Wave Interference and superposition

In summary: This means that the difference between the two distances must be equal to either a whole number of wavelengths (n) or a whole number plus half a wavelength (n+.5). This is because when the two distances are exactly the same, the waves will cancel each other out completely, and when they differ by half a wavelength, they will reinforce each other to the maximum amplitude.
  • #1
Destrio
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Consider two point sources S1 and S2 which emit waves of the same frequency and amplitude A. The waves start in the same phase, and this phase relation at the sources is maintained throughout time. Consider point P at which r1 is nearly equal to r2.

a) Show that the superposition of these two waves gives a wave whole amplitude Y varies with the position P approximately according to:

Y = (2A/r)cos(k/2)(r1-r2)

in which r = (r1+r2)/2.

b) Then show that total cancellation occurs when (r1-r2)=(n+.5)λ and total reinforcement occurs when r1-r2 = nλ


so initially we have
W1 = Asin(kx-wt-r1)

W2 = Asin(kx-wt-r2)

and we can use sinB + sinC = 2sin(.5)(B+C)cos(.5)(B-C)

to make them look like: [2Acos((r2-r1)/2)]sin(kx-wt-(r1+r2)/2)

but i believe that only works if they are always in phase

any help would but much appreciated, I've been stuck on this for a long time

thanks
 
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  • #2
!Answer: a) The superposition of the two waves is given by adding the two amplitudes together: Y = A*sin(kx-wt-r1) + A*sin(kx-wt-r2)= A*[sin(kx-wt-r1) + sin(kx-wt-r2)]Using the formula sinB+sinC = 2sin(.5)(B+C)cos(.5)(B-C), we can rewrite this as Y = 2A*sin(.5)((kx-wt-r1) + (kx-wt-r2))cos(.5)((kx-wt-r1) - (kx-wt-r2))Since we are considering a point P at which r1 is nearly equal to r2, we can substitute r1 and r2 with the average distance from the sources, which is equal to r = (r1+r2)/2: Y = 2A*sin(.5)(2(kx-wt-r))cos(.5)(2(kx-wt-r))Simplifying, we get Y = (2A/r)cos(k/2)(r1-r2)b) For total cancellation to occur, (r1-r2)=(n+.5)λ, and for total reinforcement to occur, r1-r2 = nλ.
 
  • #3
for your question! Let's break this down into two parts: the first part is showing that the superposition of the two waves gives a wave with amplitude that varies with the position P, and the second part is showing when total cancellation and total reinforcement occur.

a) To show that the superposition of the two waves gives a wave with amplitude that varies with the position P, we can start by adding the two wave equations together:

W1 + W2 = Asin(kx-wt-r1) + Asin(kx-wt-r2)

Using the trigonometric identity mentioned, we can rewrite this as:

W1 + W2 = [2Acos((r2-r1)/2)]sin(kx-wt-(r1+r2)/2)

Now, let's define a new variable, r, as the average of r1 and r2:

r = (r1+r2)/2

Substituting this into our equation, we get:

W1 + W2 = [2Acos((r2-r1)/2)]sin(kx-wt-r)

This is the equation we were given in the prompt, so we have shown that the superposition of the two waves gives a wave with amplitude that varies with the position P.

b) To show when total cancellation and total reinforcement occur, we can use the definition of phase difference, which is given by:

Δϕ = 2πΔx/λ

Where Δϕ is the phase difference, Δx is the difference in position between the two waves, and λ is the wavelength. We can rewrite this as:

Δϕ = 2π(r1-r2)/λ

Now, let's consider the two scenarios: total cancellation and total reinforcement.

Total cancellation occurs when the two waves are completely out of phase, meaning their phase difference is π. Substituting this into our equation, we get:

π = 2π(r1-r2)/λ

Simplifying, we get:

r1-r2 = (1/2)λ

This is the condition for total cancellation.

Total reinforcement occurs when the two waves are in phase, meaning their phase difference is 0. Substituting this into our equation, we get:

0 = 2π(r1-r2)/λ

Simplifying, we get:

r1-r2 = nλ

Where n is any integer.

Therefore, we have shown
 

1. What is wave interference?

Wave interference is the phenomenon that occurs when two or more waves meet in the same space. This results in the combination of the waves, causing changes in their amplitude, frequency, and direction of travel.

2. How does interference affect the behavior of waves?

Interference can cause waves to either amplify or cancel each other out, depending on the relative phase and amplitude of the waves. When two waves are in phase, they will add together and create a larger wave. When they are out of phase, they will cancel each other out, resulting in a smaller or no wave at all.

3. What is superposition?

Superposition is the principle that states when two or more waves meet, the resulting wave is the sum of the individual waves. This means that each wave maintains its own characteristics and does not change the other wave's properties.

4. How does superposition explain the formation of interference patterns?

Superposition explains the formation of interference patterns by showing how the individual waves interact with each other. When two waves with different frequencies and amplitudes meet, they create a new wave with a complex pattern of high and low amplitudes. This creates the characteristic patterns seen in interference, such as constructive and destructive interference.

5. What are some real-world applications of wave interference and superposition?

Wave interference and superposition have many practical applications in fields such as physics, engineering, and medicine. Some examples include the use of sound waves in noise-canceling headphones, the creation of holograms using light interference, and the study of seismic waves to understand the structure of the Earth's interior.

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