Propagation of transient leaking modes in a stratified elastic waveguide
Authors: Gilbert Freeman 
Published: February 2008

By using a technique first applied to seismological problems by Haskell, it is a simple matter to obtain the secular function for any given elastic waveguide, when the waveguide is composed of homogeneous layers with plane parallel boundaries. Roots of the secular function are usually exhibited as dispersion curves. In this paper the independent variable is dimensionless wave number, K, and the dependent variable is dimensionless frequency, F. A plot of complex F versus K yields the diagnostic (F, K) diagram—a dispersion curve. The procedure employed is to expand the secular function about K = 0 and then to trace the behavior of each root, F(K), for increasing values of K. In each of the specific problems treated the initial position of each root, F(0), admits of a simple physical description. For example, in a simple continental waveguide (solid layer/solid half‐space), there are three sets of roots: the Lamb roots, consisting of
and
modes; the shear ‘Organ pipe’ roots; and the compressional ‘organ pipe’ roots. There is an infinite number of the two kinds of organ pipe roots. The low‐frequency PL wave arises from one of the
roots, and the normal shear modes arise from transitions of all three kinds of roots. In a simple oceanic waveguide (fluid layer/solid half‐space) the shear organ pipe roots are absent. The low‐frequency PL wave again arises from one of the
roots, and the normal modes arise from transitions of the Lamb roots and the compressional organ pipe roots. In a simple acoustic waveguide (fluid layer/fluid half‐space) only the compressional organ pipe roots are present, and the normal modes arise from transitions of these roots. The PL wave is absent. Its disappearance is clearly traced as the half‐space of the oceanic waveguide approaches a Poisson ratio of 0.5.

The treatment of waveguides composed of more than one layer offers only the additional difficulty of finding the initial positions of the organ pipe roots. When these positions have been found, the analysis proceeds in a manner similar to that for simple waveguides. The initial positions of the organ pipe roots are complex, a situation that may be interpreted physically as radiation into the half‐space of the waveguide. It is the presence of such radiation that leads one to speak of leaking modes. The
roots also have complex initial positions, if the half‐space is soft enough. In addition to being complex, the dispersion curves for the leaking modes sometimes have regions of negative group velocity. If one equates group velocity to velocity of energy transport along the waveguide, then one must conclude that there is an inward energy flux when the group velocity is negative. But when the group velocity is demonstrably not the velocity of energy transport, a simple physical picture of negative group velocity is sometimes unavailable. Plots of group velocity versus F show a banded structure in some cases but are generally quite complex. The situation can be clarified somewhat by rejecting those modes with weak excitation functions, large decay parameters, or both. Still, the (F, K) diagram is clearer. Now that seismic array processing procedures are being developed, one hopes that experimental (F, K) diagrams will become a standard tool in seismic analysis, leading to a clearer picture of seismic dispersion and propagation characteristics.