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The Boundary Element Method (BEM) is used to solve for the fields from high-permeability, finite pole-tip gapped heads, with and without a permeable layer beneath the recording medium. The underlayer enhances the recording and playback performance for perpendicular magnetization by increasing both perpendicular fields and field gradients. Reciprocity is used to find the spectral (frequency) response for both sine-wave and square-wave perpendicular magnetizations. Regions of low spectral response, geometrically induced by the finite pole-tips, are more pronounced with the underlayer than without, and are explored for various pole-lengths.