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Effect of combination of two waves of same frequency travelling ino opposite direction |
| Answer» Two waves (with the same amplitude, frequency, and wavelength) are travelling in opposite directions. Using the principle of superposition, the resulting wave amplitude may be written as:y\xa0(\xa0x\xa0,\xa0t\xa0)\xa0=\xa0y\xa0m\xa0sin\xa0(\xa0kx\xa0-\xa0ωt\xa0)\xa0+\xa0y\xa0m\xa0sin\xa0(\xa0kx\xa0+\xa0ωt\xa0)\xa0=\xa02\xa0y\xa0m\xa0sin\xa0(\xa0kx\xa0)\xa0cos\xa0(\xa0ωt\xa0)This wave is no longer a travelling wave because the position and time dependence have been separated. The the wave amplitude as a function of position is\xa02ymsin(kx). This amplitude does not travel, but stands still and oscillates up and down according to cos(ω t). Characteristic of standing waves are locations with maximum displacement (antinodes) and locations with zero displacement (nodes)..If two sinusoidal waves having the same frequency (and wavelength) and the same amplitude are travelling in opposite directions in the same medium then, using superposition, the net displacement of the medium is the sum of the two waves.when the two waves are 180° out-of-phase with each other they cancel, and when they are exactly in-phase with each other they add together. As the two waves pass through each other, the net result alternates between zero and some maximum amplitude. However, this pattern simply oscillates; it does not travel to the right or the left, and thus it is called a "standing wave". | |