Dispersion (water Waves) - Waves On A Mean Current: Doppler Shift

Waves On A Mean Current: Doppler Shift

Water waves on a mean flow (so a wave in a moving medium) experience a Doppler shift. Suppose the dispersion relation for a non-moving medium is:

with k the wavenumber. Then for a medium with mean velocity vector V, the dispersion relationship with Doppler shift becomes:

where k is the wavenumber vector, related to k as: k = |k|. The inner product kV is equal to: kV = kV cos α, with V the length of the mean velocity vector V: V = |V|. And α the angle between the wave propagation direction and the mean flow direction. For waves and current in the same direction, kV=kV.

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