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Chapter 16: Q10Q (page 471)

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?

Short Answer

Expert verified
  1. For the seventh harmonic, there are 鈥榚ight鈥� nodes.
  2. There willbe an鈥榓ntinode鈥� at the midpoint.
  3. For the sixth harmonic, there are 鈥榮even鈥� nodes.
  4. There will be a 鈥榥ode鈥� at the midpoint.

Step by step solution

01

Given

The set up has the seventh harmonic on a string

02

Determining the concept

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

03

(a) Determining the number of nodes for the seventh harmonic

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 鈥榥ode鈥� at the midpoint and for the odd number of harmonic; there is 鈥榓ntinode鈥� at the midpoint.

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

Hence, for the seventh harmonic, there

04

(b) Determining is there a node, antinode or some intermediate state at the midpoint of the seventh harmonic

For an even number ofharmonics, there is 鈥榥ode鈥� at the midpoint and for the odd number of harmonic, there is 鈥榓ntinode鈥� at the midpoint.

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

Hence, there will be an 鈥榓ntinode鈥� at the midpoint.

05

(c) Determining the number of nodes for the sixth harmonic

Using similar logic of part a), for the sixth harmonic, there are 鈥楽even鈥� nodes.

Hence, for the sixth harmonic, there are 鈥榮even鈥� nodes

06

(d) Determining is there a node, antinode or some intermediate state at the midpoint of the sixth harmonic

Applying the same logic of part b), thereis a鈥榥ode鈥� at the midpoint.

Hence, there willbe a鈥榥ode鈥� at the midpoint.

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

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