Revise Stationary waves for Cambridge 9702 (8.1)

A stationary wave forms when two progressive waves, possessing the exact same frequency, wavelength, and amplitude, travel through the same medium in opposite directions. They combine or 'superpose' at every point. This often happens when a wave reflects off a boundary and interferes with the incident wave.

At a node, the two progressive waves always interfere destructively, resulting in zero or minimum displacement. Particles at nodes never move from their equilibrium position. Conversely, at an antinode, the waves always interfere constructively, leading to maximum displacement. Particles here oscillate with the largest amplitude.

When a string or air column vibrates, it can form various stationary wave patterns called harmonics. Each harmonic corresponds to a specific resonant frequency, which is an integer multiple of the lowest possible frequency, known as the fundamental frequency (or first harmonic).

The fundamental frequency produces a stationary wave with just one antinode between the fixed ends (e.g., a string). The n-th harmonic will feature 'n' antinodes within the vibrating section of the medium.

$f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$

Where:

• $f$ is the frequency of vibration (Hz)
• $L$ is the length of the vibrating string (m)
• $T$ is the tension in the string (N)
• $\mu$ (mu) is the mass per unit length of the string (kg m⁻¹)

Stationary waves are also fundamental to the operation of wind instruments. They are formed in a column of air within a pipe, caused by sound waves reflecting from the ends of the pipe. The nature of the stationary wave depends on whether the ends of the pipe are open or closed.

Pipe Closed at One End: For a pipe of length L that is closed at one end and open at the other, the fundamental mode of vibration (first harmonic) has one node at the closed end and one antinode at the open end. This means the length of the pipe is a quarter of a wavelength ($L = \lambda/4$). Because of this boundary condition, only odd-numbered harmonics can be produced ($f_1, 3f_1, 5f_1$, etc.).

Pipe Open at Both Ends: For a pipe of length L that is open at both ends, there must be an antinode at each end. The fundamental mode of vibration has an antinode at each end and one node in the middle. This means the length of the pipe is half a wavelength ($L = \lambda/2$). All integer harmonics can be produced ($f_1, 2f_1, 3f_1$, etc.).