Initial condition

An initial condition prescribes the state of a time-dependent problem at a chosen initial time.

Table of contents
- Initial velocity Initial condition
- Initial displacement Initial condition

Initial velocity ( $\partial_{t} u (x, t_{0})$ )

Initial velocity specifies the time derivative of the displacement field at the starting time. A second-order wave equation requires it independently of initial displacement. Both vanish for a causal scattered field before an incident pulse reaches its scatterer.

Initial displacement ( $u (x, t_{0})$ )

Initial displacement specifies the configuration of a wave equation at the starting time. For a second-order evolution equation it is paired with the initial velocity. Specifying only the boundary forcing does not determine these initial data.

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Initial condition by

Ciro Santilli  40 Updated 2025-07-16

Basically a subset of the boundary condition for when one of the parameters is time and we are specifying values for the time 0.

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