Chapter 12, Problem 45P

To determine

To Calculate: The possible frequencies of A and C and the beat frequencies that are possible when A and C are sounded together.

Explanation of Solution

Given:

Frequency of fork B, $f_{\text{B}} =441\text{ Hz}\text{.}$

Beat frequency of A and B, $f_{\text{AB }}\text{= 3 Hz}.$

Beat frequency of B and C, $f_{\text{BC }}\text{= 4 Hz}.$

Formula used:

  $f_{\text{BC}}=\left|f_{\text{B}}-f_{\text{C}}\right|$

Calculation:

The frequency of fork B $f_{\text{B}} =441\text{ Hz}\text{.}$

The possible frequencies can be found by considering the beat frequencies when they are sounded together.

If A and B sounded together, beat frequency is 3 Hz, then the frequency of A is will be,

  $\begin{array}{l}{f}_{\text{A}}=\left|441\pm 3\right|\\ \text{ = }338\text{ Hz}\text{. or 444 Hz}\text{.}\end{array}$

Similarly, the frequency of C can be found by considering the beat frequency sounded together by B and C

  $\begin{array}{l}{f}_{\text{C}}=\left|441\pm 4\right|\\ \text{ = }337\text{ Hz}\text{. or 445 Hz}\text{.}\end{array}$

The possible beat frequencies, when A and C are sounded together

  $\begin{array}{l}{f}_{\text{AC}}=\left|f_{\text{A}}-f_{\text{C}}\right|\\ \text{ = }1\text{ or 7}\end{array}$

There two beat frequencies that are possible are $\text{1 and 7}\text{.}$

Conclusion:

The frequency of A is will be,

  $f_{\text{A}}\text{ = }338\text{ Hz}\text{. or 444 Hz}\text{.}$

The frequency of C is

  $f_{\text{C}}\text{= }337\text{ Hz}\text{. or 445 Hz}\text{.}$

The possible beat frequencies between A and C are $\text{1 Hz}\text{and 7}\text{Hz}$