# amplitude

Amplitude is a term used in mathematics as well as physics and technology to describe vibrations . It can be used for quantities such as an alternating voltage and its course over time. It is defined as the maximum deflection of a sinusoidal variable from the position of the arithmetic mean . The term can also be applied to waves when the oscillation spreads locally at a constant speed (sine wave).

Sinusoidal alternating voltage:
1 = amplitude,
2 = peak-valley value,
3 = effective value ,
4 = period duration

In DIN 40110-1 a distinction is made between

• Peak value of a periodic alternating voltage and${\ displaystyle {\ hat {u}}}$
• Amplitude of a sinusoidal alternating voltage.${\ displaystyle {\ hat {u}}}$

For other terms that are not restricted to alternating quantities, but are generally used for periodic processes, e.g. B. for mixed voltage , see under peak value .

In the case of vibrations, the distance between maximum and minimum is referred to as the vibration width or also as the peak-valley value (previously called the peak-peak value).

## Mathematical representation

An undamped sinusoidal or harmonic oscillation is caused by

${\ displaystyle y (t) = {\ hat {y}} \ cos (\ omega t + \ varphi)}$

described with the amplitude , angular frequency and zero phase angle . The amplitude is independent of time and therefore constant. ${\ displaystyle {\ hat {y}}}$ ${\ displaystyle \ omega}$ ${\ displaystyle \ varphi}$

Another possibility of description is the complex representation using Euler's formula (with the symbol for the imaginary unit that is common in electrical engineering :) ${\ displaystyle \ mathrm {j}}$

${\ displaystyle {\ underline {y}} (t) = {\ hat {y}} \; \ mathrm {e} ^ {\ mathrm {j} (\ omega t + \ varphi)} = {\ hat {y} } \; \ mathrm {e ^ {j \ varphi}} \ cdot \ mathrm {e} ^ {\ mathrm {j} \ omega t}}$ .

This form makes many calculations easier, see Complex AC Calculation . The expression

${\ displaystyle {\ underline {\ hat {y}}} = {\ hat {y}} \; \ mathrm {e ^ {j \ varphi}}}$

is the complex amplitude , whose magnitude is equal to the amplitude and whose argument is equal to the zero phase angle . ${\ displaystyle {\ hat {y}}}$${\ displaystyle \ varphi}$

In certain contexts, the amplitude can also change slowly compared to the associated oscillation, e.g. B. at damping or modulation .

A weakly damped, non-periodic oscillation is caused by the decay coefficient ${\ displaystyle \ delta}$

${\ displaystyle y (t) = {\ hat {y}} \; \ mathrm {e} ^ {- \ delta t} \ cos (\ omega t + \ varphi)}$

described. The expression

${\ displaystyle A (t) = {\ hat {y}} \; \ mathrm {e} ^ {- \ delta t}}$

is the time-varying amplitude function .

For specific influencing of the amplitude, see amplitude modulation .

## Examples

The amplitude is illustrated with mechanical examples, especially on the pendulum .

In the ideal case, a spring pendulum (undamped) performs a sinusoidal oscillation. The distance between

• the turning point at which the pendulum has the greatest deflection, and
• the point of rest from which the pendulum cannot oscillate without energy input,

is the amplitude.

A plane physical pendulum swings in a sinusoidal manner neither in the angle nor in the horizontal deflection, even with undamped movement. The horizontal distance between the turning point and the point of rest is a peak value . Only with a small deflection, when the peak value is much smaller than the pendulum length, i.e. if the small-angle approximation can be used, the oscillation becomes sinusoidal and the peak value becomes the amplitude.

## Demarcation

The limit values ​​of the deviations from the respective mean value in other curves in graphical representations are also referred to as amplitude in the broader sense. Sometimes the amplitude is also assigned a different meaning, such as the difference between the minimum and the maximum . Here, the technical term has been adopted in the technical terminology of other specialist sciences that do not use it according to the standard defined above , so that the special meaning is occasionally uncertain, for example in pulmonology in spirometry , in seismology in the seismogram or in the Meteorology and climate geography in the climate diagram .

## literature

• Ilja N. Bronstein, Konstantin A. Semendjaev, Gerhard Musiol, Heiner Mühlig: Taschenbuch der Mathematik. 5th, revised and enlarged edition, unaltered reprint. Harri Deutsch, Thun et al. 2001, ISBN 3-8171-2005-2 .
• Christian Gerthsen: Physics , Springer-Verlag