Consider two interfering waves with identical amplitude \( A \), and intensity \( I_0 \propto A^2 \).
The combined amplitude at a location with a phase difference \( \phi \) is calculated as:
\[A_R = 2A \cos\left(\frac{\phi}{2}\right)\Rightarrow I = A_R^2 = 4A^2 \cos^2\left(\frac{\phi}{2}\right)\Rightarrow I = 4I_0 \cos^2\left(\frac{\phi}{2}\right)\]
Combined Intensity:
\[\boxed{I = 4I_0 \cos^2\left(\frac{\phi}{2}\right)}\]
Peak Intensity:
For phase differences \( \phi = 0, 2\pi, 4\pi, \ldots \), where \( \cos\left(\frac{\phi}{2}\right) = 1 \):
\[I_{\max} = 4I_0\]
Lowest Intensity:
For phase differences \( \phi = \pi, 3\pi, \ldots \), where \( \cos\left(\frac{\phi}{2}\right) = 0 \):
\[I_{\min} = 0\]