Question

A 90 cm -long steel string with a linear density of 1.3 g/m is under 100 N tension. It is plucked and vibrates at its fundamental frequency.

What is the wavelength of the sound wave that reaches your ear in a 20 C room?

Answer 1

the frequency f of the vibrating string is

f = sqrt (T / u) / (2 * L)
where T is the tension, u is the linear density which is 0.0013 kg/m , and L is the length which is 0.9 meters, so

f = sqrt (100 / 0.0013) / (2 * 0.9) = 154.08 hertz

That frequency goes through the air with a velocity which depends on the temperature, using v as the temperature in celsius, as

cair = 331.3 * sqrt (1 + v / 273.15) in meters/second
cair = 331.3 * sqrt (1 + 20 / 273.15)
cair = 343meters/second

The wavelength is given by speed = wavelength * frequency, so

wavelength = cair / frequency
wavelength = 343 meters/second / 154.08 /sec
wavelength = 2.226 meters

I hope help you !!