Take the 2-minute tour ×
Sound Design Stack Exchange is a question and answer site for sound engineers, producers, editors, and enthusiasts. It's 100% free, no registration required.

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

share|improve this question

migrated from avp.stackexchange.com Jan 27 '14 at 15:10

This question came from our site for engineers, producers, editors, and enthusiasts spanning the fields of video, and media creation.

Might be okay here, but might be better off over on Physics.SE –  Rory Alsop May 6 '13 at 7:29
I think this is relevant here - managing SPL is part of audio studio work. –  Warrior Bob May 6 '13 at 16:05

4 Answers 4

up vote 1 down vote accepted

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.

share|improve this answer
@JoshP - if you are going to edit make sure you do it correctly - the character "Ω" is WRONG - it should be "w" –  Andy aka May 6 '13 at 12:50
How is this answer relates to the question? –  Eugene S May 6 '13 at 13:47
Andy, fixed. My apologies. Also, if you see an incorrect edit, by all means, please correct it. –  JoshP May 6 '13 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 May 6 '13 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 May 9 '13 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

share|improve this answer

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.

share|improve this answer

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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.