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Physics · University

Wave equations, normal modes and boundary conditions

The ideal one-dimensional wave equation is ytt=c²yxx. The parameter c is propagation speed for this nondispersive model.

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Prerequisites

Partial derivatives and trigonometric functions.

Study notes

Wave equation

The ideal one-dimensional wave equation is ytt=c²yxx. The parameter c is propagation speed for this nondispersive model.

Boundary data

Fixed endpoints require zero displacement at each end. They permit standing waves whose spatial wave numbers are nπ/L.

Superposition

Solutions of a linear homogeneous wave equation can be added. The result need not resemble one single travelling sinusoid.

Energy

A standing wave can have stationary nodes while its material elements oscillate. Node locations should not be confused with material travelling at wave speed.

Worked examples

A fixed-fixed string has L=2 m and wave speed 20 m/s. Find its fundamental frequency.

f1=c/(2L)=20/4.

5 Hz

For the same ideal fixed-fixed model with f1=5 Hz, find f3.

fn=nf1; the third harmonic is 15 Hz.

15 Hz

Check your understanding

  1. What does a fixed endpoint impose on transverse displacement?

    1. It is zero at that endpoint
    2. The tension is zero
    3. Its derivative in time is always one
    4. It is maximal at all times

  2. Which property justifies adding two wave solutions?

    1. Identical initial phases
    2. A moving boundary in every case
    3. Large amplitude
    4. Linearity of the governing homogeneous equation

  3. A wave has angular frequency 12 rad/s and wave number 3 rad/m. Find phase speed.

  4. Explain why normal-mode frequencies depend on boundary conditions as well as the wave equation.

Answers and marking guidance

Answers and marking notes
  1. It is zero at that endpoint

    The boundary condition fixes displacement, not the whole field.

  2. Linearity of the governing homogeneous equation

    The linear operator applied to a sum equals the sum of its actions.

  3. 4 m/s

    For a sinusoid, vp=ω/k.

  4. The equation determines possible local motion, while boundary conditions select permitted spatial patterns and wave numbers. Different endpoint constraints select different spectra.

    Credit both the field equation and allowed-mode selection.

Further reading and review references

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