Introduction to Computer Music: Volume One

6. What is Amplitude? | page 2

Amplitude, power and intensity are correlative components of sound over time, area and distance. Below, we will see how these three phenomena correspond.

Amplitude is a measurement of the magnitude of displacement (or maximum disturbance) of a medium from its resting state, as diagramed in the peak deviation example below (it can also be a measurement of an electrical signal's increased or decreased strength above or below a nominal state). If you are watching your speaker woofer create a minute disturbance of air molecules by moving in and out a fraction of an inch from its resting state, and it suddenly starts moving in and out by several inches, you can assume that the amplitude has increased, because the maximum disturbance of the air molecules has increased (and you'd better cover your ears as well!). A perplexing element in many discussions of amplitude is the confusion between mechanical or electronic measurements of amplitude, and amplitude expressed as an acoustic measure of sound pressure. Since this is an acoustics chapter, let's go with the latter, particularly since our ears transfer sound received as pressure waves. The amplitude of sound pressure is frequently measured in pascals (Pa). 1,000 pascals is a kilopascal (kPa). For reference, sea-level air pressure (1 atmosphere) is approximately 100 kPa, close to 15 pounds per square inch. A micropascal (μPa) is one millionth of a Pascal.

Causing air molecules to move in a direction and with a velocity other than their normal state of motion to create a sound pressure wave requires force.
One newton (N) is the amount of force it takes to accelerate a 1-kilogram object by one meter per second per second (m/s2).
One pascal (Pa) is equivalent to the pressure of one newton per square meter (N/m2).

The benchmark threshold of hearing, the smallest perceptible amplitude, is approximately 0.00002 N/m2, or 20μPa, for a 1 kHz tone in laboratory conditions. 60 N/m2 (60 Pa) is considered by some to be the threshold of pain, but as we will see, this is also subjective and varies greatly by individual and age.

Discussions of amplitude depend largely on measurements of the oscillations in barometric pressure or electrical energy from one extreme (or peak) to equilibrium and not normally measured peak to peak (PP). The degree of change above OR below an imaginary center value is referred to as the peak amplitude, or peak deviation, of that waveform.
If we tried to calculate the average amplitude over time of a sine wave, it would unfortunately equal zero, since it rises and falls symmetrically above and below the zero reference. This would not tell us very much about its amplitude, since low-amplitude and high-amplitude sine waves would appear equivalent. A more meaningful reference has been developed to measure the average amplitude of a wave over time: the root-mean-square or rms method. You may also see the rms measurement applied to the power output of an amplifier.
The rms value of a waveform represents a squaring of the instantaneous amplitude for each point in time (or a sufficient number of equally spaced points) of a waveform, calculating the average (arithmetic mean) of those values, and finally taking the square root of this average.

This is extremely useful information for those using averaging level meters with audio equipment or software.

Example: The rms of a sine wave with a hypothetical peak-to-peak value of –1 to 1 will be 0.707. This can be used to extrapolate that any sine wave rms amplitude = 0.707 x peak amplitude. Peak amplitude = 1.414 x rms amplitude. How does this work? If we took 4 equally-spaced samples of a sine wave with an amplitude of 1, we would get 0, 1, 0, -1. squaring each we get 0,1,0,1. We then take the arithmetic average to get .5. We then take the square root of .5 and get approximately 0.707. If we took the rms of a sine wave with an amplitude of 3 with 4 sample values, we would get 0,3,0,-3. When squared (0,9,0,9) and averaged, we'd get 4.5, and when we take the square root, we'd get an rms of ~2.12, which is the same as multiplying the sine wave's amplitude (3) by the 0.707 rms of a 1 amp sine wave.

A square wave's rms would equal it's peak amplitude, since the wave is always at one peak or the other. Example: A square wave with a peak amplitude of 1 would have samples of 1, 1, -1, -1; when squared, averaged and sq root'ed, the wave would have an rms value of 1 as well as a peak amp of 1.

When using audio gear or software, it is important to know whether your meter is a peak-reading meter or averaging meter (or neither). While there are many good reasons to keep an eye on a signal’s peaks, the rms average is far more akin to the way we hear. In the real world, most signals don't make pretty little pictures like sines, triangles or squares, and their amplitudes over time are continuously fluctuating. So rms measurements are made as a moving average where the time domain period of an averaging meter (how far back in the signal's history they look) may be selected by choosing the meter's fast response or slow response option. Once you have an understanding of dB’s described below, the markings on the meters should/might make more sense.

Interesting Factoid: Wind turbine efficiency is determined using a root-mean-cubed method as the cube of wind speed is proportional to the energy produced over time.


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