Traveling Waves and Superposition -- Exercise Set 1

by Loren Winters

 Please wait for the Java applet to load completely before clicking on anything. Once loaded, you'll see three wave forms. Make them travel my clicking on the Forward tab at the bottom of the applet. Try the other tabs, too. Note that Step lets you advance the wave in time increments of 0.1. (The units of time are arbitrary.) The Reset tab restores the waves to their original settings.We'll assume that the top two windows display waves that are traveling simultaneously in the same medium (same string, spring, air column, etc). Note that the two waves are traveling at the same speed in opposite directions and that they have the same amplitude and wavelength. It is, of course, possible that the two waves could have different amplitudes and wavelengths. However, they must have the same speed. Why? (See Problem 1.) applet used with permission of Wolfgang Christian (see Help tab) The wave in the bottom window is the superposition (algebraic sum) of the two component waves in the upper windows. If the component waves were traveling on a spring, for example, the superposition is what you would actually see. You wouldn't see the component waves. The superposition shown here is a special type called a standing wave. The wave appears to be oscillating vertically but not horizontally. There are some points, called nodes, where there is no displacement, and other points, called antinodes, where the displacement is greatest. In order to see how the superposition is formed from the components, click Stop and then Reset. Note that the component waves have the same phase at this point. (Their maxima and minima occur at the same place.) The superposition has its maximum amplitude in this situation, and the component waves are said to interfere constructively. Now step through the motion. Note that the more the component waves differ in phase, the less is the amplitude of the superposition. When the components are one-half cycle out of phase, the superposition has no amplitude. The component waves are said to interfere destructively. Write your answers to the numbered problems below. Why must the two waves have the same speed? (Think in terms of what influences wave speed in the medium.) Stop the top wave and measure its wavelength in units of divisions along the horizontal axis. Sketch the wave, showing the two points between which you measured the wavelength. Now measure the period of the top wave in time units. Describe your method for doing this. Calculate the speed of the top wave. Show your work. Go to Exercise Set 2