We present theory and numerical simulations of strong nonlinear effects in standing shear Alfv¿n waves (SAWs) in the Earth's magnetosphere, which is modeled as a finite size box with straight magnetic lines and (partially) reflecting boundaries. In a low &bgr; plasma it is shown that the ponderomotive force can lead to large-amplitude SAW spatial harmonic generation due to nonlinear coupling between the SAW and a slow magnetosonic wave. The nonlinear coupling leads to secularly growing frequency shifts, and in the case of driven systems, nonlinear dephasing can lead to saturation of the driven wave fields. The results are discussed in the context of their possible relevance to the theory of standing ionospheric cavity wave modes and field line resonances in the high-latitude magnetosphere. ¿ American Geophysical Union 1994