The concept of mean free path has been in use since 1858 as a way to determine the average path length in which a collision and related reactions will occur. The derivation of a collision mean free path typically treats the average path length for a single collision. There are a number of instances where a series of collisions and reactions produce a product of interest, yet a means to determine the average path length for a series of reactions is not apparent in the literature. This paper will review the derivation of mean free path for a single collision, then extend the derivation to the mean free path for two and three collisions in sequence, and finally discuss how this derivation can be extended to an arbitrary number of collisions.

A proof of the optical theorem (also known as the optical cross-section theorem) is presented, which reveals the intimate connection between the forward scattering amplitude and the absorption-plus-scattering of the incident wave within the scatterer. The oscillating charges and currents as well as the electric and magnetic dipoles of the scatterer, driven by an incident plane wave, extract energy from the incident beam. The same oscillators radiate electromagnetic energy into the far field, thus giving rise to well-defined scattering amplitudes along various directions. The essence of the proof presented here is that the extinction cross-section of an object can be related to its forward scattering amplitude using the induced oscillations within the object without knowledge of the form assumed by these oscillations.

This paper reports on the second part of an investigation of student understanding of wave behavior. A previous paper focused on the reflection of pulses from fixed and free ends. In this paper the focus is on the more complicated behavior of periodic waves at a boundary between two different media. We identified several erroneous reasoning approaches. Many students treated the relationship ??=??f as a mathematical identity, and did not recognize how the variables wavelength, frequency, and propagation speed can be manipulated experimentally. Many difficulties do not appear to be limited to the context of refraction of waves at a boundary, but seem to hinder student understanding of more advanced physics phenomena such as interference and diffraction.