# Formula for the sound pressure of a pure tone

What is the formula for the sound pressure of a pure tone of 500Hz, ex-pressed as a function of time?

• Might be okay here, but might be better off over on Physics.SE Commented May 6, 2013 at 7:29
• I think this is relevant here - managing SPL is part of audio studio work.
– Warrior Bob
Commented May 6, 2013 at 16:05

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 `ω` (omega) and `t` (time)

`amplitude = sin(ωt)` --- "sin" is the mathematical operator you did in trigonometry at school and t is time.

`ω = 2 * π * f` --- π is 3.141592654 (approx) and f is 500

So for 500Hz, `ω = 3141.5927` (approx)

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.

• @JoshP - if you are going to edit make sure you do it correctly - the character "Ω" is WRONG - it should be "w"
– Andy aka
Commented May 6, 2013 at 12:50
• How is this answer relates to the question?
– Eugene S
Commented May 6, 2013 at 13:47
• Andy, fixed. My apologies. Also, if you see an incorrect edit, by all means, please correct it. Commented May 6, 2013 at 13:58
• @EugeneS the OP asked for a formula for a pure tone and the sound pressure is directly related to the amplitude of a sinewave
– Andy aka
Commented May 6, 2013 at 14:19
• The formula you gave here (`W=2*pi*f`) is an an angular frequency which units are radians per second. I don't really understand how this relates to sound pressure level which is measured in dB..
– Eugene S
Commented May 9, 2013 at 2:13

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

This was a question asked in a Digital Sound and Music exam which has been copy-pasted here (even the "ex-pressed" has been left intact). The question only gives two marks. I think this is the answer:

``````f(t) = Amplitude * sin(2 * π * Frequency * t)
``````

So in this case the Frequency would be replaced with 500.

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.

``````0dB SPL = 20 micro Pascal
``````

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 `10^{-16}` watts per centimeter squared (i.e., there is no consideration of the frequency domain here at all).

Reference and more complete explanation here.