Page:Elementary Text-book of Physics (Anthony, 1897).djvu/166

153 the principle inyolved is known as Huygens' principle. Let $$a$$ (Fig. 44) be a centre at which sound originates. At the end of a certain time it will have reached the surface $$mn$$. From the preceding discussion it is evident that each particle of the surface $$mn$$ has a vibratory motion similar to that at $$a$$. Any one of those particles would, if vibrating alone, be, like $$a$$, the centre of a system of spherical waves, and each of them must, therefore, be considered as a wave centre from which spherical waves proceed. Suppose such a wave to proceed from each one of them for the short distance $$cd$$. Since the number of the elementary spherical waves is very great, it is plain that they will coalesce to form the surface $$m'n'$$ which determines a new position of the wave surface. In some cases the existence of these elementary waves need not be considered, but there are many phenomena of wave motion which can only be studied by recognizing the fact that propagation always takes place as above described.

131. Graphic Representation of Wave Motion.—In order to study the movements of a body in which a wave motion exists, especially when two or more systems of waves exist in the same body, it is convenient to represent the movement by a sinusoidal curve. Suppose the layer $$a$$ (Fig. 45) to move with a simple harmonic motion of which the amplitude is $$a$$ and the period $$T$$, and let time be reckoned from the instant that the particles pass the position of equilibrium in a positive direction. A sinusoidal curve may be constructed to represent either the displacements of the various layers from their positions of equilibrium, or the velocities with which they are severally moving at a given time.

To construct the first curve let the several points along $$OX$$ (Fig. 45) represent points of the body through which the wave is moving. Let $$Oy = a$$ be the amplitude of vibration of each particle. The displacement of the particle at $$O$$ at any instant $$t$$ after