posted 2006.08.01

Spherical geometry

Assume the speakers are arranged on a spherical shell sitting on the floor. Where do we place the speakers such that their locations correspond to those of the recording stations on the Earth? Consider this diagram, which represents both the Earth and the listening space:

where:

S	indicates the seismometer on Earth or the speaker in the model space
θ	(geographic) latitude
φ	longitude
R	radius
d	axial distance

From the figure: d = Rcosθ and h = Rsinθ. The speaker is placed at a height h+R above the floor, at a distance d from the center of the space. The speaker is tilted at an angle of -θ to the horizontal, thus pointing towards the center.

Cylindrical geometry

Assume the speakers are arranged on a cylindrical shell tangent to the equator of the spherical model space. Each speaker is situated at the point at which the station would be projected onto the cylinder (Mercator projection):

where:

S	seismometer location on Earth
S’	speaker location in model space
θ	geographic latitude
h’	speaker height above equator

From the figure: h’ = Rtanθ and h = Rsinθ. The speaker is thus located at a height R(1+tanθ) above the floor.

By moving the speaker from S to S’ the audio level experienced by the listener at O decreases due to inverse-square losses. Because the ratio of the distances O-S’ to O-S is simply h’/h, or 1/cosθ, the audio level must be increased by 20log(1/cos²θ) dB, or -40log(cosθ) dB.

Here is a simple JavaScript implementation of all this.

A simple speaker placement calculator