Problem no. 14 "Wheat Waves"

Let us at first look what happens when a sudden blast of wind passes a field of wheat.

Sketch of Wheat Waves for a sudden blast of wind

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

Formula for the period of flexural vibration
llength
rhodensity
Across section
EYoung`s modulus
Jmoment of inertia

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".

Formula for the wavelength

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:

Formula for constant windforces

Solving the equation we get the result that the period would be constant in that case.

Furthermore we consider possible ways of coupling:

  1. Domino-effect: One ear might hit another. This might happen only if the distances between the ears are very short.
  2. At the ear: Vortexes may develop which may have an effect on the period. However the effect of vortexes is expected to be very small.
  3. The main effect: When one ear goes downward, the way to the next one is freed for the wind. This phenomenon is described by the following equation:
differential equation

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.