9.2.2 Phase Relationship in Graphs

We must be able to calculate the phase difference when the information is presented to us in graphs.

Displacement-Distance Graph

Sometimes you’re given a displacement-distance graph (like the one below) and asked to calculate the phase difference between two oscillations (at two different positions of the wave).

First, let’s agree that the oscillation at A leads the oscillation at B. How can we tell? Because the wave is coming from the left, meaning the wave will hit A first before B. The one further from the wave source must be lagging since it is the delayed version.

Now back to the calculation. The phase difference between 2 points on a wave depends on how far apart they are along the wave. The larger the separation Δx, the larger the phase difference Δθ. We know that if 2 points are one wavelength apart, the phase difference between them would be 2π rad. By simple proportion, if they are separated by a distance of Δx, the phase difference Δθ would be

$\displaystyle \Delta \theta =\frac{{\Delta x}}{\lambda }\times 2\pi$

Two Displacement-Time Graphs

Other times, you’re given the displacement-time graphs of two oscillations (see below), and asked to calculate the phase difference between them.

Firstly, let’s agree that A leads B. This is easy to tell since B is a delayed version of A. Whatever A did, B will do at a later time.

As for the calculation, you must first obtain the misalignment in time Δt between the two oscillations. Just look for the delay between two crests, or two troughs, or any two corresponding points from each of the two graphs. Since a misalignment of one period corresponds to a phase difference of 2π rad, a misalignment of Δt would correspond to phase difference of

$\displaystyle \Delta \theta =\frac{{\Delta t}}{T}\times 2\pi$