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Interactive Java Tutorials

Complex Waveforms and Beat Frequencies in Superposed Waves

In general, the process of describing interference through the superposition of sine waves generates simple waveforms that can be adequately represented by a resultant sine curve in a plot of amplitude, wavelength, and relative phase displacement. If the recombined waves have appreciably different frequencies, the resulting waveform is often complex, yielding a contour that is no longer a sine function with a simple, single harmonic. This interactive tutorial explores the complex waveforms and beat frequencies generated by the superposition of two light waves propagating in the same direction with different relative frequencies, amplitudes, and phases.

The tutorial initializes with two parallel light waves, labeled Wave 1 and Wave 2, assumed to be propagating parallel to each other from left to right in the two upper frames of the tutorial window. The resultant wave arising from the summation of the two waves by interference is presented as the Wave Sum in the bottom-most frame of the tutorial window. In order to operate the tutorial, use the Wavelength, Phase, and Amplitude sliders to vary these parameters for each of the input waves, and observe how the resultant wave is affected.

In cases where the superposed waves have closely matched wavelengths and amplitudes (for example, 400 and 430 nanometers), the resultant waveform displays several harmonics, including the classical beat frequencies so commonly observed in sound waves. This effect was first demonstrated with light waves in 1955, prior to the invention of the laser, and is quite useful in a number of applications, including the Doppler effect that accounts for the frequency shift when light is reflected from a moving surface. A precise measure of the speed of an object can be derived by scattering light from the target, and then beating the original and reflected waves.

Adding together two waves that have significantly different wavelengths (400 and 650, for example) produces complex waveforms that deviate significantly from a simple sine function. If the resultant wave is composed of visible light, the human eye experiences the sensation of a mixture of two colors, which are the same regardless of the phase difference. The examples presented in the tutorial are monochromatic in nature, but white light can be considered as an extreme case of superposition between waves, where a large number of simple waveforms have wavelengths that differ by only a few nanometers. In this case, Fourier analysis is necessary to describe the complex interplay that occurs between waveforms in such a composite mixture of wavelengths.

Contributing Authors

Robert T. Sutter and Michael W. Davidson - National High Magnetic Field Laboratory, 1800 East Paul Dirac Dr., The Florida State University, Tallahassee, Florida, 32310.



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