Circulation in inviscid gas flows with shocks

In this note, we show that the circulation $\Gamma=\int_C\mathbf{u}\cdot\mathbf{dx}$ around a closed material curve $C(t)$ in an inviscid gas flow develops according to the equation $\frac{d\Gamma}{dt}=\int_CT\,dS$, even when the curve may cross shocks, with the entropy jumps at the shocks excluded from the right-hand side

Mobile radio propagation prediction using ray tracing methods

The basic problem is to solve the two-dimensional scalar Helmholtz equation for a point source (the antenna) situated in the vicinity of an array of scatterers (such as the houses and any other relevant objects in 1 square km of urban environment). The wavelength is a few centimeters and the houses a few metres across, so there are three disparate length scales in the problem. The question posed by BT concerned ray counting on the assumptions that: (i) rays were subject to a reflection coefficient of about 0.5 when bouncing off a house wall and (ii) that diffraction at corners reduced their energy by 90%. The quantity of particular interest was the number of rays that need to be accounted for at any particular point in order for those neglected to only contribute 10% of the field at that point; a secondary question concerned the use of rays to predict regions where the field was less than 1% of that in the region directly illuminated by the antenna. The progress made in answering these two questions is described in the next two sections and possibly useful representations of the solution of the Helmholtz equations in terms other than rays are given in the final section