Graphical Representation of Superposition

Linearity: The principle works because wave equations in most media are linear, meaning the sum of two solutions is also a valid solution.

Algebraic Summation: At any horizontal position $x$, the resultant vertical displacement $Y$ is calculated as $Y = y_1 + y_2$. This requires careful attention to the sign (positive for peaks, negative for troughs).

Independence of Waves: Waves pass through each other without being permanently altered; they only interact at the specific region and time where they overlap.

Check the Extremes: Always start by plotting the points where both waves are at their maximum or minimum, as these define the boundaries of the resultant wave.

Zero-Crossing Rule: If one wave is at the equilibrium line (displacement = 0), the resultant wave's displacement is exactly equal to the displacement of the second wave at that point.

Symmetry Check: If the two waves are identical but shifted, the resultant wave should maintain the same frequency as the source waves, even if the amplitude changes.

Verify Anti-Phase: If waves have equal amplitude and are in anti-phase, the resultant graph must be a flat line along the equilibrium axis.

Adding Amplitudes Only: A common error is simply adding the maximum heights (amplitudes) without considering the actual displacement at specific points. You must sum the values at the same $x$-coordinate.

Ignoring Signs: Students often forget that troughs have negative displacement. Adding a peak ($+5$) and a trough ($-5$) results in zero, not ten.

Frequency Confusion: Superposition changes the amplitude and phase of the resultant wave, but it does not create a new frequency unless the source waves had different frequencies to begin with.