Page:Elementary Text-book of Physics (Anthony, 1897).djvu/168

154 $$y = 0.$$ For $$x = \tfrac{1}{4}\lambda, y = -a.$$ For $$x = \tfrac{1}{2}\lambda, y = 0.$$ For $$x = \tfrac{3}{4}\lambda, y = a.$$ For $$x = \lambda, y = 0,$$ etc. Laying off these values of $$x$$ on $$OX$$ and erecting perpendiculars equal to the corresponding values of $$y$$, we have the curve $$Obcde$$....

The above expression for y may be put in the form Hence, if any finite value be assigned to $$t$$, we shall obtain for $$y$$ the same values as were obtained above for $$t = 0$$, if we increase each of the values of $$x$$ by $$\frac{t\lambda}{T}\cdot$$ For instance, if $$t$$ equal $$\tfrac{1}{4}T$$, we have $$y = 0$$ for $$x = \tfrac{1}{4}\lambda, y = -a,$$ for $$x = \tfrac{1}{2}\lambda$$, etc., and the curve becomes the dotted line $$b'c'd'$$.... The effect of increasing $$t$$ is to displace the curve along $$OX$$ in the direction of propagation of the wave.

The formula for constructing the curve of velocities is derived in the same way as that for displacements. It is Fig. 46 shows the relation of the two curves. The upper is the curve of displacement, and the lower of velocity.

132. Composition of Wave Motions.—The composition of wave motions may be studied by the help of the curves explained above. If two systems of waves coexist in the same body, the displacement of any particle at any instant will be the algebraic sum of the