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Dix defines an average medium in the infinite frequency/bandwidth limit, thus ending up with a paraxial approximation, the rms velocity. His exposition is based on scalar-wave traveltimes and is purely kinematic. Let this be the ``ray average'', because rays see well defined interfaces in the medium. Schoenberg&Muir [(Schoenberg and Muir,1989)] on the other hand are operating at the very low frequency (long wavelength) spectrum of observations. Within this limit, their approach lends itself to anisotropic media and even dipping interfaces. Their theory can be cast in terms of an equivalent medium scheme that preserves well defined properties. It extends beyond kinematics. Let this be the ``wave average'', because it is more closely related to basic elastic quantities, namely stress and strain. In reality, observations lie in a frequency regime that might be closer to one or the other frequency limit. Here I compare both schemes.