Abstract. The radial, oscillatory motion of the Earth's magnetopause has been found to occur predominantly with some distinct, sometimes called "magic" frequencies, which have been attributed to magnetospheric wave guide modes, typical solar wind variations or, more recently, surface waves on the magnetopause standing between the northern and southern ionospheres. In this paper we present for the first time a derivation of these surface waves, denominated as Kruskal-Schwarzschild-modes (KS-modes), in the approximation of the ideal, single-fluid magnetohydrodynamic theory for incompressible plasmas. The calculations are performed in the simplified geometry of the box magnetosphere with the magnetopause being a plane between two plasma regimes of homogeneous conditions. The reflection of the KS-modes at the ionospheres is being discussed. Under the given assumptions and realistic conditions the validity of the calculations is shown to be limited to cases of parallel or anti-parallel background magnetic fields on both sides of the magnetopause, respectively. For these cases a detailed discussion of the mode structure is presented. The magnetopause when affected by a KS-mode is found to resemble a membrane under tension with respect to its motion; the ionospheres act as supporting points of the membrane and the KS-modes correspond in this picture to their eigenmodes of oscillation. Localized pressure enhancements in the magnetosheath are discussed as possible excitation mechanism for the KS-modes.