All subjects · Physics · WPH12 (Unit 2)

Standing waves and waves on strings

A standing (stationary) wave forms when two waves travel in opposite directions. Standing waves on strings set the notes of musical instruments.

How a standing wave forms

Two waves of the same frequency (and similar amplitude) travelling in opposite directions superpose. Usually one is a reflection of the other. The result is a pattern that does not travel:

Adjacent nodes are λ/2 apart.

Standing versus progressive

In a standing wave no energy is transferred along the wave. All points between two adjacent nodes oscillate in phase, with different amplitudes; points in neighbouring segments are in antiphase. In a progressive wave, every point has the same amplitude and the phase changes along the wave.

Strings fixed at both ends

The ends are nodes. In the fundamental (first harmonic) there is one loop, so λ = 2L and f = v / 2L. The next modes have 2, 3, ... loops, with frequencies 2f, 3f, ...

Speed of a wave on a string

v = sqrt(T / μ), where T is the tension (N) and μ is the mass per unit length (kg/m). Tighter or lighter strings carry faster waves and so give higher notes.

Worked example

A string of length 0.65 m has tension 70 N and mass per unit length 1.0 x 10-3 kg/m. Find its fundamental frequency.

  1. v = sqrt(70 / 1.0 x 10-3) = sqrt(70000) = 265 m/s
  2. Fundamental: λ = 2L = 1.30 m
  3. f = v / λ = 265 / 1.30

Answer: f = 204 Hz

Key idea: Nodes λ/2 apart; fixed string fundamental λ = 2L; v = sqrt(T/μ).

Check yourself

  1. A standing wave has wavelength 0.80 m. Find the distance between adjacent nodes, in m.
    Show answer

    0.4 m. Adjacent nodes are λ/2 = 0.40 m apart.

  2. A string has tension 45 N and mass per unit length 2.5 x 10-3 kg/m. Find the wave speed in m/s, to 3 s.f.
    Show answer

    134 m/s. v = sqrt(45 / 2.5 x 10-3) = sqrt(18000) = 134 m/s.

  3. Waves travel at 200 m/s on a string of length 0.50 m fixed at both ends. Find its fundamental frequency in Hz.
    Show answer

    200 Hz. λ = 2L = 1.0 m; f = 200 / 1.0 = 200 Hz.

  4. A 1.5 m length of wire has a mass of 3.0 g. Find its mass per unit length in kg/m.
    Show answer

    0.002 kg/m. μ = 0.0030 / 1.5 = 0.0020 kg/m.

  5. A string of length 0.60 m vibrates with two loops (the second harmonic). Find the wavelength in m.
    Show answer

    0.6 m. Two loops = two half-wavelengths = one wavelength, so λ = 0.60 m.

  6. What is a node in a standing wave?
    • A point of maximum amplitude
    • The point where the wave is reflected
    • A point where the wave travels fastest
    • A point of zero amplitude
    Show answer

    A point of zero amplitude. At a node the two travelling waves always cancel.

  7. The tension in a string is increased by a factor of 4. What happens to the wave speed?
    • It increases by a factor of 4
    • It doubles
    • It halves
    • It increases by a factor of 16
    Show answer

    It doubles. v is proportional to sqrt(T), and sqrt(4) = 2.

  8. How do points between two adjacent nodes of a standing wave oscillate?
    • In phase with each other, with different amplitudes
    • With the same amplitude but different phases
    • Not at all
    • In antiphase with each other
    Show answer

    In phase with each other, with different amplitudes. Within one loop all points move together, but the amplitude is largest at the antinode.

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Last updated 7 November 2026. Spotted a mistake? Tell us.