Chapter 11, Problem 57P

To determine

To Find: The fundamental frequency and frequency of first two overtones.

Answer to Problem 57P

The fundamental frequency is $290.77\text{ Hz}$ and frequency of first two overtones are $\text{581}\text{.54 Hz}$ and $\text{871}\text{.31 Hz}$ respectively.

Explanation of Solution

Given:

Mass of guitar string $m=3.6\text{ g = }0.0036\text{kg}$

Length of the guitar string $l=90\text{ cm = }0.90\text{ m}$

Tension in the string $T=520\text{ N}$

Distance from bridge to support post $L=62\text{ cm = }0.62\text{ m}$

Formula used:

Speed of the wave on string is calculated by following formula,

  $v=\sqrt{\frac{T}{\left(\frac{m}{L}\right)}}$

Here,

  $T$ is string tension,

  $m$ is string mass,

  $L$ is string length.

The fundamental frequency of string with both end fixed is calculated by following formula,

  $f_n=\frac{n\times v}{2\times L}$

Here $f_n$ is the frequency of $n-1$ overtone,

  $v$ is speed of wave,

  $l$ is the length of vibrating string.

  $n$ = no. of fixed end - $1$ .

Calculation:

Substitute the given values in the above equation,

  $\begin{array}{l}{f}_n=\frac{n}{2\text{}\times L}\times \sqrt{\frac{T}{\left(\frac{m}{l}\right)}}\\ f_n=\frac{1}{2\text{}\times 0.62}\times \sqrt{\frac{520}{\left(\frac{0.0036}{0.9}\right)}}\\ f_n=290.77\text{ Hz}\end{array}$

Frequency of first overtone is calculated by following way,

  $f_2=2\times f_1=2\times 290.77=581.54\text{ Hz}$

Frequency of second overtone is calculated by following way,

  $f_3=3\times f_1=3\times 290.77=872.31\text{ Hz}$

Conclusion:

Thus, the fundamental frequency is $290.77\text{ Hz}$ and frequency of first two overtones are $\text{581}\text{.54 Hz}$ and $\text{871}\text{.31 Hz}$ respectively.