A string of length L vibrates at its fundamental frequency. The amplitude at a point 1/4L...

A string of length L vibrates at its fundamental frequency. The amplitude at a point 1/4L from one end is 5.80cm .

-What is the amplitude of each of the traveling waves that form this standing wave?

Thanks in Advance!!

Expert Solution

The question is referring to 2 different amplitudes.

E.g. If you set up a standing wave by plucking a guitar string, each PART of the string performs simple harmonic motion (SHM). The parts of the string have the same frequency but different amplitudes. At the middle, the amplitude of SHM is large. Near the nodes (ends) the amplitude of SHM is small. I'll refer to these amplitudes as 'local amplitudes',

The standing wave on the string arises because of superposition of the 2 travelling waves being reflected from each end of the string. By itself, each of the traveling waves has its own 'wave amplitude', 'A'.

In the centre where the reflected waves meet in phase, they makes the string vibrate with a local amplitude of 2A.

At the ends, the 2 waves are always out of phase and give an overall zero local amplitude.

The standing wave is a sine curve shape. The maximum local amplitude is 2A at the peak of the sine curve. This maximum corresponds to a phase angle of 90degrees, so the local amplitude is 2Asin(90deg) = 2A

L/4 is halfway between the node and antinode, so its phase angle is 45degrees,
The local amplitude at L/4 = 2Asin(45deg) = 1.414A
1.414A = 5.8cm, so A = 5.8/1.414 = 4.101 cm

genius_generous answered 6 hours ago

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