## 6. What is amplitude?

Amplitude is the objective measurement of the degree of change (positive or negative) in atmospheric pressure (the compression and rarefaction of air molecules) caused by sound waves. Sounds with greater amplitude will produce greater changes in atmospheric pressure from high pressure to low pressure. Amplitude is almost always a comparative measurement, since at the lowest-amplitude end (silence), some air molecules are always in motion and at the highest end, the amount of compression and rarefaction though finite, is extreme. In electronic circuits, amplitude may be increased by expanding the degree of change in an oscillating electrical current. A woodwind player may increase the amplitude of their sound by providing greater force in the air column i.e. blowing harder.

Amplitude is directly related to the acoustic energy or intensity of a sound. Both amplitude and intensity are related to sound's power. All three of these characteristics have their own related standardized measurements and will be discussed below.

Amplitude is measured in the amount of force applied over an area. The most common unit of measurement of force applied to an area for acoustic study is the Newtons per square meter (N/m2).

 One Newton is the amount of force it takes to accelerate a 1-kilogram object by one meter per second (m/s)

The benchmark threshold of hearing, in other words the smallest perceptible amplitude, is approximately 0.00002 N/m2 for a 1 kHz tone in laboratory conditions (this is actually contradicted by loudness curves discussed below). 60 N/m2 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 from one extreme (or peak) to the other. The degree of change above or below and imaginary center value is referred to as the peak amplitude or peak deviation of that waveform.
 If we tried to calculate the average amplitude 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, called the root-mean-squared 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 the amplitude of each point of a waveform and then taking its mathematical average.

The function of the squaring is to eliminate negative values, since all the negative values square to positive ones. 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 rms amplitude = 0.707 x peak amplitude. Peak amplitude = 1.414 x rms amplitude.

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 peak, the rms average is far more akin to the way we hear. Once you have an understanding of dB’s described below, the markings on the meters should make more sense.

Power and Intensity:

If we picture a sound wave as an expanding sphere of energy, power is the total amount of kinetic energy contained on the sphere’s surface. By examining the formula below, you can see how power is a measurement of amplitude over time.

 The unit of measurement for power is the watt, named after James Watt. 1 watt = 1 Newton of work per second

The power of the original sound source along with distance of measurement from the sound source combine to form the intensity, Intensity can be measured as watts per square meter or w/m2. Intensity can be seen as amplitude over time over an area. As the surface area of the sound sphere expands, the amount of energy generated by the sound source is distributed over an exponentially increasing surface area. The amount of energy in any given square meter of the expanding sphere's surface decreases exponentially by the inverse square law, which states that the energy drops off by 1/distance2. So acoustic energy twice the distance from the source is spread over four times the area and therefore has one-fourth the intensity, or simply put, relative intensity is the reciprocal of the change in distance squared.

 You may recall from your grade-school math that the , so as the radius of a sound sphere increases arithmetically, its surface area increases geometrically. The intensity of the source signal energy is distributed over the broadening surface area so that the , where s = the source intensity.

The inverse square law is extremely useful to remember in microphone placement, where even small changes in distance can have a significant impact on the resultant signal strength.

A few more relationships between amplitude, intensity and power: intensity equals the square of the amplitude, so if the amplitude of a sound is doubled, its intensity is quadrupled. Power is also proportional to amplitude squared, therefore power and intensity are proportional to each other.

Decibels:

While power is measured in watts, the most-used acoustic measurement for intensity is the decibel (dB). Named in honor of Alexander Graham Bell, a decibel = 1/10 of a bel. A decibel is a logarithmic measurement that reflects the tremendous range of sound intensity our ears can perceive and closely correlates to the physiology of our ears and our perception of loudness. There are many different forms of decibel measurement and it is not always clear which method of computation is being used unless it is labeled properly.

I must admit that I was once intimidated by logarithms, but with cheap calculators to do the math (one previously used log tables), just a simple understanding of how they work is all that is necessary for decibel calculations.

 A logarithm primer can be thought of as "what power of 10 will result in x." For example, because 102 = 100. Decibels are often used to measure very minute values, which can also be expressed by logs of negative numbers. For example, and , a value we will use for our threshold of hearing measurement below. If it is expressed then .

