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Q10Q

Expert-verifiedFound in: Page 471

Book edition
10th Edition

Author(s)
David Halliday

Pages
1328 pages

ISBN
9781118230718

**If you set up the seventh harmonic on a string, (a) how many nodes are present, and (b) is there a node, antinode, or some intermediate state at the midpoint? If you next set up the sixth harmonic, (c) is its resonant wavelength longer or shorter than that for the seventh harmonic, and (d) is the resonant frequency higher or lower?**

- For the seventh harmonic, there are ‘eight’ nodes.
- There will be an ‘antinode’ at the midpoint.
- For the sixth harmonic, there are ‘seven’ nodes.
- There will be a ‘node’ at the midpoint.

The set up has the seventh harmonic on a string

**Determine the number of nodes by knowing the number of harmonics of the standing wave.**

Determine the number of nodes for the sixth and seventh harmonic as, for the nth harmonic there are (n+1) nodes. Also, for an even number of harmonics, there is ‘node’ at the midpoint and for the odd number of harmonic; there is ‘antinode’ at the midpoint.

for the n^{th} harmonic, there are (n+1) number of nodes. So, for the seventh harmonic there will be eight nodes.

** **

Hence, for the seventh harmonic, there

For an even number of harmonics, there is ‘node’ at the midpoint and for the odd number of harmonic, there is ‘antinode’ at the midpoint.

So, for the seventh (odd) harmonic, there is an antinode at the midpoint.

Hence, there will be an ‘antinode’ at the midpoint.

Using similar logic of part a), for the sixth harmonic, there are ‘Seven’ nodes.

** **

Hence, for the sixth harmonic, there are ‘seven’ nodes

Applying the same logic of part b), there is a ‘node’ at the midpoint.

Hence, there will be a ‘node’ at the midpoint.

Therefore, determine the number of nodes by knowing the number of harmonics of the standing wave.

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