Irreg ularities in the ionosphere which are sma ll compared wit h one wavele ngth may modify the constitutive relations , and hence, may affect the refractive indices for electromagnetic waves . The modifications arc in somc ways similar to t hose which would be introduced into the Appleton-Hartree formula b y a Lorentz force. The t heory is g iven first for the case when the irregula rities exte nd only in one dimension, and it is found that even in a loss-free medium t he re fractive ind ex now has an imagina ry part which m igh t be associated with loss of energy from the wave by scattering. The theory for t hree-dimensiona l irregulari ties is then disc ussed but is m ore diffi cult, a ll d a m ethod of su ccesRive approximations is used . The results ind icate t h at s ma ll irreg ulari ties m ay p lay a n important pa r t in the propagation of ver y-low-frequ ency r adio waves in the ionos phere. In particu lar, t h ey may explain why " whis tlers" a re observ ed only on comparatively rare occasions.
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K. G. Budden (1959) studied this question.