain is headacheing question. However, sound absorption treatment on the walls will contribute. This would be a different subject and discussed separately.

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