Time-harmonic interfacial waves on a composite elastic membrane supported by a weak Winkler foundation are studied. The membrane is comprised of two identical semi-infinite parts connected by a narrow strip having distinct material properties. The associated dispersion relation is derived, turning out to be a relatively simple one compared to those known for interfacial waves. The existence of the sought for localised solution over the long-wave low-frequency domain is addressed, revealing a dependence solely on the ratios of material parameters. The effect of elastic foundation on the lowest cut-off frequency is investigated.