The theory of ray propagation of VLF waves breaks down at larger distances because in the sum of these waves successive multihop sky waves are involved, and the sum diverges. In addition, it becomes necessary to take into account the spherical Earth. Mode theory which is the sum of eigen-modes in the Earth–ionosphere waveguide is valid in this range of distances. The wave modes have fixed vertical structures of their vertical electric field components with maximum amplitudes at the bottom and zero amplitudes at the top of the waveguide. In the case of the fundamental first mode, it is a quarter wavelength. With decreasing frequency, the eigenvalue becomes imaginary at the cutoff frequency, where the mode changes to an evanescent wave. For the first mode, this happens at