Time average

In physics, the time mean value or temporal mean value is a special mean value of a physical quantity or function that is dependent on time . It is often used u. a. in statistical physics for the ergodic hypothesis and in electrical engineering for calculating the equivalent value , but it is a general tool for many physical applications. ${\displaystyle {\overline {f(t)}}}$ ${\displaystyle f(t)}$

Mathematically, the time average is formed by integrating any time-dependent variable over a time interval and then normalizing it over the duration of this interval: ${\displaystyle [t_{0},t_{0}+\Delta t]}$ ${\displaystyle \Delta t}$

${\displaystyle {\overline {f(t)}}=\int \limits _{t_{0}}^{t_{0}+\Delta t}\!\!\!f(t)\,\mathrm {d} t\cdot {\frac {1}{\Delta t}}}$

Only if the observed quantity is linearly dependent on time:

${\displaystyle f(t)=at+b}$,

the time average is the arithmetic mean (the " average ") of the start and end value:

${\displaystyle {\begin{aligned}\Rightarrow \quad {\overline {f(t)}}&={\frac {f(t)+f(t+\Delta t)}{2}}\\&=f\left(t+{\frac {\Delta t}{2}}\right)\\&=a\cdot \left(t+{\frac {\Delta t}{2}}\right)+b\end{aligned}}}$

If the function under consideration is continuous and differentiable (predominantly given in physical processes), the limit value of the temporal mean for very short time intervals is identical to the instantaneous value :

${\displaystyle f(t)=\lim \limits _{\Delta t\rightarrow 0}\,{\overline {f(t)}}}$

This connection is important because the instantaneous value is often not accessible by measurement, but only the time average z. B. for a speed measurement by determining the distance and time interval. A good match between the measured value and the instantaneous speed can only be obtained if the measurement interval can be selected as short as possible.

Examples

• In electrical engineering, the equivalent is the time average of a periodically changing variable. For example, the electrical voltage is composed of a DC voltage component (the time average) and an AC voltage component . The equivalent of a pure alternating voltage is zero.
• The average speed over time is called the average speed . The following applies: . ${\displaystyle v(t)}$ ${\displaystyle {\overline {v}}}$ ${\displaystyle {\overline {v}}={\frac {\Delta s}{\Delta t}}}$
• The mean value of the air temperature over time is called the annual mean temperature . For practical reasons, measured values ​​are recorded every hour and calculated over a measuring interval of one year with the arithmetic mean.