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CHAPTER 1

5 SECT IO

N 5

15-5 INTERFERENCE

The moment that two waves meet while traveling along the same medium is defined as wave interference. This creates a new shape of the medium, in that instant of interference, from the net effect of the two individual waves on the particles.

One dimensional (1D) waves are simpler than two dimensional (2D) and three dimensional (3D) waves due to the fact that two 1D waves with the same frequency and wavelength have a fixed phase difference.

This phase difference depends on the change in phase constant and is not subject to changes due to time or position.

CONSTRUCTIVE INTERFERENCE

Constructive interference occurs when the waves are perfectly in phase

[Image demonstrating constructive interference from phys.uconn.edu]

Constructive interference occurs when waves with amplitudes in the same direction overlap

At the point of interference, the waves overlap and the medium is displaced upward or downward by the sum of the amplitudes

Waves that have identical frequency and wavelength are produced by two point sources in phase.

When two sources are at large negative amplitude, a whole integer of the wavelength will be between each source and point.

Key Notes

MATHEMATICAL CONDITIONS

d1 = path length from source 1 √x2-(y-1)2

d2 = path length from source 2 √x2+(y+1)2

d1=mλ and m = 1,2,3….

d2=nλ and n = 1,2,3….

So,

Δd=d2-d1=(n-m)λ=pλ and p=0, ±1,±2,±3….

Therefore an integer multiple of the wavelength must be the path difference.

SPHERICAL WAVES

Oscillate in time and space Amplitude is constant over any spherical

surface (centered on source) Spatial variation (r) is the distance from the

source

s (r,t) = smrcos(kr-ωt+ϕ)

The wave front spreads over a larger area as it propagates and depends on r

As the distance from the source increases, the amplitude decreases

SPHERICAL WAVES

Image from h2physics.org

TWO WAVES IN PHASE

The cosine functions of two waves in phase differ by 2π

We neglect ϕ since the waves are in phase and achieve the equation

(kd2-ωt)-(kd1-ωt)=k(d2-d1)=n2π

Therefore d2-d1=n(2π/k)=nλ and n=0,±1,±2,±3…

[Image demonstrating when waves are in phase and out of phase from Nelson- Physics for Scientists and Engineers pg. 433]

“Two plane waves with the same wavelength travelling in slightly different directions. The lines represent the crest of the waves, the black circles indicate where the waves are perfectly in phase, and the open circles indicate where the waves are exactly out of phase.”

DESTRUCTIVE INTERFERENCEDestructive Interference occurs when the waves are out of phase and the amplitudes are in opposite directions of each other. One path is a whole integer of the wavelength and the other is ½ an integer of the wavelength

Δd=d2-d1=[(2n+1)⁄2]λ=(n+½)λ n=0,±1,±2,±3……

Image of Destructive Interference from phsy.uconn.edu

At the moment of destructive interference, the net effect of the maximum amplitude of the individual waves being in opposite direction cause the wave to ‘cancel out’, but not permanently. It is important to note that this is only momentarily, in the instant of the interference, and has no permanent effect on the particles of the wave. The positive displacement of the wave ‘pulls’ the particles of the negative displacement of the wave, and vice versa, resulting in a displacement of the medium that is smaller than both of the waves maximum displacements. The amplitudes do not need to be equal. For example, if the amplitudes are equal then the waves will cancel out, but if one wave is +2 units and the other wave is -1 unit then the resulting wave will be 1 unit.

REVIEW QUESTIONS

#1) Bobby and his father go fishing. Waves propagate in the water from the fishing lines being thrown into the water. If the white areas demonstrate crests and the dark areas demonstrate troughs, where in the image is a) constructive interference occurring and b) deconstructive interference occurring

SOLUTION (#1)

Symbolizes areas of destructive interference

Symbolizes areas of constructive interference

Question # 2 True or False

a) A sound wave is emitted from a speaker in the middle of the room. As the wave oscillates away from the source, the amplitude increases in proportion to the distance that the wave is away from the speaker.

b) When two waves with equal amplitudes in opposite directions collide, they will always cancel out.

c) If a wave with the equation s(x,t)=12cos(3t-2π+½π) is passing through the same medium as a wave with the equation s(x,t)=6cos(2t-2π+π) constructive interference will occur and create a momentarily new maximum amplitude (where x is in m and t is in s) of 18 m.

SOLUTIONS #2

a) False; for spherical waves as the spatial distance from the source increase, the amplitude decreases. There is also no information to suggest a connection between proportionality.

b) True

c) False; constructive interference cannot occur if the waves are not perfectly in phase.

QUESTION #3

If the first path length is 5λ and the second is 5.5λ, does interference occur? If so, what type and what would ‘n’ be

SOLUTION #3

Since the path lengths differ by one-half of a wavelength, the interference will be destructive.

d1=5λ and d2=5.5λ

=(5+0.5)λ

=(n+½)λ

So n = 5

SOURCES

Physics for Scientists and Engineers “An Interactive Approach”. J. Iqbal, F. Mansour, M. Milner-Bolotin, P. Williams and R. Hawkes. Nelson Education Ltd. 2014

The Physics Classroom

http://www.physicsclassroom.com/class/waves