N masses on an infinite string and related one-dimensional scattering problems

P.A. Martin / Wave Motion 51 (2014) 296–307 297 transmission coefficient is calculated exactly, as is the average transmission coefficient. These results do not assume that ε is small (compared to the spacing or wavelength) but they are complicated. We then approximate these results for small kε, and we show that they can be obtained more easily by assuming that kε is small from the outset. This latter approach is then developed for the more difficult problem where each scatterer is displaced by a small distance, independently of all the others (Section 8). Again, explicit results are obtained for the average reflection and transmission coefficients, correct to second order in kε. The purpose of this work is to obtain some benchmark solutionswithminimal assumptions. Themethods used are rather elementary. The main tool used is the transfer matrix for each scatterer; their properties are reviewed in Section 2. Each scatterer can be quite general: we do not assume point scatterers. 2. Transfer matrices Consider one scattering region or ‘‘cell’’, |x| < a. Outside the cell, the governing differential equation is u′′(x)+k2u(x) = 0. Thus, we can write u(x) = { Ae + Be, x < −a, Ce + De, x > a, where A, B, C and D are constants. (At this stage, we do not have to say anything about what is in the cell, except we shall assume that there are no losses.) The suppressed time-dependence is e−iωt , so that the eikx terms give waves going to the right (x increasing) whereas the e−ikx terms give waves going to the left. The amplitudes on the right are related to those on the left using a transfer matrix T: