1-10 Standing Waves

Two sinusoidal waves traveling in opposite directions interfere to produce a standing wave described by the wave function

where x and y are in meters and t is in seconds.

a. What is the amplitude of one of the interfering waves?

 b. Find the second smallest value of the position x   corresponding to an antinode.

c. Find the third smallest value of the position x         corresponding to a node.

a. The amplitude  = m

a. In the case of standing waves, the resultant wave is given by

where y_m is the amplitude of the two interfering waves. Therefore, the two interfering waves have equal amplitudes of (2.5 m) /2 = 1.25 m each.

b. Antinodes are positions of maximum displacement  and given by

 

In this case the wavelength is

 

The second smallest value of x corresponding to an       antinode is for n = 1, which gives x₂ = 3.9 m.

b. The second antinode is at  x = m

c. Nodes are positions of zero displacement and given by

 

 

The third smallest value of x corresponding to a node is for n = 2, that is x₃ =  5.2 m.

b.The third node is at x = m