Sound: Standing Waves, Resonance, Harmonics, and Intensity

A standing wave is formed when two waves of the same frequency, amplitude, and wavelength travelling in opposite directions superpose. This typically occurs when a wave reflects back on itself and interferes with the incident wave. In a standing wave, certain points (nodes) remain permanently at rest, while other points (antinodes) vibrate with maximum amplitude. Energy is not transferred along the medium in a standing wave.

Nodes are points on a standing wave where the displacement is always zero (no vibration). Antinodes are points on a standing wave where the displacement is maximum (maximum vibration). The distance between two consecutive nodes or two consecutive antinodes is half a wavelength (λ/2). The distance between a node and an adjacent antinode is a quarter wavelength (λ/4).

Resonance occurs when a vibrating system is subjected to a periodic force at a frequency equal or very close to its own natural frequency. When this happens, the system absorbs energy efficiently, leading to a rapid increase in the amplitude of its oscillations. Examples include a child on a swing, a tuning fork resonating with another, or the shattering of a glass by a specific sound frequency.

When a string fixed at both ends vibrates, it can form standing waves. The ends of the string must be nodes. The simplest mode of vibration is the fundamental frequency (or first harmonic), where the string vibrates with one antinode in the middle. Higher modes of vibration are called overtones or harmonics. The nth harmonic has n antinodes and (n+1) nodes. The speed of a wave on a string depends on the tension (T) and its mass per unit length (μ).

An open pipe is open at both ends. When air in an open pipe vibrates, standing waves are formed. The open ends of the pipe must be antinodes (as air particles can move freely there). The simplest mode of vibration is the fundamental frequency (or first harmonic), with an antinode at each end and one node in the centre. All harmonics (odd and even multiples of the fundamental) are present.

A closed pipe is open at one end and closed at the other. The open end must be an antinode, and the closed end must be a node (as air particles cannot move there). The simplest mode of vibration is the fundamental frequency (or first harmonic), with a node at the closed end and an antinode at the open end. Only odd harmonics are present in a closed pipe.

The intensity of sound (I) is defined as the rate at which sound energy is transmitted per unit area perpendicular to the direction of propagation. It is the power (P) carried by the sound wave per unit area (A). Intensity is measured in watts per square metre (W m⁻²). The intensity of sound decreases with the square of the distance from the source (assuming an isotropic source in a non-absorbing medium).

The decibel (dB) scale is a logarithmic scale used to measure sound intensity level (L) because the human ear perceives sound over a very wide range of intensities. It compares the intensity of a sound (I) to a reference intensity (I₀), which is the threshold of human hearing (I₀ = 1.0 × 10⁻¹² W m⁻²).