Acoustic Theory

1. What a sine wave is

A sine wave is the simplest possible sound. At any moment its value is

y = A · sin(2πft + φ)

where A is the amplitude, f is the frequency in hertz, t is time in seconds and φ is the phase in radians. Amplitude sets how far the wave swings from zero, which you hear as level. Frequency sets how many cycles pass each second, which you hear as pitch. Phase sets where in its cycle the wave starts, which you hear only when two waves combine.

The cycle repeats every period T = 1 / f. A 440 Hz tone repeats 440 times a second, so one period lasts 1 / 440 ≈ 2.273 ms.

Why sine is the building block

The Fourier theorem states that any repeating signal, however complex, can be written as a sum of sine waves at whole-number multiples of a base frequency. The reverse is also true: a complex sound is nothing but a set of sines added together. This is why a sine is the reference signal for calibration and testing. It contains energy at exactly one frequency and nothing else, so anything the sine picks up is a property of the system under test, not of the signal.

2. Frequency, period and wavelength

Frequency and period are reciprocal:

f = 1 / T

Wavelength is the physical distance the wave travels through one cycle:

λ = c / f

where c is the speed of sound. In dry air at 20 °C, c ≈ 343 m/s. The speed is not constant. It rises with temperature at roughly 0.606 m/s per °C, so a rough model is c ≈ 331.3 + 0.606 · θ m/s for air temperature θ in °C. At 0 °C the speed is about 331 m/s; at 30 °C it is about 349 m/s. Humidity changes it by a smaller amount. All wavelengths in this reference use 343 m/s.

Period and wavelength at reference frequencies (c = 343 m/s)

Frequency	Period	Wavelength (m)	Wavelength (mm)
20 Hz	50 ms	17.15 m	17 150 mm
100 Hz	10 ms	3.43 m	3 430 mm
440 Hz	2.273 ms	0.780 m	780 mm
1 kHz	1 ms	0.343 m	343 mm
10 kHz	0.1 ms	0.0343 m	34.3 mm
20 kHz	0.05 ms	0.01715 m	17.15 mm

The pattern to remember: raise the frequency by a factor of ten and the wavelength drops by the same factor. A 20 Hz wave is longer than a room. A 20 kHz wave is smaller than a fingertip.

3. Amplitude and the decibel

Human hearing spans an enormous range of pressures, from the threshold of hearing to the threshold of pain. A linear scale would be unusable, so level is measured in decibels, which are logarithmic.

For a ratio of two powers, dB = 10 · log10(P2 / P1). For a ratio of two voltages or amplitudes, dB = 20 · log10(A2 / A1). The factor of two appears because power is proportional to amplitude squared.

The numbers worth memorising

• 6 dB is a doubling of voltage or amplitude, 20 · log10(2) ≈ 6.02 dB.
• 10 dB is a doubling of perceived loudness, a rule of thumb that holds over the mid range.
• 3 dB is a doubling of power, 10 · log10(2) ≈ 3.01 dB.

The three decibel scales used here

dB SPL is sound pressure level relative to 20 µPa, the nominal threshold of hearing. It is an absolute measure of the acoustic pressure in air. A quiet room is about 30 dB SPL; normal speech is about 60 dB SPL.

dBFS is decibels relative to digital full scale. 0 dBFS is the largest value a sample can hold. Everything below it is negative. A signal at -6 dBFS peaks at half the amplitude of full scale. Digital clipping happens when a value tries to exceed 0 dBFS.

dBu is decibels relative to 0.775 V, a reference inherited from telephone and line-level equipment. It describes an electrical voltage, not sound pressure, so it is only meaningful for the analogue part of a chain.

4. Harmonics and timbre

A pure sine is one frequency. A real instrument radiates many frequencies at once, and the mix of those frequencies is what we call timbre. The components above the lowest one, the fundamental, are called harmonics, at whole-number multiples of the fundamental.

Each classic waveform has a fixed recipe of harmonics:

Waveform	Harmonics present	Amplitude of harmonic n	Character
Square	Odd only	1 / n	Hollow and buzzy
Sawtooth	All harmonics	1 / n	Bright and string-like
Triangle	Odd only	1 / n²	Mellow and soft

A square wave keeps only the odd harmonics, at 1, 3, 5, 7 and so on, each with amplitude 1 / n. A sawtooth keeps every harmonic, also at 1 / n, so it has roughly twice the number of partials and sounds brighter. A triangle keeps only the odd harmonics but rolls them off as 1 / n², so the high partials fade much faster and the result is softer than a square.

A square and a sawtooth sound bright because their high harmonics stay strong as 1 / n, which falls slowly. A sine has no harmonics at all, and a triangle loses them as 1 / n², which falls fast. This is also why a square or saw reads as a dense comb of peaks in a spectrum display, while a sine reads as a single peak.

