Superposition, Interference, and Standing Waves

Superposition and Interference

Two waves can occupy the same place at the same time, adding point by point before walking away unchanged. The superposition principle states that the total displacement is the vector sum: $y_{\text{total}} = y_1 + y_2$.

When waves meet, interference occurs based on their phase difference $\Delta\phi$:

Beats occur when two waves of slightly different frequencies combine, producing a modulation frequency of $f_{\text{beat}} = |f_1 - f_2|$.

Common pitfall: At perfect destructive overlap, the string looks completely flat, but energy has not vanished. It hides in the string's velocity. Displacement cancels; energy does not.

Standing Waves and Boundaries

Confining identical waves travelling in opposite directions forms a standing wave: $y(x,t) = 2A\sin(kx)\cos(\omega t)$. The pattern features fixed nodes (zero displacement) and antinodes (maximum displacement).

Standing waves only form at specific frequencies called natural modes. For a string of length $L$, boundary conditions dictate the allowed wavelengths $\lambda_n$ and resonant frequencies $f_n$:

Resonant frequency depends on string properties: $f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$, where $T$ is tension and $\mu$ is mass per unit length.