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Physics · lesson 4 of 4

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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Learn the ideas

Prerequisites: Partial derivatives and trigonometric functions.

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

Example 1

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

Example 2

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

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

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

Show answer and explanation

5 Hz

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

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

Show answer and explanation

15 Hz

fn=nf1; the third harmonic is 15 Hz.

3 What does a fixed endpoint impose on transverse displacement?

Show answer and explanation

It is zero at that endpoint

The boundary condition fixes displacement, not the whole field.

4 Which property justifies adding two wave solutions?

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Linearity of the governing homogeneous equation

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

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

Show answer and explanation

4 m/s

For a sinusoid, vp=ω/k.

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

Sample answer and marking guidance

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.

7 At a node of an ideal standing transverse wave, what happens to displacement?

Show answer and explanation

It remains zero

A node is a position with zero oscillation amplitude.

8 State why an ideal string model may fail at very large displacements.

Sample answer and marking guidance

The small-slope approximation and approximately constant tension can fail, introducing nonlinear terms and changing wave speed and mode behaviour.

Credit explicit model assumptions, not merely saying all equations fail.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.