Stationary Waves

The formation of stationary waves relies on the Principle of Superposition, which states that when two waves meet, the resultant displacement is the vector sum of the individual displacements.

When the incident wave and the reflected wave are in phase, they interfere constructively to create an antinode. When they are $180^\circ$ out of phase, they interfere destructively to create a node.

The distance between two adjacent nodes (or two adjacent antinodes) is exactly half a wavelength ($\frac{\lambda}{2}$), while the distance between a node and the adjacent antinode is a quarter wavelength ($\frac{\lambda}{4}$).

For a string of length $L$ fixed at both ends, stationary waves can only form at specific resonant frequencies called harmonics. The simplest pattern is the first harmonic (fundamental frequency), where $L = \frac{\lambda}{2}$.

The speed $v$ of a wave on a stretched string depends on the tension $T$ and the mass per unit length $\mu$ (linear density), expressed by the formula: $v = \sqrt{\frac{T}{\mu}}$

By combining the wave equation $v = f\lambda$ with the harmonic condition, the fundamental frequency $f_0$ is calculated as: $f_0 = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$

Identify the Harmonic: Always count the number of 'loops' in a diagram. One loop equals half a wavelength ($\frac{\lambda}{2}$). If there are 3 loops, then $L = 1.5\lambda$.

Unit Consistency: When calculating mass per unit length ($\mu$), ensure mass is in kilograms (kg) and length is in meters (m). Students often forget to convert grams or centimeters, leading to power-of-ten errors.

Boundary Conditions: Remember that a fixed end must always be a node, while an open end (in air columns) or a driving oscillator usually acts as an antinode or near-antinode.

Verify Proportionality: If tension is quadrupled ($4T$), the frequency only doubles ($2f$) because frequency is proportional to the square root of tension.