Multiple scattering of scalar waves by point scatterers in one dimension. I

We discuss the problem of scalar wave multiple scattering in one dimension on a target of $n$ identical fixed scatterers with delta-function potentials. We consider in detail a statistical ensemble of configurations of scatterers whose positions are uniformly distributed throughout the scattering region. We succeed in analytically performing a configurational average (over all scatterer positions) for the wave function for the problem of a transmitted wave with constant amplitude. We discuss the relationship between this problem and the standard problem of an incident wave with constant amplitude. From the simple closed form for the average of the wave function, the optical potential for the system is obtained. We then present the large and small incident particle wavelength limits [with respect to the length ($L$) of the scattering region] for both the average of the wave function and the optical potential. We also examine the question as to where the optical potential can be approximated by the form it takes in the limit of infinite $n$. (The question of where in parameter space this occurs and how well the transmitted and reflected waves can be predicted with this form is discussed in the following paper.) Furthermore, we consider the large incident particle wave number limit for the average wave function and the optical potential for a general distribution of the scatterer positions in the limit of both $n$ and $L$ approaching infinity but with $\frac{n}{L}$ remaining fixed. Lastly, knowing the simple closed form for the average of the wave function, we prove that the effective field approximation becomes exact in the limit of infinite $n$ with all other parameters held fixed.

NUCLEAR REACTIONS Multiple scattering, randomly distributed point scatterers, one dimension; configurational average wave function, optical potential; low and high energy limits.