Theory of the Propagation of Nonstable Waves

The general theory of the propagation of pulse disturbances in an unstable, slightly inhomogeneous, and time-dependent medium is derived in complementary fashion to the eikonal theory of the propagation of stable waves. For a dispersion $D(\mathrm{k}, \omega ; \mathrm{a})$, where k is the wave number and $\omega$ is the (complex) frequency which is related to k for given values of the auxiliary parameters a (temperature, density, etc.) by the vanishing of $D$, the rate of change of propagation velocity of an element of the disturbance is found to be $\frac{\mathrm{dV}}{\mathrm{da}}=\mathrm{Re}(-\frac{D_{\mathrm{a}}}{k{D}_{\omega }})$ in one dimension. A similar equation holds in the case of a general $n$-dimensional system. This relation makes possible the construction of the complete pulse shape in fairly simple fashion for the case of a general slowly varying medium. The theory is illustrated by application to a particular unstable dispersion.