In a homogeneous elastic half space, plane harmonic waves give rise to standing waves only when on reflection there is no conversion from one wave-type to another. The existence of standing waves in a horizontally-layered half space is established for vertically propagating plane harmonic P, SV or SH waves. Expressions are derived for the particle displacements at the free surface and at any given depth. The layered system acts as a complicated filter suppressing certain periods while amplifying others. The results obtained may be helpful in an understanding of the ‘ground factor’, ambient seismic body-wave noise, and the vibration problem of a structure due to earthquake motion.