Stationary wavesEdexcel International A Level Physics: Revision notes

A stationary (standing) wave is formed when two progressive waves of the same frequency and similar amplitude, travelling in opposite directions, superpose. In practice the second wave is usually the first one reflected at a boundary, such as the fixed end of a string.

The result is a wave pattern that does not move along the string. Unlike a progressive wave, a stationary wave does not transfer energy along the medium.

Nodes are points of zero amplitude, where the two waves always arrive in antiphase and cancel. Antinodes are points of maximum amplitude, where the two waves always arrive in phase and reinforce.

Adjacent nodes (or adjacent antinodes) are half a wavelength apart, so the distance between a node and the next antinode is a quarter of a wavelength. A string fixed at both ends must have a node at each end.

For a string of length L fixed at both ends, a whole number of half-wavelengths must fit along it: L = nλ/2. The fundamental (first harmonic) has one antinode, so λ = 2L and f₁ = v/2L.

Higher harmonics have frequencies f = nf₁ with n antinodes: the second harmonic has two antinodes, λ = L, and twice the fundamental frequency. A string can only resonate at these frequencies.

The speed of a transverse wave on a stretched string is v = √(T/μ), where T is the tension in newtons and μ is the mass per unit length in kg m⁻¹.

Worked example. A string has T = 36 N and μ = 4.0 × 10⁻⁴ kg m⁻¹, length 0.60 m. Then v = √(36 / 4.0 × 10⁻⁴) = 300 m s⁻¹. The fundamental has λ = 1.2 m, so f = 300 / 1.2 = 250 Hz.

Combining the equations gives f = (1/2L)√(T/μ): frequency rises with tension and falls with length and with mass per unit length.

Set up a string with one end on a vibration generator driven by a signal generator, passing over a pulley with a hanging mass that provides the tension T = mg. Increase the frequency slowly until the string forms a single large loop with a node at each end: this is the fundamental.

Investigate one variable at a time: length (move the pulley or a bridge), tension (change the mass) and mass per unit length (use strings of different thickness, measuring μ by weighing a known length). Graphs of f against 1/L (gradient √(T/μ)/2) and f² against T (gradient 1/(4L²μ)) should be straight lines through the origin.