Question

Learning Goal:

To gain an understanding of constructive and destructive interference.

Consider two sinusoidal waves (1 and 2) of identical wavelength ?, 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.

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.

Answer 1

A) $y_{max}=2A$

B) $y_{max}=A$

C) $y_{max}=0$

D) $y_{max}=A$

So, the order is,

${\color{Blue} A),\, B)=D), \, C)}$