Let us at first look what happens when a sudden blast of wind passes a field of wheat.
What we can see from this sketch, the wavelength is not dependent on x, the distance between two ears. Moreover, we can see that this isn't a harmonic wave. The pattern is clearly inharmonic.
Then we found out, that wheat vibration can be considered as flexural vibration. The known formula for the period of flexural vibration is
So knowing the period we can calculate the wavelength. By doing so we found out that the wavelength is dependent on vp (the so called "phase velocity"). This phenomenon, that the wavelength is dependent on the phase velocity is called "dispersion".
Then we wanted to know what would happen if there was not only a sudden blast of wind but a constant wind-force.
That case leads us to a differential equation:
Solving the equation we get the result that the period would be constant in that case.
Furthermore we consider possible ways of coupling:
We can solve this equation for different ratios d/v and then plot the result for the force on the ears and the elongation of the ears versus time and the actual wave pattern as you can see in the following diagrams.