Question

Please show all work! thanks!

A 80.0 cm long guitar string has mass of 75.0g and it is kept taut with force of tension equal to 800N.

 Determine: (1) speed of propagation of a transverse wave in this string, (2) frequency of the fourth harmonic (of a transverse standing wave on this string), and (3) wavelength of the fourth harmonic.

Sketch the fourth harmonic normal mode.

 How many nodes (N) and antinodes (A) are in the fourth harmonic? Label them above.

 If incident wave has the amplitude of 5.5 cm, what is the amplitude of the standing wave?

 If the reflected wave has a vertical displacement of +2.3 cm (at some time instant) at the left end of the string, what is the vertical displacement (at the same time instant) of the incident wave at the same end of the string?

Answer 1

The Data

We will assumed that both ends of the string are fixed.

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The linear mass density is given,

(1) the speed of propagation of the wave is given by

(2) The frequency of the n´th harmonic is given by

where

hence

the frequency is given by

(3) the wavelength of the fourth harmonic is

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(The fourth harmonic) there are four nodes and five antinodes

here is a plot of the fourth normal mode (using Matlab)

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The standing wave is produced by two progresive waves of the same amplitude same frequency and

wavelength and traveling in opposite directions,

thus

,

thus

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We are dealing with standing waves, this is a standing wave with a free extreme, so the vertical

displacement of the incident wave is also +2.3 cm.

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