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A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. [1] The variable denoting time is usually written as t {\displaystyle t} .
The units of group-delay dispersion are [time] 2, often expressed in fs 2. The group-delay dispersion (GDD) of an optical element is the derivative of the group delay with respect to angular frequency, and also the second derivative of the optical phase:
Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency .
Dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency. [1] Sometimes the term chromatic dispersion is used to refer to optics specifically, as opposed to wave propagation in general. A medium having this common property may be termed a dispersive medium.
The behavior of temperature when the sides of a 1D rod are at fixed temperatures (in this case, 0.8 and 0 with initial Gaussian distribution). The temperature approaches a linear function because that is the stable solution of the equation: wherever temperature has a nonzero second spatial derivative, the time derivative is nonzero as well.
Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. (Velocity is on the y-axis and time on the x-axis. Multiplying the velocity by the time, the time cancels out, and only displacement remains.)
Using the backward difference at time + and a second-order central difference for the space derivative at position (The Backward Time, Centered Space Method "BTCS") gives the recurrence equation: u j n + 1 − u j n k Δ t = u j + 1 n + 1 − 2 u j n + 1 + u j − 1 n + 1 h 2 . {\displaystyle {\frac {u_{j}^{n+1}-u_{j}^{n}}{k\Delta t}}={\frac {u ...