Intermediate · 13 min
Standing waves on strings
Identify nodes and calculate the fundamental frequency and harmonics of an ideal string.
A stationary pattern from travelling waves
A standing wave can form when waves of the same frequency and suitable amplitudes travel in opposite directions and overlap. Nodes stay at zero displacement; antinodes have the largest oscillation amplitude. For a string fixed at both ends, the endpoints are nodes. The lowest-frequency standing pattern has one antinode between them and fits half a wavelength into the string length.
Worked example
For a fixed-end string of length L = 0.8 m: Fundamental wavelength = 2L = 1.6 m
Harmonics fit whole half-wavelengths
For an ideal uniform string fixed at both ends, with wave speed v unchanged, the fundamental frequency is f₁ = v/(2L). Harmonic number n has frequency fn = n × f₁ and wavelength 2L/n. The second harmonic fits two half-wavelengths along the string, the third fits three, and so on. These relationships assume the same tension and mass per unit length, so the wave speed remains the same.
Worked example
For L = 0.5 m and v = 80 m/s: f₁ = 80 ÷ (2 × 0.5) = 80 Hz Third harmonic frequency = 3 × 80 = 240 Hz
Sketch the first two modes
- Draw the fundamental and second harmonic patterns for a string fixed at both ends. Mark every node and antinode.
- For a 1 m string with wave speed 100 m/s, calculate the frequencies of these two modes.