The energy levels of two simple examples of screened Coulomb potentials have been analyzed using nonperturbative methods. The analysis indicates that the energy levels as a function of the perturbation parameter ) have a branch cut along the negative real axis; starting from the origin. Furthermore, there are singularities on the second sheet, along~X~e '' for~X~0. As a consequence of these singularities, the energy levels have an asymptotic series in X, which means that orie cannot use a power series in X to describe the energy levels to an arbitrary accuracy, The approximate but nonperturbative expression for the energy levels, which has been obtained by using dispersion relations, predicts energy levels which are in good agreement with those obtained from variational calculations.

Hadronic multiparticle reactions at very high energies are reviewed with emphasis on current theoretical pictures and models.

We have studied the bound states of the potential V(r) = -Z/(r + P). The S-wave energy level E(P), .as a function of the parameter p, has a logarithmic singularity at the origin, The imaginary part of E(p) for P~0 goes as Z + ' P ', where I is the angular momentum.

A theory is developed for the propagation of pressure waves in a two‐phase medium where one phase consists of spherical inclusions distributed randomly in the second phase. The theory is based on an integral equation of Foldy (1945) and Twersky (1970) for the average wave which includes almost all multiple scattering processes, but it ignores correlations among inclusions. In the low‐frequency limit, this equation is solved exactly for an analytical expression for the refractive index of compressional waves in terms of elastic parameters of the matrix and the inclusions. The theory is then applied to fluid‐solid suspensions and to fluid‐saturated porous rocks. In the former case, velocities measured by Kuster and Toksöz (1974b) as a function of the concentration of inclusions are compared with theoretical predictions of seven different models. The closest agreement is obtained for the present theory. This is attributed to systematic inclusion of multiple scattering effects including near‐field scattering interactions. Theoretically computed attenuation also agrees well with experiments. In the case of fluid‐saturated porous rocks, comparison between theoretically computed velocities and observed velocities in sandstones (Wyllie et al, 1962) suggests the importance of microcracks and nonspherical inclusions, factors which are not included in the present model. Attenuation caused by viscosity of inclusions is shown to arise in two different ways: losses caused by generation of shear waves in the fluid, and those caused by Darcy‐type flow of the fluid relative to the matrix. The former increases with viscosity, while the latter decreases. The sum, however, remains negligible at frequencies relevant to seismic prospecting and earthquakes.

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