驻波

Standing waves form when two waves traveling in opposite directions with the same frequency and amplitude interfere with each other. When a wave reflects off a fixed boundary, it travels back along the string and superimposes with the incoming wave. This interference and diffraction pattern creates the stationary appearance where the string appears to vibrate up and down without linear movement.

Nodes are points of minimum amplitude where waves have opposite phases and cancel each other out completely. Antinodes are points of maximum amplitude where waves have the same phase and their amplitudes combine constructively. The pattern of nodes and antinodes determines the harmonic structure of the standing wave.

The wavelength of an nth harmonic standing wave equals twice the string length divided by n. For the first harmonic, the wavelength equals twice the string length. As the harmonic number increases, the wavelength decreases proportionally, creating shorter and shorter wave patterns with more nodes and antinodes.

Since wavelength and frequency have an inverse relationship, shorter wavelengths correspond to higher frequencies. As harmonic number increases, wavelength decreases, so frequency increases. The frequency of each harmonic is the nth multiple of the fundamental frequency, meaning the second harmonic has twice the frequency of the first harmonic.

Simple harmonic motion is periodic oscillation where the restoring force is proportional to displacement, following Hooke's Law. Standing waves exhibit simple harmonic motion because the string's restoring force pulls displaced portions back toward equilibrium. This proportional relationship between force and displacement creates the characteristic oscillating motion observed in vibrating strings and other finite media.

Acoustophoresis uses standing waves in microfluidic devices to displace and focus microscopic particles in flowing liquid. When a standing wave with specific frequency forms within a microchannel, it focuses particles into a controlled stream. This technique enables researchers to rapidly separate or concentrate microscopic entities for biomedical analysis and processing.

A plucked guitar string produces only specific notes because only certain standing waves can form on that string. These standing waves are integer multiples of the string's fundamental frequency, determined by the string's length, tension, and density. Musicians can create new sets of harmonics by shortening the string length, producing different pitches.