Non-Doppler frequency shift of Newton waves

The Doppler effect has widely applications in optics, physics, acoustics and cosmology. It is verified in vacuum and linear media such as air and water but the validity in complex materials is rarely tested. Now we propose a counter-example. There is an extra term proportional to the square of relative velocity u2 for the Newton wave whose dispersion relation is ω∝k2. Actually, the frequency shift depends upon the dispersion relation and a universal expression does not exist.