# Difference between sound and high temperature

Thermodynamic principles say that molecular motions should produce a white noise background, but a back-of-envelope calculation shows that the resulting white noise will be far below the threshold of audition for any given tone.

The physiological threshold of audition for a well-defined tone a little over 0 dB, or 1 pW/m² in air. If the capture area of the ear shell is roughly 1 cm² , the minimum audible signal would be -160 dBW.

The thermal-noise background would produce a signal *kTB*, where *B* denotes the effective bandwidth of cochlear detector cells, reciprocal to their time resolution, say 125 ms, the duration of notes in a fast run on a piano. (This crude estimate of *B* is also consistent with quarter-tone pitch discrimination in the range of 256-512 Hz, where the threshold is lowest.) Using *kT* = -204 dBJ and *B* = +9 dBHz, we get *kTB* = -195 dBW.

The actual frequency range of audition is almost 10 kHz, but there is not a shred of evidence that neural machinery integrates subliminal signals over such a broad bandwidth. (If it could do so efficiently, the thermal noise might be almost audible.) It is important to understand the difference between coherent and non-coherent integration. Coherent integration sums amplitudes; non-coherent integration sums powers. Signals with different frequencies (hence random phase relationships) could in principle be summed non-coherently, but the inefficiency of non-coherent integration is well known to radar engineers. Pulsed radars typically use signal bandwidths around 1 MHz (reciprocal to pulse width and/or time resolution), but integration bandwidths as low as 100 Hz (reciprocal to beam dwell time). Coherent integration of *N* pulses gives an integration gain of *N*, non-coherent integration an integration gain ~ $\sqrt{N}$.