What is the speed of a wave on this string, and what is the wavelength and frequency of the standing wave?

Question

A 2.00 meter long string is clamped at both ends and has a mass of 7.00 grams.  The tension in the string is 125N.

 

What is the speed of a wave on this string?

 

Determine the wavelength and frequency of the standing wave?  

Answer 1

The wave speed is given by v = √(T/μ) where T is the tension and μ = m/L is the mass per unit length.

 

v = √(125/(0.007/2.00)) = 189 m/s

 

Integer half-wavelengths have to fit within the length of the string, L = 2.00 m, because the ends are clamped.

 

L = n(λ_n/2) or λ_n = 2L/n for n = 1, 2, 3, ...

 

 

The frequency is related to the wavelength through v = λ_nf_n.

 

Using the result above, the possible frequencies are f_n = (n/2L)v.

 

 

For n = 1:

 

λ₁ = 2L/1 = 4.00 m

 

f₁ = (1/2L)(189) = 47.25 Hz