So, “traveling deformation waves” – what is it?
We are sure you have been watching them all your life: by looking at wave motions in the water from a thrown stone, or at creeping caterpillar with bend (i.e., wave) rolling along its body. We, the authors of this site, started research of traveling deformation waves by observing and studying wave motion of such creatures as caterpillar, earthworm, snake, and snail. In this site we will tell you how that has helped us to make some discoveries in different fields of science.
Let’s take a closer look on how these land creatures move, even though they have no legs, but only an elongated, deformable body, able to carry deformation waves.
We will prepare and place video films and computer animations of moving mechanisms of mentioned living creatures.
Let’s focus on these two typical and simple models: long, deformable strips lying on solid base. One of them can bend (“caterpillar”), while another one can stretch (“earthworm”).
If we periodically bend one end section of such a strip (i.e., create a transversal wave) and move it towards another strip’s end, the whole strip will “crawl” along the base – this is a model of caterpillar’s movement. If we stretch some end fragment of the strip (i.e., create a longitudinal wave) and move it towards another strip’s end, the strip will “crawl” as well – a model of earthworm’s movement.
But do not rush to conclusion that those “well-known” phenomena are simple and trivial. This is what we thought when began studying those methods of locomotion from positions of mechanics. Let’s ask ourselves a question: “Why a caterpillar crawls in the same direction as moving deformation wave on its body, but an earthworm moves in the direction opposite to deformation wave’s motion?” Finding answers led us to understanding that there exist only two kinds of traveling deformation waves:
• wave of surplus of mass (example: bend (transversal) wave on caterpillar’s body);
• wave of shortage of mass (example: elongation (longitudinal) wave on earthworm’s body);
Later we will show that all other traveling deformation waves concerned in this site (traveling deformation waves in all media – solid, liquid and gas, and waves of different scales – up to global atmospheric and geospheric waves) are waves of these two kinds – surplus of mass and shortage of mass.
Let’s go back to our “basic biomechanical examples” of traveling deformation waves we started with – methods of transportation of caterpillar and earthworm. From positions of theoretical mechanics, those mechanisms, invented by Nature itself, have many characteristics that are not achievable in man-made devices.
First, this is an extremely economical method of transportation. Because of having massive body and small power for its move, those creatures managed to move themselves “by parts”. As you can see from our models, at every given moment only the wave part of body is moving, while the rest is motionless (“resting”).
Second, there is no sliding friction. This fact seems improbable. How can any legless creature, crawling on solid surface, avoid friction? However, strict analysis shows that:
• for caterpillar, moving part of its body is raised from the ground, thus avoiding friction;
• for earthworm, moving (i.e. stretched) part of its body is raised from the ground as well, because longitudinal stretching of a deformable body leads to its narrowing, i.e., reduction of cross-section (Poisson’s law).
We call such body’s wave movement method, when some of its particles are moving, while others are resting, “discrete-wave” movement.
Let’s define now a concept of mass content for transversal and longitudinal deformation waves. Considering that wave is moving along x-axis, let’s assume a wave length l as a length of the interval along x-axis containing the wave. If mass of deformable body fragment in the wave interval l is , and mass of regular body fragment (not in wave) of the same length l is m, then the mass contents of the wave is
= – m
Another words, mass content of the wave is a difference of mass of body in the wave fragment and mass of non-wave fragment of the same length.
There are two other important characteristics of waves we will need in chapters to follow.
• Wave mass transfer equals to product of mass content of wave and its traversed path L:
• Momentum (impulse) of traveling deformation wave equals to product of mass content of wave and its velocity .