Audio Engineering

46 Chapter 2

2.7 Inverse Square Law ..................................................................................................

If we double the radius of the sphere to 0.564 m, the surface area of the sphere quadruples
because the radius is squared in the area equation ( A  4 πr^2 ). Thus our intensity will
drop to one-fourth its former value. (Note, however, that the total acoustic power is still
1 W so the LW still is 120 dB.) Now an intensity change from 1 W to 0.25 W/m^2 can be
written as a decibel change. The acoustic intensity (i.e., the power per unit of area) has
dropped 6 dB in any given area:

LI 10

025

1

2

2

log

. (W/m )(new measurement)
(W/m)at the shorter roriginal referenceaadius
dB.

⎛

⎝

⎜⎜⎜ ⎞

⎠

⎟⎟⎟

602.

Therefore our LP had to also drop 6 dB and would now be approximately 114 dB.

A1 m^2

A4 m^2

A 4 πr^2

r0.564 m r0.282 m

(a) Sphere and radius (b) Area increases with the square of the radius

A A

(d) Area increases as the radius

increases when only one angle diverges

A A

(c) Area increases with the square of the

radius when both angles diverge

Figure 2.4 : Relationship of spherical surface area to radius.