Consider the case where a branch is bent down from its original angle with the gravitational field. The new set of wavelengths no longer "match" the cell lengths and wavelengths for the large amplitude wavelengths formally present. In this case the plant produces substances that I hypothesize increase the wave velocity and they migrate and/or fall to the lower side of the branch (in other cases substances may decrease the wave velocity). A new set of standing waves form which have longer mean wavelengths because of the increased wave velocity. The internodal distances (speaking of waves here) are larger at a larger velocity which result in new cell growth. Longer cells are grown on the lower side of the branch and the branch tends to be pushed up to its original angle.

Many of you are probably unfamiliar with standing waves. For fun you can produce standing waves in your own home. Just attach a small rope to a door knob. Stretch the rope tight and wiggle your end up and down with different frequencies. With the proper up and down motion and tension in the rope one can get a pattern on the taut rope where some points in the rope are almost stationary (including the rope at your hand) and parts of the rope in between are moving up and down with large amplitude. The almost stationary points are called nodes and points in between anti nodes of the standing wave pattern. The motions in the rope are not the same as standing waves in a plant because the waves in the rope are called transverse waves while plant waves are longitudinal waves. Sound waves are an example of longitudinal waves. Standing waves in general are due to waves traveling in one direction which are reflected back on themselves with the proper phase. W-waves seem to be very standing wave prone. W-waves are almost always observed as standing waves.

Gravitropism and phototropism are responses of a plant to gravity and light respectively. Actually no complete explanation has been given in the literature for either of these phenomena. Plant physiologists have been trying to determine what is going on for the past 125 years (see Frank B. Salisbury, Gravitropism: Changing Ideas in Horticultural Reviews volume 15 pages 233-278.1993. Phototropism is also discussed). The conclusions have been contradictory.

In phototropism a plant structure bends toward the light because the cells on the lighted side either are grown shorter or cease to grow. On the shaded side cells continue to grow longer.

In the wave approach to a plant's response to gravity, specific cell dimensions and internodal spacings are associated with a set of wavelengths for each angle that the plant part makes with the gravitational field. In new growth the plant must obey the wave rules. These rules permit much latitude for the implementation of genetic requirements. If there is any change in effective cell dimensions due to compression, extension, or displacement of starch a mismatch occurs between wavelengths and dimensions. A mismatch also occurs if the angle that a plant part makes with the gravitational filed is changed. One of these mismatches is sufficient cause for the plant to initiate corrections or change growth patterns. Thus one can see that the whole plant is sensitive to gravity. Certain parts of plants, such as the root cap, are specialized to be more sensitive to gravity changes so that growth can be controlled appropriately. (see Anisotropy of Wave Velocities in Plants: Gravitropism by Wagner published in Physiological Chemistry and Physics 28:173-196 (1996). Longer wave lengths make vertical cells and vertical internodal spacings longer as is found to be the case. The velocity anisotropy (different in different directions) provides a reference for the plant. This reference is missing in space craft grown plants so cells tend to be all of one mean length or even odd shaped (see Halstead et. al. in Plants in Space. Annual Reviews of Plant Physiology volume 38 pages 317-345. 1987). See the included articles for data proving the wave related nature of a plant's response to gravity and microgravity.

See the book Waves in Dark Matter and the article by Wagner mentioned above for further details. See the last few pages of the included paper showing that plant growth angles are quantized with respect to the gravitational field.