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Interference occurs when two waves collide with each other. There are two types: constructive and destructive. Waves that collide do not have to have the same wavelength or frequency to interfere with each other.
Constructive interference occurs when two wave crests or two wave troughs headed in opposite directions come in contact with each other. The result is the sum of their amplitudes; that is, if two waves of 1 m amplitudes each collide, for the moment at which they overlap, the bump in the rope or other medium will be 2 m high. Then the waves will continue on their separate ways, each with a 1 m amplitude again. Thus, as all interference is just the sum of the two interfering waves, constructive interference is specifically marked by an increase in amplitude.
Destructive interference occurs when one wave's crest collides with another wave's trough (again the waves are headed in opposite directions). The result is again the sum of the amplitudes, but the amplitude of the trough has a negative sign. This means that if a crest of 1 m amplitude and a trough of -1 m amplitude collide, for the moment at which they overlap, the resulting amplitude will be 1+
(-1) = 0 m. The rope would lie flat at a state of equilibrium. Then the waves will each continue on in opposite directions retaining the same amplitudes they had before the collision. Thus destructive interference results in a decrease in amplitude.
In the context of sound, this phenomenon can be seen in relation to the subject of beat frequencies. When two sound waves of different frequencies both hit your ear at the same time, they interfere with each other. Where constructive interference has occured, the sound is amplified, and is thus louder. The opposite is true of the destructive interference. The result is a pattern of loud and soft that you will hear, much like a series of beats in a constant rhythem. To find the beat frequency (how often you hear the beats), you need only to take the absolute value of the difference of the two frequencies of the waves that interfered. For example, if one wave has a frequency of 425 Hz, while another 428 Hz, and they collide, the sound produced will have a beat frequency of 3 Hz. And that is physics in action!