In the time domain, amplitude, the dependent variable, is shown in volts relative to the Y-axis. In an oscilloscope display, for example, we may see branch circuit voltage as 325 V peak-to-peak, but the meaningful figure is 115 to 120 V, depending on your distance from the transformer, wire size and loading. One hundred and twenty is the RMS value. That stands for root-mean-square, and it is used in a variety of disciplines including statistics, water flow, weather forecasting, etc.

The reason we are interested in the RMS value of alternating current that conforms to a sine wave is that it is equal to the amount of direct current that would dissipate the same amount of power in a resistive load. RMS values for non-sinusoidal waveforms are different. RMS is defined in mathematics as the square-root of the mean square.

In the time domain, amplitude is usually shown as volts on a linear scale. In the frequency domain, when you press Math>FFT or send a signal into the RF input, amplitude along the Y-axis displays as power rather than as volts as in the time domain. This power, moreover, is denoted in decibels relative to the Y-axis. The scale is logarithmic rather than linear. The advantage is that it permits the user to see high-amplitude spikes juxtaposed with the low-amplitude noise floor, both in the same display. Otherwise, the oscilloscope screen might have to be 30 ft tall.

Decibels are used to compare two quantities in acoustics and elsewhere. This convention works well because it is how we perceive sound and light. The decibel is 0.10 bel, an obsolete term. The ratio R of two signals in decibels is:

R = 10 log₁₀ P₂/P₁, where P₂ and P₁ are the amount of power in the two signals.

When considering amplitudes A of two identical waveforms, the ratio is
R = 20 log₁₀ A₂/A₁.

If one signal is twice the amplitude of another signal, it is + 6 dB relative to it because log₁₀ 2 = 0.3010. A signal ten times as large is +20 dB. A signal one-tenth  as large is -20 dB.

As you can see, a quantity in the decibel scale takes off like a rocket.

In addition to representing the ratio between two signals, the decibel scale can represent the absolute value of one signal. For this to make sense, it is necessary to assume some reference point. It cannot be the intersection of X and Y axes in an oscilloscope display, because zero multiplied by any number is zero. One volt is commonly used as this level. Another reference level is the minute amount of thermal noise generated within a resistive load at room temperature.

When the number of decibels represents an absolute value rather than a ratio between two signals, this is indicated by appending a letter corresponding to the units involved. Accordingly, if the reference quantity is one volt, V is added after dB (dBV). If the reference quantity is one milliwatt, m is added after dB (dBm).

Decibel notation is frequently used in acoustics, electronics and control theory. Amplifier gain, signal attenuation and signal-to-noise ratios are denoted in this manner.

Decibel terminology began with investigations into signal attenuation in telegraph and telephone lines. The first figure to quantify this loss was Miles of Standard Cable (MSC), which referred to power loss perceived by a human listener in a transmission line operating at just under 800 Hz. The cable has an assumed resistance of 88 Ω/mile with capacitance amounting to 0.054 μF. This figures to 19 AWG wire.

This procedure evolved in the twentieth century, Transmission Unit (TU) superseding MSC. One TU was set at 10× log₁₀ of the measured power with respect to a fixed reference. The way it worked out was that one TU was quite close to one MSC. TU was soon redefined as exactly one decibel, which is 0.1 Log₁₀ of the power ratio. An equivalent equation is:
One decibel = P₁ – P₂
Where P₁ and P₂ are in the ratio 100:1.
Furthermore
N decibels = P₁ – P₂
when P₁ and P₂ are in the ratio 10N(0.1).

To summarize, the number of Transmission Units corresponding to the ratio of two amounts of power is 10 times the logarithm of that ratio. All of this can be readily worked out by consulting logarithm tables or using a hand-held scientific calculator. Power gain and loss in telephone circuits are calculated by addition and subtraction of the units involved. Decibel is now recognized by the IEC, which permits its use for field quantities in addition to power. Other standards organizations define voltage ratios in terms of decibels.

The logarithm of intensity, rather than a linear relationship, describes the human perception of both sound and light. If this were not so, we could not process the vast range of sensory inputs. In acoustics, the decibel refers to sound pressure level. For sound in air, the reference level is equal to the human perception threshold. Because sound pressure is a field quantity, RMS is used. The reference sound pressure in air is 20 μPa, far less than in water where it is 1 μPa.

Sound perception in humans has an enormous dynamic range, approximately one trillion. This is the ratio of the faintest sound we can detect to the amount that causes physical damage. Only a logarithmic scale as measured in decibels can conveniently represent this range. And because we are not equally sensitive throughout the sound spectrum, frequency weighting is required.

Regarding damage to human hearing, it is not solely the intensity of the sound that is relevant, but also its duration.

The conventional sound level meter is essentially a microphone connected to a display or indicating gauge. To factor in duration, an integrating meter is required. But even that is not the whole story. A valid evaluation also depends upon averaging the intensity from beginning to end of the time cycle. Additionally, frequency must be considered.

To do a comprehensive analysis, it is necessary to have an exponentially averaging sound-level meter in which the oscillating electrical signal from the microphone is converted to dc by an RMS circuit, providing a time-constant of integration, a form of time weighting.

Sound can be described in terms of Pascals, but it is more efficient to apply logarithmic conversion to display sound pressure. Then, 0 dB sound pressure level is equal to 20 μPa.

Light can also be measured in linear or decibel scaling. The usual instrument is the lux meter. The total amount of light emitted by a source is known as the luminous flux. Light is measured in lumens. One lux is 1 lm/m². Light intensity appears across a large gradient. In our immediate experience, it ranges from 100,000 lux on a bright summer day to less than one lux in full moonlight. This is an unbounded logarithmic scale with perceivable amounts above and below these levels.

Light meters, frequently used to measure workplace illumination, are constructed and configured to respond to incandescent, fluorescent or LED lighting, each of which has a unique spectral distribution.