Question

Learning Goal:

To gain an understanding of constructive and destructive interference.

Consider two sinusoidal waves (1 and 2) of identical wavelength λ, period T, and maximum amplitude A. A snapshot of one of these waves taken at a certain time is displayed in the figure below. (Figure 1) Let y1(x,t) and y2(x,t) represent the displacement of each wave at position x at time t. If these waves were to be in the same location (x) at the same time, they would interfere with one another. This would result in a single wave with a displacement y(x,t)given by

y(x,t)=y1(x,t)+y2(x,t).

This equation states that at time t the displacement y(x,t) of the resulting wave at position x is the algebraic sum of the displacements of the waves 1 and 2 at position x at time t. When the maximum displacement of the resulting wave is less than the amplitude of the original waves, that is, when ymax<A, the waves are said to interfere destructively because the result is smaller than either of the individual waves. Similarly, when ymax>A, the waves are said to interfere constructively because the resulting wave is larger than either of the individual waves. Notice that 0≤ymax≤2A.

Figure 1

Figure 2

figure3

figure 4

figure 5

Part A

To further explore what this equation means, consider four sets of identical waves that move in the +x direction. A photo is taken of each wave at time t and is displayed in the figures below.

Rank these sets of waves on the basis of the maximum amplitude of the wave that results from the interference of the two waves in each set.

Rank from largest amplitude on the left to smallest amplitude on the right. To rank items as equivalent, overlap them.

Part B

Now consider a wave which is paired with seven other waves into seven pairs. The two waves in each pairing are identical, except that one of them is shifted relative to the other in the pair by the distance shown:

A. −(1/2)λ

B. 2λ

C. −5λ

D. (3/2)λ

E. 0

F. (17/2)λ

G. (6/2)λ

Identify which of the seven pairs will interfere constructively and which will interfere destructively. Each letter represents a pair of waves.

Enter the letters of the pairs that correspond to constructive interference in alphabetical order and the letters of the pairs that correspond to pairs that interfere destructively in alphabetical order separated by a comma. For example if pairs A, B and D interfere constructively and pairs C and F interfere destructively enter ABD,CF.

Consider what water waves look like when you throw a rock into a lake. These waves start at the point where the rock entered the water and travel out in all directions. When viewed from above, these waves can be drawn as shown(Figure 2) , where the solid lines represent wave peaks and troughs are located halfway between adjacent peaks.

Part C

Now look at the waves emitted from two identical sources (e.g., two identical rocks that fall into a lake at the same time). The sources emit identical waves at the exact same time.(Figure 3)

Identify whether the waves interfere constructively or destructively at each point A to D.

For points A to D enter either c for constructive or d for destructive interference. For example if constructive interference occurs at points A, C and D, and destructive interference occurs at B, enter cdcc.

Each wave travels a distance d1 or d2 from its source to reach Point B. (Figure 4) Since the distance between consecutive peaks is equal to λ, from the picture you can see that Point B is 2λ away from Source 1 and 3λ away from Source 2. The path-length difference, ΔdB, is the difference in the distance each wave travels to reach Point B:

ΔdB=d1−d2=2λ−3λ=−1λ.

Part D

What are the path-length differences at Points A, C, and D (respectively, ΔdA, ΔdC, and ΔdD)? (Figure 3)

Enter your answers numerically in terms of λ separated by commas. For example, if the path-length differences at Points A, C, and D are 4λ, λ/2, and λ, respectively, enter4,.5,1.

Part E

What are the path-length differences at Points L to P? (Figure 5)

Enter your answers numerically in terms of λ separated by commas. For example, if the path-length differences at Points L, M, N, O, and P are 5λ, 2λ, 32λ, λ, and 6λ, respectively, enter 5,2,1.5,1,6.

Answer 1

Part-A) The correct ranking cannot be determined

Part-B) BCEG,ADF

Part-C) ccdd

Part-D) ?dA, ?dC, ?dD = 0,1.5,0.5 ?, ?, ?

Part-E) ?dL, ?dM, ?dN, ?dO, ?dP = 1,1,1,1,1 ?, ?, ?, ?, ?