In this case, a sine trough with an amplitude of -1 unit (negative means a downward displacement) interferes with a sine trough with a displacement of -1 unit. These two troughs are drawn in red and blue. The resulting shape of the medium is a sine trough with a maximum displacement of -2 units.
Destructive interference is a type of interference which occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction. For instance, when a sine crest with an amplitude of +1 unit meets a sine trough with an amplitude of -1 unit, destructive interference occurs. This is depicted in the diagram below.
In the situation in the diagram above, the interfering pulses have the same maximum displacement but in opposite directions. The result is that the two pulses completely destroy each other when they are completely overlapped. At the instant of complete overlap, there is no resulting disturbance in the medium. This "destruction" is not a permanent condition. In fact, to say that the two waves destroy each other can be partially misleading. When it is said that the two pulses "destroy each other," what is meant is that when overlapped, the effect of one of the pulses on the displacement of a given particle of the medium is "destroyed" or canceled by the effect of the other pulse. Recall from that waves transport energy through a medium by means of each individual particle pulling upon its nearest neighbor. When two pulses with opposite displacements (i.e., a crest and trough) meet at a given location, the upward pull of the crest is balanced (canceled or "destroyed") by the downward pull of the trough. Once the two pulses pass through each other, there is still a crest and a trough heading in the same direction which they were heading before interference. Destructive interference leads to only a momentary condition in which the medium's displacement is less than the displacement of the largest-amplitude wave.
The two interfering waves do not need to have equal amplitudes in opposite directions for destructive interference to occur. For example, a crest with an amplitude of +1 unit could meet a trough with an amplitude of -2 units; the resulting displacement of the medium during complete overlap is -1 unit.
This is still destructive interference since the two interfering waves have opposite displacement. In this case, the destructive nature of the interference does not lead to complete cancellation.
Interestingly, the meeting of two waves along a medium does not alter the individual waves or even deviate them from their path. This only becomes an astounding behavior when it is compared to what happens when two billiard balls meet or two football players meet. Billiard balls might crash and bounce off each other and football players might crash and come to a stop. Yet waves meet, produce a net resulting shape of the medium, and then continue on doing what they were doing before the interference.
The task of determining the shape of the resultant demands that the principle of superposition is applied. The principle of superposition is sometimes stated as follows:
In the cases above, the summing the individual displacements for locations of complete overlap was mad out to be an easy task - as easy as simple arithmetic:
In actuality, the task of determining the complete shape of the entire medium during interference demands that the principle of superposition be applied for every point (or nearly every point) along the medium. As an example of the complexity of this task, consider the two interfering waves at the right. A snapshot of the shape each of the individual waves at a particular instant in time is shown. To determine the precise shape of the medium at this given instant in time, the principle of superposition must be applied to several locations along the medium. A short-cut involves measuring the displacement from equilibrium at a few strategic locations. Thus, approximately 20 locations have been picked and labeled as A, B, C, D, etc. The actual displacement of each individual wave can be counted by measuring from the equilibrium position up to the particular wave. At position A, there is no displacement for either individual wave; thus, the resulting displacement of the medium will be 0 units. At position B, the smaller wave has a displacement of approximately 1.4 units; the larger wave has a displacement of approximately 2 units; thus, the resulting displacement of the medium will be approximately 3.4 units. At position C, the smaller wave has a displacement of approximately 2 units; the larger wave has a displacement of approximately 4 units; thus, the resulting displacement of the medium will be approximately 6 units. At position D, the smaller wave has a displacement of approximately 1.4 units; the larger wave has a displacement of approximately 2 units; thus, the resulting displacement of the medium will be approximately 3.4 units. This process can be repeated for every position. When finished, a dot (done in green below) can be marked on the "graph" to note the displacement of the medium at each given location. The actual shape of the medium can then be sketched by estimating the position between the various marked points and sketching the wave. This is shown as the green line in the diagram below.
Check Your Understanding
1. Several positions along the medium are labeled with a letter. Categorize each labeled position along the medium as being a position where either constructive or destructive interference occurs.
