Superposition, Interference, and Standing Waves

Two waves can occupy the same place at the same time — try that with matter. They add point by point, sometimes reinforcing, sometimes erasing, then walk away unchanged. Confine waves between two walls and this superposition selects a discrete menu of standing patterns: the physics behind every musical instrument.

The superposition principle states that when two waves meet, the displacement is the sum:

$$y_{\text{total}} = y_1 + y_2$$

Interference

• Constructive: Waves in phase ($\Delta\phi = 0, 2\pi, \ldots$) $\Rightarrow$ larger amplitude.
• Destructive: Waves out of phase ($\Delta\phi = \pi, 3\pi, \ldots$) $\Rightarrow$ cancellation.

Standing waves form when two identical waves travel in opposite directions:

$$y = 2A\sin(kx)\cos(\omega t)$$

Boundary conditions on a string

Resonant frequencies:

$$f_n = \frac{nv}{2L} = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$$

where $T$ is tension and $\mu$ is mass per unit length.

Beats — Two waves of slightly different frequencies produce:

$$f_{\text{beat}} = |f_1 - f_2|$$

Key insight: Standing waves only form at specific frequencies — these are the natural modes of the system. This is why musical instruments produce distinct pitches.

Common pitfall: At an instant of perfect destructive overlap the string can look completely flat — but the energy has not vanished. It is hiding in the string’s velocity. Displacement can cancel; energy cannot.

Standing Waves

When two identical waves travel in opposite directions, they form a standing wave:

$$y(x,t) = 2A\sin(kx)\cos(\omega t)$$

The pattern has fixed nodes (zero displacement) and antinodes (maximum displacement).

For a string of length $L$ fixed at both ends:

$$\lambda_n = \frac{2L}{n}, \quad f_n = \frac{nv}{2L}$$

Standing waves appear wherever boundaries reflect travelling waves — strings, pipes, and resonant cavities.