ETM Learn · 01 / 08
Connect particle motion, pressure, audible sound and the numbers on a meter.
A moving speaker changes nearby air pressure. The disturbance travels to your ear; air particles oscillate around their positions. They do not fly from the speaker to you. Their motion in this model is greatly enlarged.
A cycle is one repetition. Frequency counts cycles per second: 100 Hz means 100 cycles each second. Amplitude is the size of the oscillation. A sine is a smooth, simple periodic wave. In this sine experiment, raising frequency raises pitch; raising amplitude changes signal level. Phase describes position within a cycle. Wavelength is the distance between matching points in neighbouring cycles.
Dotted terms offer a definition on hover, keyboard focus or tap. Click to keep it open; Escape or an outside click closes it. All definitions are also available here.
A decibel describes a ratio on a logarithmic scale. Identify the quantity and reference; the number alone does not tell you perceived loudness.
A standard for measuring digital audio equipment. In the RMS convention discussed here, a sine reaching full-scale peak amplitude is the 0 dBFS reference.
Sound pressure level relative to 20 micropascals RMS in air. Zero on this scale is a reference level, not an absence of pressure fluctuations.
Size of an oscillation. Specify peak or effective value; this is not perceived loudness.
Decibels relative to digital full scale. For sample peaks, peak amplitude 1 is 0 dBFS. For RMS, check the reference convention.
Root mean square: square samples, average them, then take the square root. Specify the measurement window. For a steady sine it is peak divided by the square root of 2.
Start with frequency and amplitude, then add two waves. Compare level definitions last.
Predict → change one control → compare → explain the result.
Track one dot and increase frequency with amplitude fixed. More cycles pass each second. Slow motion changes the display speed, not the audible pitch.
Use two sines with equal frequency and amplitude. Show their sum and move phase from 0° to 180°: amplitude changes from double to zero. Exact cancellation requires matched frequency and amplitude.
Amplitude 0.25 → 0.5 doubles. Amplitude 0.5 → 1 doubles too. Both changes are about +6.02 dB.
For gain, 0 dB means a ratio of 1: no change. Negative gain in dB means reduction, not a negative waveform. A dBFS reading uses a digital full-scale reference; it does not specify the sound pressure at your ears.
In the level experiment, predict what doubling amplitude will do. Try ×2 and compare the waveform and meter. The model stops at its −60 and 0 dBFS boundaries, so doubling may be limited there.
Divide the new amplitude by the original amplitude, take its base-ten logarithm, then multiply by 20. For a ratio of 2 this gives about 6.02 dB. The logarithm turns equal multiplicative changes into equal additive steps.
A power ratio uses 10 log₁₀. An amplitude ratio uses 20 log₁₀ when power is proportional to amplitude squared, with the same impedance. Doubling amplitude does not imply doubling perceived loudness.
ΔL = 20 log₁₀(A₂ / A₁)
A₁ is the original amplitude, A₂ the new amplitude, and ΔL the change in dB. Both amplitudes must be positive and measured using the same method and units. Example: 0.5 divided by 0.25 is 2. Zero amplitude is a limiting case with no finite logarithm.
Change signal level and compare linear amplitude with the decibel scale. Try multiplication and division and note the model limits. Once the relationship is clear, open the comparison to explore measurement conventions.
Two identical, aligned signals are added. What happens to amplitude?
Amplitudes add: A + A = 2A. Adding powers of uncorrelated signals is a different case, approximately +3.01 dB for equal powers.For a sinusoidal plane travelling wave, λ = c/f and particle-displacement amplitude , where p is pressure amplitude and ρ is air density. At fixed pressure in the same medium, 20 Hz requires exactly ten times the displacement of 200 Hz. Sound speed is approximately 343 m/s in air at 20°C. A room can reproduce wavelengths longer than its dimensions; boundaries and modes change the response. A wavelength does not have to fit.
For amplitude ratios with equal impedance, ΔL = 20 log₁₀(A₂/A₁); for power ratios, ΔL = 10 log₁₀(P₂/P₁). Twice the amplitude is +6.02 dB; twice the power is +3.01 dB. The RMS convention does not make a full-scale sine’s sample peak +3 . uses an RMS pressure reference of 20 μPa in air.
Equal-frequency, equal-amplitude sines separated by phase φ sum to amplitude . A phase shift of one isolated steady sine is a time shift. Changing relative phases in a complex signal can change wave shape and transients; audibility depends on signal and phase response. Phase is not audible only when mixing two tracks.
λ = c / f
Divide sound speed c by frequency f. The result λ is the distance of one cycle. Example: 343 metres per second divided by 343 Hz gives one metre.
ΔL = 10 log₁₀(P₂ / P₁)
Divide new power P₂ by original power P₁, take the base-ten logarithm and multiply by 10. Both powers are positive and use the same units. Doubling power gives about 3.01 dB.
ξ = p/(ρc·2πf)
Divide pressure amplitude p by density ρ, sound speed c and angular frequency 2πf. This gives displacement amplitude ξ in metres with SI units. It assumes an ideal plane travelling wave; at fixed pressure in the same medium, doubling frequency halves displacement.
2A|cos(φ/2)|
A is each sine's amplitude and φ their phase difference. Halve the phase difference, take its cosine and absolute value, then multiply by 2A. Frequencies and amplitudes must match. At 0° the result is 2A; at 180° it is zero.
Use 2A|cos(φ/2)|; evaluate the cosine in degrees.
2 × 0.5 × |cos 60°| = 0.5. Equal amplitudes do not guarantee +6 dB: relative phase matters. The formula assumes equal, stable frequencies.
Sine RMS = peak/√2. Convert that ratio to dB.
−12 − 20 log₁₀(√2) = −15.0103 dBFS. Under the AES17 full-scale-sine RMS reference, the reading is −12 dBFS instead. Always state the reference with the number.
Further reading: AES: digital level terminology