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Physics · Advanced university

Group velocity from a dispersion relation

The group velocity generally differs from phase velocity ω/k; the dispersion constants carry compatible SI units.

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Method and conditions

The group velocity generally differs from phase velocity ω/k; the dispersion constants carry compatible SI units.

vg = dω/dk

Worked examples

A wave has dispersion ω(k)=7k + 6k³ with SI-compatible constants. Find group velocity at k=10 m⁻¹. Round the final result to 4 decimal places if needed. Answer in m/s.

Differentiate before substituting: vg=7+18k².

1807 m/s

A wave has dispersion ω(k)=20k + 5k³ with SI-compatible constants. Find group velocity at k=4 m⁻¹. Round the final result to 4 decimal places if needed. Answer in m/s.

Differentiate before substituting: vg=20+15k².

260 m/s

Try it

  1. A wave has dispersion ω(k)=4k + 7k³ with SI-compatible constants. Find group velocity at k=11 m⁻¹. Round the final result to 4 decimal places if needed. Answer in m/s.

  2. A wave has dispersion ω(k)=4k + 2k³ with SI-compatible constants. Find group velocity at k=9 m⁻¹. Round the final result to 4 decimal places if needed. Answer in m/s.

Check your work

Answers and marking notes
  1. 2545 m/s

    Differentiate before substituting: vg=4+21k².

  2. 490 m/s

    Differentiate before substituting: vg=4+6k².

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