Chapter 12, Problem 28P

To determine

To Calculate: The fundamental frequency of a tight guitar string.

Answer to Problem 28P

The guitar’ string is fingered at a length of the $60\%$ of its original length then its fundamental frequency is $300\text{Hz}$ .

Explanation of Solution

Given:

The frequency of a guitar string is, $f=540\text{Hz}$

Formula used:

The frequency of the $n^{th}$ harmonic for a fixed string is given as,

  $f_n=n{f}_1$

The fundamental frequency of the vibrating string is,

  $f_1=\frac{v}{2L}$

Where, L is the length of the string

M is the masse of the vibrating portion of the string

Calculation:

By substituting the values of given information in the formula. The fundamental for the string is calculated as,

  $\begin{array}{l}{f}_3=3f_1\\ \Rightarrow f_1=\frac{f_3}{3}\\ \Rightarrow f_1=\frac{5\text{40Hz}}{3}=180\text{Hz}\end{array}$

Guitar has a new length of $60\%$ of its original length if it is fingered.

Therefore, the fundamental frequency of a vibrating string is given as,

  $f_1=\frac{v}{2L}$

Let us consider, the tension in the string is constant and also $v$ is also constant for the string. So, the fundamental frequency is,

  $\begin{array}{l}{f}_{{\text{1}}_{\text{fingered}}}=\frac{v}{2L_{\text{fingered}}}\\ =\frac{v}{2\times \left(0.60L\right)}=\frac{1}{\left(0.60\right)}\left(\frac{v}{2L}\right)\\ =\frac{1}{\left(0.60\right)}{f}_1\\ =\frac{180\text{Hz}}{0.60}=300\text{Hz}\end{array}$

Conclusion:

The fundamental frequency is $300\text{Hz}$ .