How do you calculate SPL at one distance to SPL at another distance?
As everyone knows the further away a sound source is, the quieter the sound will be. This is due to the surface expansion of the affected space. One can imagine an ideal point source producing a sound field that moves like an expanding sphere into all 3 dimensions. This means that for doubling the distance the occupied surface gets quadrupled, reducing the energy per m² by a factor of 4.
If we put this into an equation we first must state the equation for the SPL depending on the air pressure:
being the actual sound pressure.
being the reference sound pressure (20µPa), which is the average hearing threshold at 1kHz of humans.
Or in “nice” mathematical writing:
Sidenote: This whole chapter is only relevant if the sound is moving like a wave through the air. It does not apply in case of In-ear / artificial ear measurements. There the speaker directly compresses the air between speaker and microphone/eardrum, rendering distance irrelevant as a parameter.