Chapter 11, Problem 56P

To determine

To Find: The fundamental frequency of a vibrating string.

Answer to Problem 56P

The fundamental frequency of a vibrating string $\Delta f=70\text{ Hz}$ .

Explanation of Solution

Given:

The vibrating string has two successive overtones,

First overtone= $\text{= }280\text{ Hz}$

Second overtone $\text{ = }350\text{ Hz}$

Formula used:

The expression for the frequency.

  $f_n=n\times f_1$

Here,

  $f_1$ is frequency of 1^st harmonic.

The expression for the change in the frequency in successive.

  $\Delta f=f_{n+1}-f_n$

Here,

  $f_{n+1}$ is frequency of (n+1)th harmonic,

  $f_n$ is frequency of n^th harmonic.

Calculation:

Two successive overtones differ by the frequency. Taking the difference of the two successive overtones in the above formula,

  $\begin{array}{l}\Delta f=f_{n+1}-f_n\\ \Delta f\text{}=\left(n+1\right)\times f_1-\left(n\times f_1\right)\\ \Delta f=n\times f_1+f_1-n\times f_1\\ \Delta f=f_1=350\text{ Hz - }280\text{ Hz = }70\text{ Hz}\end{array}$

Conclusion:

Thus, the fundamental frequency of a vibrating string $\Delta f=70\text{ Hz}$ .