Speed of a Wave in a Stretched String

The speed of a wave in a stretched string depends on the elasticity of the string. It varies directly on the tension of the string and inversely on the inertial factor or the mass per unit length which is also called as the linear density of the string.

These are the reasons why there are different sound or notes released by the guitar and other string instruments. Heavier string pertains to slower speed of sound, giving out a lower musical note. Hence, lighter strings means higher note. Vibrating strings include the string musical instruments such as guitar, violin and harp.

The general equation for the speed of transverse wave in a stretched string is

$$v=\sqrt {\frac {T}{\mu}}$$

where $$v$$ is the speed of the wave in $$m/s$$, $$T$$ is the tension on the string with unit $$N$$ and $$\mu$$ is the inertial factor or the mass per unit length of a string expressed as $$\mu=\frac ml$$. The unit for the inertial factor is $$kg/m$$.

Example 1.

A $$0.5-kg$$, $$3-m$$ long clothesline is connected from post to post. If stretching the string produces a tension of $$250\;N$$, what is the speed of wave on the string?

Given:

$$m=0.5\;kg\\ l=3\;m\\ T=250\;N$$

Solution 1.

Solve first for the initial factor using the equation $$\mu=\frac ml$$.

$$\mu=\frac {0.5\;kg}{3\;m}=0.17\;kg/m$$

Solution 2.

The speed of wave in the string can be calculated with the use of the equation $$v=\sqrt{\frac {T}{\mu}}$$.

$$v=\sqrt{\frac {250\;N}{0.17\;kg/m}}=\sqrt {1,470.59\;m^2/s^2}=38.35\;m/s$$.

The speed of wave in the string is $$38.35\;m/s$$.

Example 2.

A guitar string has a mass of $$3.18 \times 10^{-3}\;kg$$ and length $$0.81\;m$$. If the speed of wave of the string after vibration is $$134.4\;m/s$$, what is the tension on the string?

Given:

$$m= 3.18 \times 10^{-3}\;kg\\ l=0.81\;m\\ v=134.4\;m/s$$

Solution 1.

Solve for the inertial factor of the string.

$$\mu={\frac ml}={\frac {3.18 \times 10^{-3}\;kg}{0.81\;m}}=3.93\times 10^{-3}\;kg/m$$.

Solution 2.

Derive the formula for tension from the general equation of the wave on the stretched string.

$$v=\sqrt {\frac {T}{\mu}} \implies v^2=\frac {T}{\mu} \implies T=v^2 \mu$$.

$$T=(134.4\;m/s)^2(3.93\times 10^{-3}\;kg/m)=71\;N$$

The tension on the string is $$71\;N$$.