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The figure shows the circular wave fronts emitted by two

sources.

A) Give the distances from points P, Q and R to the point 1 in

indicated order as multiples of lambda.

B) Give the distances from points P, Q and R to the point 2 in

indicated order as multiples of lambda.

C) Indicate whether the interference at the point P is

constructive or destructive.

D) Indicate whether the interference at the point Q is

constructive or destructive.

E) Indicate whether the interference at the point R is

constructive or destructive.

Interference of sound waves is the solution to this problem.

In terms of wavelength, calculate the distances between points P, Q and R and point 1 or 2. Later, determine if the interference at points Q and P is constructive or destructive. Finally, determine whether interference at point Q is constructive or destructive.

Wave interference– This is when two waves are superimposed as they travel along the same medium.Constructive interfer: When the amplitudes of sound waves are superimposed so that they add up, then this is called constructive interference.Destructive Interference:When the amplitudes of sound waves are superimposed in such an way that they are out of phase, then the interference is called constructive interference.(A)

The distance between two circular waves fronts is closest to,

[katex]\beta = \frac{{\lambda D}}{d}[/katex]

Here, [katex]\beta [/katex]

The distance between point P and the source 1 is proportional to the wavelength.

[katex]{r_{1P}} = 3\lambda [/katex]

Distance between point

Qand point 1:[katex]{r_{1Q}} = 3.5\lambda [/katex]

Distance between point

Rand point 1:[katex]{r_{1R}} = 2.5\lambda [/katex]

(B)

The distance between two circular waves fronts is closest to,

[katex]\beta = \frac{{\lambda D}}{d}[/katex]

Here, [katex]\beta [/katex]

The distance between point P and the source 2 is proportional to the wavelength.

[katex]{r_{2P}} = 4\lambda [/katex]

Distance Q from source 2 is

[katex]{r_2}_Q = 2\lambda [/katex]

The distance between point R and the source 2 is also,

[katex]{r_{2R}} = 3.5\lambda [/katex]

(C)

If the path difference between wave fronts from points 1 and 2 equals an integer multiple of wavelength, the waves are considered constructively interfered. However, if it equals the odd multiple, the waves can be said to be destructively interfering.

The difference in the path between the waves that reach point P is called,

[katex]{\rm{Path difference = }}{r_{{\rm{2p}}}} – {r_{1p}}[/katex]

Here, [katex]{r_{{\rm{2p}}}}[/katex]

Substitute $equaTag12

[katex]\begin{array}{c}\\{\rm{Path difference = }}\left| {{r_{{\rm{2p}}}} – {r_{1p}}} \right|\\\\ = 4\lambda – 3\lambda \\\\ = \left( 1 \right)\lambda \\\end{array}[/katex]

The path difference is an integer multiple of the wavelength so the interference is constructive.

(D)

The difference in the path between waves that reach Q at point Q is called,

[katex]{\rm{Path difference = }}{r_{{\rm{2Q}}}} – {r_{1Q}}[/katex]

Here, [katex]{r_{{\rm{2Q}}}}[/katex]

Substitute $equaTag16

[katex]\begin{array}{c}\\{\rm{Path difference = }}\left| {{r_{{\rm{2Q}}}} – {r_{1Q}}} \right|\\\\ = \left| {3.5\lambda – 2\lambda } \right|\\\\ = \left( {1.5} \right)\lambda \\\end{array}[/katex]

The path difference equals half the wavelength of interference, so the interference is destructive.

(E)

The difference in the path between the waves that reach point R is

[katex]{\rm{Path difference = }}{r_{{\rm{2R}}}} – {r_{1R}}[/katex]

Here, [katex]{r_{{\rm{2R}}}}[/katex]

Substitute $equaTag20

[katex]\begin{array}{c}\\{\rm{Path difference = }}\left| {{r_{{\rm{2R}}}} – {r_{1R}}} \right|\\\\ = \left| {2.5\lambda – 3.5\lambda } \right|\\\\ = \left( 1 \right)\lambda \\\end{array}[/katex]

The path difference equals an integer multiple of wavelength so the interference is constructive.

Part A – Ans

Distances from points P, Q and R to point 1 are [katex]3\lambda ,{\rm{ 3}}{\rm{.5}}\lambda {\rm{, and 2}}{\rm{.5}}\lambda [/katex]Part A

[katex]4\lambda ,{\rm{ 2}}\lambda {\rm{, 3}}{\rm{.5}}\lambda [/katex]Part B. The distance between points P, Q, and R are the following:

The interference at point P is constructive.

Part D

Interference at Q can be destructive.

Part E

The interference at point R is constructive.