Given the laws of individual collision events ( in the form of absorption coefficients and scattering kernels / phase functions ) the problem of linear transport theory is then to determine the result of a large number of random collisions governed by these laws.
12.
Consequently, approximate expressions for a ( t ) and ( t ) can be obtained when the spectrum, including its phase function ( ) is given and, similarly, approximate expressions for | S ( | and ( ) can be obtained when the signal characteristics are given.
13.
This development was given a more mathematical form by Einstein, who noted that the phase function for the waves : \ theta ( J, x ) in a mechanical system should be identified with the solution to the Hamilton Jacobi equation, an equation which even Hamilton considered to be the short-wavelength limit of wave mechanics.
14.
Example of spectral phase functions include the case where " ? " ( " ? " ) is a constant, in which case the pulse is called a bandwidth-limited pulse, or where " ? " ( " ? " ) is a quadratic function, in which case the pulse is called a chirped pulse because of the presence of an instantaneous frequency sweep.
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