New simplified formulas for ELF subsurface-to-subsurface propagation

New formulas for the electric and magnetic fields produced by the four elementary dipole antennas have been developed for the subsurface-to-subsurface propagation case. The formulas are of rather simple, but useful, form and are completely general (i.e., the air can easily be replaced by the sea bottom). They are valid at any frequency and at any range beyond a certain minimum distance for the flat-earth case. They are especially applicable at extremely low frequency (ELF) for the air/sea and sea/ seabed propagatipn cases. The main restrictions on these formulas are 1) the square of the index of refraction is greater than or equal to 10 and 2) the horizontal separation is greater than or equal to 3 times the sum of the depths of burial of the transmitting and receiving point sources. With these new formulas, computer evaluation can be reduced to fractions of a minute, compared with hours for the complete numerical evaluation of the exact Sommerfeld integrals. The formulas also exhibit an interference pattern set up under certain close-range conditions. The three waves (direct, modified mirror image, and lateral) may interfere, either constructively or destructively, with each other.