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for k = ±∞ . In terms of ω$_{m}$ , ω$_{p}$ and ω$_{r}$ Eq. (43) becomes 

where ω$_{r}$ = ( ω 2 afm cos 2 θ + Ω 2 me ) 1 / 2 represents the pole frequency of permeabilities given by Eq.(13-16) and is located slightly below ω$_{m}$ so that ω$_{r}$ < ω$_{m}$ < ω$_{p}$. Assume that the asymptotic frequency for the surface is located in the region above the pole frequency ω$_{p}$ . For k = + ∞ , the second term in Eq. (44) is always larger than the first term. Hence, Eq. (44) is never satisfied and the surface modes do not exist in this wavelength region. 

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A surface mode at k = −∞ may exist for some values of ω$_{r}$, ω$_{m}$ and ω$_{p}$ . For a solution to exist in this region, the function f ( µ ) of Eq. (44) should cross zero. This gives a condition 

