| a + b |
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| 2 |
| a - b |
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| 2 |
When two waves of the same frequency and amplitude are traveling in opposite directions in a medium, the result is a standing wave - a wave that does not travel one way or the other.
If the waves are identical except from their direction of propagation, they can be described by the equations:
y1 = A sin(kx - ωt)
y2 = A sin(kx + ωt)
The resultant wave is the addition of these. Use the trig. identity:
| sin (a) + sin (b) | = 2 sin | ( |
|
) | cos | ( |
|
) |
the resultant wave can be written as:
y = 2A sin(kx) cos(ωt)
This is quite different in form from the equation for a traveling wave, which involved some function of (kx - ωt). The standing wave equation has the spatial part (kx) separated from the time part. It predicts that the string is totally flat at certain points in time, and it also predicts that there are certain positions where the amplitude is always zero - these points are called nodes. There are other points halfway between the nodes where the amplitude is maximum - these are the anti-nodes.