FIG. 12.
But there is one form of motion of a particle in a wave which is looked upon as the simplest and fundamental form. It is that form which is executed by the bob of a pendulum, the balance wheel of a watch, the prong of a tuning-fork, and most other vibrations where the controlling force is provided by a spring or by some other elastic solid.
It is called "Simple Harmonic Motion" or "Simple Periodic Motion," and the essential feature of it is that the force restoring the displaced particle to its undisturbed position is proportional to its displacement from the undisturbed position. A wave in which all the particles execute simple harmonic motion has the form in Fig. 10 or Fig. 11, which is therefore looked upon as the fundamental wave form or simple wave form.
Simple waves will vary only in amplitude, wave-length, and frequency, and the energy in the wave will depend upon these quantities.
Energy in a Simple Wave.—If the velocity is the same for all wave-lengths, then the frequency will evidently be inversely proportional to the wave-length and the energy will depend upon the amplitude and the wave-length. The kinetic energy of any moving body, i.e. the energy due to its motion, is proportional to the square of its velocity, and we may apply this to the motion of the particles in a wave and to show how the energy depends upon the amplitude and wave-length.
Since the distance travelled by a particle in a single period of the wave will be equal to four times the amplitude, the velocity at any point in the wave must be proportional to the amplitude and therefore the kinetic energy is proportional to the square of the amplitude.
With the same amplitude but with different wave-lengths, we see that the time in which the oscillation is completed is proportional to the wave-length and that the velocity is therefore inversely proportional to the wave-length. The kinetic energy is therefore inversely proportional to the square of the wave-length.