Duty cycle

A square wave with equal high and low halves has a 50 % duty cycle. Change the fraction of the period spent high and the harmonic content changes: a pulse narrower than 50 % develops even harmonics, and the timbre moves from hollow towards thinner and more nasal. The pulse train in the alarm generator is the audible version of this.

Aliasing of harmonics above Nyquist

Harmonics do not stop at the audible limit. A 1 kHz square has partials at 3, 5, 7 kHz and onward forever. In a digital system, any harmonic above half the sample rate, the Nyquist frequency, cannot be represented. It does not simply disappear. It folds back to a lower frequency, a mirror image of itself, and appears as an inharmonic tone that was not in the original signal. Good synthesisers limit the number of harmonics generated so that none cross Nyquist, which is why a high square wave in a digital generator can sound less rich than the maths suggests.

5. Sampling

Digital audio stores a sound as a series of measurements, or samples, taken at a steady rate, the sample rate. CD audio uses 44 100 samples per second, written 44.1 kHz. The rate must be high enough to capture the fastest change in the signal.

The Nyquist–Shannon theorem

The sampling theorem states that a signal can be reconstructed exactly only if the sample rate is greater than twice the highest frequency present:

fs > 2 · fmax

Half the sample rate is the Nyquist frequency. At 44.1 kHz it is 22.05 kHz. Any component above it folds back and is misrepresented, so a real converter filters the input to remove energy above Nyquist before sampling.

Why 44.1 kHz

44.1 kHz was chosen for the compact disc for two reasons. The audible range tops out at roughly 20 kHz, so the rate must exceed 40 kHz. The extra 4.1 kHz gives the anti-aliasing filter a transition band: a filter cannot cut off instantly, so the standard places the passband edge at 20 kHz and the stopband by 22.05 kHz, leaving a small gap to roll off in. Many hardware converters run at 48 kHz, which leaves slightly more headroom.

Quantisation and dynamic range

Each sample is stored with a fixed number of bits. 16-bit audio has 65 536 possible values. The gap between adjacent values is the quantisation step, and the error this introduces sets the noise floor. Every extra bit adds about 6 dB of dynamic range, so 16 bits give roughly 16 · 6.02 ≈ 96 dB between the loudest signal and the noise floor.

Dithering is the practice of adding a tiny amount of low-level noise before quantisation. The error from rounding a sample is correlated with the signal and sounds like harsh distortion on quiet passages. Adding a trace of noise decorrelates that error and turns it into a gentle hiss, which the ear tolerates far better. The noise is below -90 dBFS in 16-bit audio and is inaudible in most material.

6. How humans hear

A healthy young ear covers roughly 20 Hz to 20 kHz. The upper limit falls with age and with noise exposure, and the drop is gradual rather than a hard edge.

Typical upper frequency limit by age

Age	Typical upper limit	Comment
Under 20	20 kHz	Best case, undamaged hearing
20 to 29	17 kHz	Slow loss begins
30 to 39	16 kHz	High frequencies recede
40 to 49	14 kHz	Common point at which 15 kHz is lost
50 to 59	12 kHz	Presbycusis is clear
60 and over	10 kHz	Highly variable between people

These are typical values, not rules. Noise exposure, genetics and medical history move them a long way in either direction.

Loudness is not linear

The ear does not treat all frequencies as equally loud. Equal-loudness contours, first mapped by Fletcher and Munson and now standardised as ISO 226, show how much level each frequency needs to sound as loud as a reference tone. The curves bunch up near 3 to 4 kHz, where the ear canal resonates and hearing is most sensitive. This is why a quiet 3 kHz tone can seem far louder than a quiet 100 Hz tone at the same measured level, and why mixing by SPL alone is not enough.

As level rises the contours flatten: quiet sounds seem to lose their low and high end, loud sounds seem fuller. This is the basis of the loudness control on older amplifiers, and the reason a mix judged at low volume does not translate directly to a loud system.

Masking

A loud sound hides a quieter one nearby in frequency. This is masking, and it is why white or pink noise can cover a conversation in an open office. Masking is strongest for frequencies just above the masker, and it spreads with the masker's level. Lossy codecs exploit it by discarding components that a nearby strong tone would hide anyway. The noise generator is a practical masking tool.

7. Musical pitch

Modern Western instruments use twelve-tone equal temperament: the octave is divided into twelve equal steps, and each step is a frequency ratio of 2^(1/12) ≈ 1.05946. Equal steps mean instruments can play in any key without retuning, at the cost of every interval except the octave being slightly impure.

