# Amplitude is inversely proportional to distance

Our tympanic membrane (a.k.a. our eardrum) and a microphone are both devices that measure sound intensity. When a sound arrives at our eardrum or at the diaphragm of a microphone, either of which has a certain surface area, the power in that area (i.e. the intensity) is detected. However, the intensity of a sound, as measured by an eardrum or a microphone, will differ depending on the distance from the sound's source, because the sound is being emitted from the source in __all__ directions. If you think of the sound energy as radiating outward from the source in a spherical pattern, and you bear in mind that the surface of a sphere is proportional to the square of its radius (the surface area of a sphere is equal to *4πr ^{2}*), you can understand that the intensity of a sound as measured in a given surface area is inversely proportional to the square of the distance of the point of measurement from the sound source. This principle is known as the

*inverse square law*: intensity is inversely proportional to the square of the distance from the source (

*I ∝ 1/d*).

^{2}Our subjective sense of a sound's "loudness" is not the same as its intensity, but is generally roughly proportional to it. But what does that mean in terms of the __amplitude__ factor we'll use to alter a sound's intensity in digital audio? As defined in physics, the intensity of a wave is proportional to the square of its amplitude (*A ^{2} ∝ I*). So that means that if we want to emulate the effect of a sound being twice as far away, (1/4 the intensity), we would need to multiply the amplitude by one-half. Indeed, based on what we know about the relationship between distance and intensity (the inverse square law,

*I ∝ 1/d*), we can see that the relationship between distance and amplitude is simply

^{2}*A ∝ 1/d*; amplitude is inversely proportional to distance.

This patch shows how you can emulate a change of distance simply by changing its relative amplitude inversely. Of course, our sense of distance is also affected by reverberation and high-frequency rolloff, but this basic relationship between distance and ampitude is useful to know for sound spatialization.