# A string 3.8m long is fixed at both ends. A pulse at one end takes 0.60s to move through the string and then come back. Find the 3 lowest frequencies that will produce a standing wave in the string.

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The relationship between the length of the string and possible wavelengths (`lambda` of the standing wave is

`L=n/2 * lambda`

(Please see the reference link for the explanation of where this comes from.)

So the three longest possible wavelengths will be

If n = 1 then `lambda_1 = (2L)/n = (2*3.8 m)/1 = 7.6 m`

If n = 2 then `lambda_2 = (2L)/n = (2*3.8 m)/2 = 3.8 m`

If n = 3 then `lambda_3 = (2L)/n = (2*3.8 m)/3 = 2.53 m`

Since it takes the pulse time t = 0.6 seconds to travel through the string and then come back, the speed of the wave is

`v = (2L)/t = (2*3.8 m)/(0.6 s)=12.67 m/s`

The frequency f of the wave is the speed divided by the wavelength:

`f = v/lambda`

So the three lowest frequencies will be

`f_1 = v/lambda_1 = 12.67/7.6 Hz = 1.67 Hz`

`f_2 = v/lambda_2 = 12.67/3.8 Hz =3.3 Hz`

`f_3 = v/lambda_3 = 12.67/2.53 = 5 Hz`

**Sources:**