The size of an interval in cents is

cents = 1200 · log2(f2 / f1)

One semitone is 100 cents; one octave is 1200 cents. Concert pitch sets A4 at 440 Hz, defined by ISO 16. The frequency of any equal-tempered note from its MIDI number n, where A4 is 69, is

f = 440 · 2^((n − 69) / 12)

Alternative references are common. Verdi tuning puts A4 at 432 Hz, about 31.8 cents flat of concert pitch: 1200 · log2(432 / 440) ≈ -31.8 cents. Baroque pitch uses 415 Hz. The difference is audible when the two references are played back to back, but within a single piece it changes the character of the instruments, not the intervals. The 432 Hz page plays each reference directly.

8. Beats and interference

Two tones close in frequency and played together rise and fall in level at a rate equal to the difference between them:

f_beat = |f1 − f2|

A 440 Hz tone against a 444 Hz tone produces a 4 Hz beat, four pulses a second. The effect is strongest below about 20 beats per second, after which the two tones merge into a single rough or dissonant sound. Beats are the standard way to tune an instrument by ear: remove the beating and the pitches match. The multiple tone generator can play two tones at once so you can hear this directly.

Binaural and monaural beats

A monaural beat occurs when both tones reach both ears and mix in the air or in one channel. A binaural beat occurs when each ear receives only one tone, through headphones. The difference frequency is then created in the brain rather than in the room, and it can be perceived even though no single place in the air contains the beat. Binaural beats require stereo separation, which the binaural beat generator provides. Claims about their effect on brain states are not firmly established, and this site presents them as tones to listen to, not as therapy.

Standing waves and room modes

Sound reflects from walls, and a reflected wave interferes with the incoming one. At frequencies where the room dimension is a multiple of half a wavelength, the reflections reinforce into a standing wave: some positions get louder, others quieter, and the pressure swells at particular frequencies. These are room modes, given for a dimension L by

f = c / (2 · L)

for the fundamental, and integer multiples of it for the higher modes. In a room 4 m long, the first axial mode is 343 / (2 · 4) ≈ 42.9 Hz, with the next at 85.8 Hz, 128.6 Hz and so on. A tone near one of those frequencies can sound uneven as you move around the room. The subwoofer test exists to find these by ear.

9. Hearing safety

Sound exposure damages hearing by dose, not by peak alone. Damage accumulates from level multiplied by time, and it is permanent. Recognised occupational limits use a 3 dB exchange rate: every 3 dB of extra level halves the safe exposure time.

Typical exposure limits, 3 dB exchange rate

Level	Recommended maximum
80 dB	40 hours per week
85 dB	8 hours per day
91 dB	2 hours
94 dB	1 hour
100 dB	15 minutes
110 dB	Under 2 minutes

These values follow WHO and NIOSH guidance for occupational noise. A pure tone concentrates its energy at one place on the cochlea, and long exposure to a single frequency is harder on that spot than the same level of broadband sound. Tones are also unforgiving of concentration: a sustained 2 kHz tone is far more fatiguing than speech at the same level.

Protecting yourself while testing

• Start with the site level low and raise it only as far as you need to hear.
• Keep test tones short. A few seconds per frequency is enough to judge audibility.
• Never test tones in headphones at high level to "check" a frequency. Use a level you would accept for a whole album.
• Take breaks. Hearing fatigue dulls sensitivity and makes judgements unreliable.
• If a tone causes discomfort, ringing or a sense of pressure, stop immediately.

This site cannot measure the sound pressure reaching your ears. The level slider on this site controls a digital gain, and the true SPL depends on your device, headphones and the setting of your system volume. Treat the limits above as a reason for caution, not as a calibrated measurement.

10. Reference frequencies measured on this site

The table below lists the fixed frequencies the tools on this site use. They come from the standards named at the end of the page, not from estimation.

Frequency	Where it is used
50 Hz	Mains hum in Europe, and most of the world
60 Hz	Mains hum in North America and parts of Asia
300 to 3400 Hz	Speech band of a telephone channel, the classic POTS band
697 to 1633 Hz	DTMF keypad tones: rows 697, 770, 852, 941 Hz; columns 1209, 1336, 1477, 1633 Hz
1000 Hz	The pips time signal, six 1000 Hz pips on the hour
250 Hz to 8 kHz	Standard audiometric test frequencies for a hearing test
17.4 kHz	The "mosquito" tone, chosen to be audible to younger ears
20 kHz	Nominal upper limit of human hearing, and the top of this site's range

Plain-text list of the standards and results referenced above.

• ISO 16: standard musical pitch, A4 = 440 Hz.
• ISO 226: normal equal-loudness-level contours (the Fletcher and Munson work, later revised).
• IEC 61672: sound level meters and the definition of the weighting scales behind dB SPL.
• Nyquist–Shannon sampling theorem: exact reconstruction requires a sample rate above twice the highest frequency present.
• WHO and NIOSH: occupational noise exposure limits and the 3 dB exchange rate.
• ITU-T Q.23 / Q.24: the DTMF tone plan and signalling frequencies.
• IEC 60645: audiometric equipment, including the standard test frequencies.