Wave Velocity

Because
is an *eigenfunction* under differentiation
(*i.e.*, the exponential function is its own derivative), it is often
profitable to replace it with a generalized exponential function, with
maximum degrees of freedom in its parametrization, to see if
parameters can be found to fulfill the constraints imposed by differential
equations.

In the case of the one-dimensional ideal wave equation (Eq. (C.1)), with no boundary conditions, an appropriate choice of eigensolution is

(C.12) |

Substituting into the wave equation yields

Defining the

(C.13) | |||

Thus

is a solution for all . By

is also a solution, where and are arbitrary complex-valued functions of arbitrary points in the complex plane.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University