Interference of waves in a pond

1. Michelle Wu, Feb/ 28, 15
LO 5: Interference of Waves in a Pond
Question:
A wave is generated after dropping a
pebble into a pond. The pebble is dropped
at x=0, and the wave generated has an
amplitude of 1.0 cm. At t=2.0 sec, the
movement of the wave is captured with a
snapshot graph with a phase constant of π.
What would the phase constant of the
resultant wave be if another pebble of the
same size is dropped from the same height,
and that the amplitude of the resultant
wave is 1.3 cm? If a small leaf is sitting on
the surface of the pond, would it reach a
maximum displacement faster with or
without the interference of waves?
Solution:
We need to take into account that the second wave generated is identical to the first
wave, and that an interference occurs between the two waves. The relationship between
component waves and the resultant wave is described in the equation:
2A cos (φ/2) sin (kx-ωt+( φ/2))

2. A sketch of the snapshot graph with waves moving to the right would look like:
The amplitude of the resultant wave is 2A cos (φ/2). Given the amplitude of the
component wave is 1cm, and that the amplitude of the resultant wave is 1.3cm, we can find the
phase constant difference to be:
0.013m= 2(0.010m) cos (Δ φ/2)
φ1- - φ2 = Δ φ= 1.7264 radians
Therefore φ of the second component wave is:
π- φ2 = 1.7264
φ2 = 1.415 radians
The phase constant of the resultant wave at t= 2.0 sec is:
(φ1 + φ2)/2= (π+ 1.415)/2= 2.278 radians = 2.3 radians
Since the crest of the resultant wave comes slightly after the component wave, the leaf
would reach a maximum displacement faster with the first component wave than it would with
the resultant wave. Therefore, the leaf would reach a maximum displacement faster without
the interference of waves.