Section 7.7 Interference
Recall that a sinusoidal wave traveling in the positive \(x\)-direction can be written:
\begin{equation*}
y(x,t) = y_{\mathrm{max}} \sin\left(\frac{2\pi}{\lambda}x - \frac{2\pi}{T}t + \phi_i \right)
\end{equation*}
Using the relations \(\omega = \frac{2\pi}{T}\) and \(k = \frac{2 \pi}{\lambda} \text{,}\) you can rewrite this equation in the more compact form:
\begin{equation*}
y(x,t) = y_{\mathrm{max}} \sin\left(kx - \omega t + \phi_i \right)
\end{equation*}
The video below uses this form to help determine two different superpositions of traveling waves.
Definition 7.7.1. Phase Difference.
The phase difference (or the relative phase) between two sinusoidal waves, \(\Delta \phi = \phi_2 - \phi_1\text{,}\) describes the difference in where two waves are in their cycles.
If you consider two similar waves (with the same frequency and wavelength) at the same position and time, the phase difference is equal to the difference in the initial phase of the two waves
\begin{equation*}
\Delta \phi_i = \phi_{i,2} - \phi_{i,1}
\end{equation*}
Exercises Activities
Consider the two waves written below:
\begin{equation*}
y_1(x,t) = y_m \sin{(kx - \omega t)}
\end{equation*}
\begin{equation*}
y_2(x,t) = y_m \sin{(kx - \omega t + \pi/2)}
\end{equation*}
1.
Sketch a graph of each wave at \(t = 0\) showing at least two full wavelengths.
2.
From looking at the graphs, can you tell whether these waves have constructive interference, destructive interference, or neither?
3.
From looking at the equations, can you tell whether these waves have constructive interference, destructive interference, or neither?
4.
If you have not done so already, sketch a graph of the sum of the two waves at \(t = 0 \) showing at least two full wavelengths.
Exercises Check Your Answers
When two waves add together at the same location in space, the sum of the two waves causes interference. The degree of interference can be quantified using the relative phase between the two waves.
Definition 7.7.2. Maximum Constructive Interference.
Two similar waves undergo maximum constructive interference if their relative phase is an integer multiple of \(2\pi\) such as
\begin{equation*}
\Delta \phi = 0, \pm 2\pi, \pm 4\pi, ...
\end{equation*}
In maximum constructive interference, the resulting superposition of the waves is a new wave with twice the amplitude.
Definition 7.7.3. Complete Destructive Interference.
Two similar waves undergo complete destructive interference if their relative phase is an odd integer multiple of \(\pi\) such as
\begin{equation*}
\Delta \phi = \pm \pi, \pm 3\pi, ...
\end{equation*}
In complete destructive interference, the resulting superposition of the waves has a displacement of \(0\) at all instants in time.
