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Mathematics · Advanced university

Angular frequency of a wave-equation mode

For a string with fixed endpoints, length L, wave speed c, and positive integer mode n.

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Method and conditions

For a string with fixed endpoints, length L, wave speed c, and positive integer mode n.

ωₙ = cnπ/L

Worked examples

A fixed-end string of length 12 m has wave speed 2 m/s. Write the angular frequency of mode n=10 as aπ rad/s. Find a. Round the final result to 4 decimal places if needed.

The wavenumber is nπ/L. Multiply by c = 2, giving (20/12)π rad/s.

1.6667

A fixed-end string of length 2 m has wave speed 41 m/s. Write the angular frequency of mode n=10 as aπ rad/s. Find a. Round the final result to 4 decimal places if needed.

The wavenumber is nπ/L. Multiply by c = 41, giving (410/2)π rad/s.

205

Try it

  1. A fixed-end string of length 6 m has wave speed 5 m/s. Write the angular frequency of mode n=9 as aπ rad/s. Find a. Round the final result to 4 decimal places if needed.

  2. A fixed-end string of length 4 m has wave speed 7 m/s. Write the angular frequency of mode n=5 as aπ rad/s. Find a. Round the final result to 4 decimal places if needed.

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Answers and marking notes
  1. 7.5

    The wavenumber is nπ/L. Multiply by c = 5, giving (45/6)π rad/s.

  2. 8.75

    The wavenumber is nπ/L. Multiply by c = 7, giving (35/4)π rad/s.

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