A decibel is a measurement used to compare the ratio of intensities of two acoustic sounds (or electronic signal). The ratio (R) of two signals expressed by their power in watts (W1 and W2) is:

There are many different types of decibel measurements, so for the purpose of clarity, the above form, which measures power or intensity is called dBm. For the purpose of having a standardized absolute measurement of power (i.e. a comparison not to another signal, but to an industry-fixed value), the nominal reference wattage (W2)has been defined as 1 miliwatt (0.001 watt). In absolute terms, a 1-watt signal, which has 1,000 times the power of the reference wattage, will be 30 dB, computed below:
 dBm=10 log10 (1 watt/.001 watt) dBm =10 log10 (1000) dBm=10 x 3 [because log10 1000 = 3] dBm=30

dBm is the form most commonly used to evaluate power in audio circuits.

Since intensity (I) at a fixed distance of measurement is directly proportional to power, a similar measurement can be made:

In this case, a doubling of power equals an increase of +3dB. When we study filters later on, you will notice that a filter cut-off frequency is defined as the half-power point, which is calculated as –3dB.

While the original dB scale was created for comparison of intensity or power, it is also commonly used as a measurement of amplitude (A) or sound pressure as defined above. The formula for computing relative amplitude or sound pressure is:

By comparing this formula to the one for dB above, the relationship between amplitude, power and intensity becomes clear. In this case, a doubling of amplitude from one source to another equals an increase of +6 dB as shown below:

The most common acoustic ratio measures a current sound against a predetermined value of the threshold of audibility mentioned above but expressed as 2 x 10^-12 watts. This absolute measurement is referred to as the sound-pressure Level (SPL) and gives us a means of generalizing relative loudness of common acoustic sources (note that the dB is followed by SPL to indicate this mode of measurement). The logarithmic scale from the threshold of hearing to the threshold of pain ranges from 0.00002 N/m^2 to 200 N/m^2, or about 120-130 dB SPL, at which point the entire body, not just the ears sense the vibrations (NB: In preparing this article, it quickly became apparent that no “standard” for the threshold of feeling or the threshold of pain has been established, and in fact ranges in the references used from 120 dB SPL to 140 dB SPL, which is a huge variation of opinions and points out the differences between acoustic and psychoacoustic measurement). Younger people also have more effective protection mechanisms and so can tolerate louder sounds (surprise!).

If we accept 130 dB as the threshold of pain, then humans hear sounds that range from the smallest perceptible amplitude to those that are 10,000,000,000,000 as loud or 10 watts/m2. Both the dB and dB SPLscales reflect the incredible discrimination of human hearing, our most sensitive sense by far.

Here are some vague benchmarks (which of course depend on many factors, including the listener’s distance from the sound).

 Source Power (watts/m2) dB SPL Threshold of pain 10 130 Jet takeoff from 500 ft. 1 120 Medium-loud rock concert .1 110 Circular saw .01 100 New York subway .001 90 Jack-hammer from 50 ft. .0001 80 Vacuum cleaner from 10 ft. .00001 70 Normal conversation .000001 60 Light traffic from 100 ft. .0000001 50 Soft conversation .00000001 40 Whisper from 5 ft. .000000001 30 Average household silence .0000000001 20 Breathing .00000000001 10 Threshold of hearing in young .000000000001 0

Signals from microphones, most of which seek to accurately transform changes in SPL to proportional changes in voltage (V), can also be measured by the same method. If one were to change the miking distance to the sound source, the voltage differences could be measured as follows:

If measured properly, halving the distance of the mic to the source, thanks to the inverse square law should double the voltage produced by the microphone, giving a +6 dB increase in amplitude (which if you’ve been reading closely also produces four times the intensity). For a standardized comparison of voltages, 0.775 volts is used as the reference level for = 0 dB.

We have looked at two basic types of dB measurement, one for power and intensity, and the other for amplitude, SPL and voltage. Several other weighted dB scales, such as dBA are used for specific purposes, such as more closely mirroring the way we hear, but this will be discussed in further detail in the psychoacoustics sections.

Envelopes:

Dynamic envelope refers to the amplitude change over time of a sound event (usually a short one, such as an instrumental or synthesized note). As a very simple example (because there is usually much more going on in acoustic sounds), a note can have an initial attack characterized by the amount of time it takes to change from no sound to a maximum level, a decay phase, whereby the amplitude decreases to a steady-state sustain level, followed by a decay phase, characterized by the time it take the amplitude to change from the sustain level to 0.

Not only do real world (and complexly synthesized) sounds have more complex overall envelopes, but they often exhibit different envelopes for all their individual frequency components.

For further study, see Hyperphysics->Sound Level Measurement

An Acoustics Primer, Chapter 6
URL: www.indiana.edu/~emusic/acoustics/amplitude.htm