[1] They always carry energy eastward, but their 'crests' and 'troughs' may propagate westward if their periods are long enough.
The eastward speed of propagation of these waves can be derived for an inviscid slowly moving layer of fluid of uniform depth H.[2] Because the Coriolis parameter (f = 2Ω sin(θ) where Ω is the angular velocity of the earth, 7.2921 × 10−5 rad/s, and θ is latitude) vanishes at 0 degrees latitude (equator), the “equatorial beta plane” approximation must be made.
[3] With the inclusion of this approximation, the primitive equations become (neglecting friction): These three equations can be separated and solved using solutions in the form of zonally propagating waves, which are analogous to exponential solutions with a dependence on x and t and the inclusion of structure functions that vary in the y-direction: Once the frequency relation is formulated in terms of ω, the angular frequency, the problem can be solved with three distinct solutions.
[1] On a typical "m,k" dispersion diagram, the group velocity (energy) would be directed at right angles to the n = 0 (mixed Rossby-gravity waves) and n = 1 (gravity or Rossby waves) curves and would increase in the direction of increasing angular frequency.
[1] Similarly, westward-propagating mixed waves were also found in the Atlantic Ocean by Weisberg et al. (1979) with periods of 31 days, horizontal wavelengths of 1200 km, vertical wavelengths of 1 km, and downward group velocity.