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The hyperbolic / exponential formula
Area = Throat Area [ cosh ( x*2*Pi*f / c) + M * sinh ( x*2*Pi*f / c) ] ^2

where
• x = distance from throat
• f = the cutoff frequency of the horn
• M = the flare constant - M = 1 is exponential, 0 < M < 1 is hyperbolic
• c = the speed of sound, approximately 13538 inches per second or 344 m/s (depends on temperature, etc.)

Mouth Size
For a horn radiating into full space, the horn mouth should have a circumference equal to the wavelength of the low cutoff frequency. Usually, what is of interest is the area of the mouth since bass horns are not commonly made with circular mouths. Here’s how you get to an equation for the mouth area:
• Circumference = c / f
• and Circumference = Pi * diameter = Pi * 2 * radius
• setting these equations equal, Pi * 2 * radius = c / f
• radius = c / (2 * Pi * f )
• In addition, Area of a circle = Pi * radius^2
• Substituting in for radius, Area = Pi * [c / (2 * Pi * f )]^2
If you want to find the appropriate mouth area for a loading situation other than full space, you would divide this area by the appropriate factor (divide by 2 for half space, by 4 for 1/4 space, etc.).