On discontinuous electromagnetic waves and the occurrence of a surface wave

Two problems are considered: (1) The field around a dipole free in space. Contrary to the usual treatment, where the moment of the dipole is considered to vary harmonically in time, here the moment is assumed initially to be zero but at the instant
t = 0
to jump to a constant value, which it further maintains. (2) The same dipole is placed vertically on a horizontal plane separating two media of different refractive index. It is shown that the resulting disturbance on the plane is composed of two space waves and one surface wave. First the Hertzian vector at a distance
\rho
from the dipole is zero. At
t = t_{1}
the disturbance arrives there through the less dense medium, and slowly begins to rise till, at the moment
t = t_{2}
, when the disturbance has had time to reach the same distance through the second (denser) medium it reaches its final static value and further stays constant. During the transitory interval
t_{1} < t < t_{2}
the disturbance is found to be representable, apart from a constant, by a pure surface wave. The two problems are solved with the help of the modern form of the operational calculus based on the two-sided Laplace transform. The analytical tools of the operational calculus needed are explained in a separate paragraph.