What is the formula for the sound pressure of a pure tone of 500Hz, ex-pressed as a function of time?
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Sound pressure level is directly related to the amplitude of the waveform.
A pure tone is a sine-wave and sine-waves are defined by
So for 500Hz,
You will find that if you recreate the 500Hz sinewave by producing samples at (say) 20kHz you will get a string of numbers (every 50 microseconds) that rise to a peak (+1) at 500µsecs then start to fall through zero at 1000µsecs and go negative to -1 at 1500µsec then fall back to 0 at 2000µsecs. The waveform repeats.
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The instantaneous sound pressure of a pure tone equals the ambient pressure (p0) with a superimposed pressure that varies in time as a sine function, i.e.: p(t) = p0 + A sin ωt, where A is the peak amplitude of the pressure variation and ω the angular frequency ( ω = 2πf ).
The amplitude A is related to sound pressure level L in dB by the following equation:
L = 20 log10( prms / pref ), where, in the case of a sine wave, the following equality holds: prms = A / √2. The reference pressure is arbitrary, but a fairly common value is 20 µPa, which would put a SPL of 0 dB around the threshold of human hearing.
Through substitution, we obtain the following for the pressure as a function of time of a pure tone with frequency f:
p(t) = p0 + √(2) 10L/20 pref sin 2πf
Sound Pressure Level (SPL) is a dB scale defined relative to a reference that is approximately the intensity of a 1000 Hz sinusoid that is just barely audible.
Since sound is created by a time-varying pressure, sound levels computed in dB-SPL by using the average fluctuation-intensity (averaged over at least one period of the lowest frequency contained in the sound).
Generally, the intensity level of a sound wave is its dB SPL level, measuring the peak time-domain pressure-wave amplitude relative to
Reference and more complete explanation here.