The **destructive interference**, in physics, occurs when two independent waves that combine in the same region of space are out of phase. Then the crests of one of the waves meet the troughs of the other and the result is a wave with zero amplitude.

Several waves pass without problem through the same point in space and then each one continues on its way without being affected, like the waves in the water in the following figure:

Suppose two waves of equal amplitude A and frequency ω, which we will call y1 and y2, which can be described mathematically by the equations:

y1= A sin (kx-ωt)

y2 = A sin (kx-ωt + φ)

The second wave y2 has an offset φ with respect to the first. When combined, since waves can easily overlap, they give rise to a resulting wave called yR:

yR = y1 + y2 = A sin (kx-ωt) + A sin (kx-ωt + φ)

Using the trigonometric identity:

sin α + sin β = 2 sin (α+ β)/2 . cos(α – β)/2

The equation for yR becomes:

yR = [2A cos (φ/2)] sin(kx – ωt + φ/2)

Now this new wave has a resulting amplitude AR = 2A cos (φ/2), which depends on the phase difference. When this phase difference takes the values +π or –π, the resulting amplitude is:

AR = 2A cos (± π/2) = 0

Since cos (± π/2) = 0. It is precisely then that destructive interference between waves occurs. In general, if the cosine argument is of the form ± kπ/2 with k odd, the amplitude AR is 0.

[toc]

**Examples of destructive interference**

As we have seen, when two or more waves pass through a point at the same time, they overlap, giving rise to a resulting wave whose amplitude depends on the phase difference between the participants.

The resulting wave has the same frequency and wave number as the original waves. In the following animation, two waves in blue and green colors are superimposed. The resulting wave is in red.

The amplitude increases when the interference is constructive, but cancels out when it is destructive.

Waves that have the same amplitude and frequency are called *coherent waves*, as long as they maintain the same phase difference φ between them. An example of a coherent wave is laser light.

**Condition for destructive interference**

When the blue and green waves are 180º out of phase at a given point (see figure 2), it means that while they are moving, they have *phase differences* φ of π radians, 3π radians, 5π radians, and so on.

In this way, by dividing the resulting amplitude argument by 2, the result is (π/2) radians, (3π/2) radians… And the cosine of such angles is always 0. Therefore the interference is destructive and the amplitude it becomes 0.

**destructive interference of waves in water**

Suppose two coherent waves start out in phase with each other. Such waves can be those that propagate through the water thanks to two vibrating bars. If the two waves travel to the same point P, covering different distances, the phase difference is proportional to the path difference.

Since a wavelength λ is equivalent to a difference of 2π radians, then it is true that:

│d1 – d2│/ λ = phase difference / 2π radians

Phase difference = 2π x│d1 – d2│/ λ

If the path difference is an odd number of half-wavelengths, ie: λ/2, 3λ/2, 5λ/2 and so on, then the interference is destructive.

But if the path difference is an even number of wavelengths, the interference is constructive and the amplitudes add up at point P.

**destructive interference of light waves**

Light waves can also interfere with each other, as Thomas Young revealed in 1801 through his celebrated double-slit experiment.

Young passed light through a slit made in an opaque screen, which, according to Huygens’s principle, generates two secondary light sources. These sources followed their way through a second opaque screen with two slits and the resulting light was projected onto a wall.

The diagram can be seen in the following image:

Young observed a distinctive pattern of alternating light and dark lines. When light sources interfere destructively, the lines are dark, but if they interfere constructively, the lines are light.

Another interesting example of interference is soap bubbles. These are very thin films, in which interference occurs because light is reflected and refracted by the surfaces bounding the soap film, both above and below.

Since the thickness of the film is comparable to the wavelength, the light behaves in the same way as it does when it passes through the two Young slits. The result is a pattern of colors if the incident light is white.

This is because white light is not monochromatic, but contains all the wavelengths (frequencies) of the visible spectrum. And each wavelength is seen as a different color.

**solved exercise**

Two identical speakers driven by the same oscillator are 3 meters apart, and a listener is 6 meters from the midpoint of separation between the speakers, at point O.

It is then translated to point P, at a perpendicular distance of 0.350 from point O, as shown in the figure. There he stops hearing the sound for the first time. What is the wavelength at which the oscillator emits?

**Solution**

The amplitude of the resulting wave is 0, therefore the interference is destructive. You have to:

Phase difference = 2π x│r1 – r2│/ λ

By the Pythagorean theorem applied to the shaded triangles in the figure:

r1 = √1.152 + 82 m = 8.08 m; r2 = √1.852 + 82 m = 8.21 m

│r1 – r2│= │8.08 – 8.21 │m = 0.13 m

The minima occur at λ/2, 3λ/2, 5λ/2… The first one corresponds to λ/2, then, from the formula for the phase difference we have:

λ = 2π x│r1 – r2│/ Phase difference

But the phase difference between the waves must be π, so that the amplitude AR = 2A cos (φ/2) is zero, then:

λ = 2π x│r1 – r2│/ π = 2 x 0.13 m = 0.26 m

**References**

Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 7. Waves and Quantum Physics. Edited by Douglas Figueroa (USB).

Physicalab. wave interference. Retrieved from: physicalab.com. Giambattista, A. 2010. Physics. 2nd. Ed. McGraw Hill. Serway, R. Physics for Science and Engineering. Volume 1.7ma. Ed. Cengage Learning.

Wikipedia. Interference in thin sheets. Source: en.wikipedia.org.