Question: (II) A guitar string produces 3 beats/s when sounded with a 350-Hz tuning fork and 8 beats/s when sounded with a 355-Hz tuning fork. What is the vibrational frequency of the string? Explain your reasoning.

The beat frequency is the difference in frequencies of two overlapping sound waves. The equation can be given as \({f_B} = \left| {{f_2} \pm {f_1}} \right|\).

Given data:

The beat frequency at 350 Hz tuning fork frequency is 3 beats/s .

The beat frequency at \(355\;{\rm{Hz}}\) tuning fork frequency is 8 beats/s .

When the guitar string produces \(3\;{\rm{beats/s}}\) at \(350\;{\rm{Hz}}\) tuning fork frequency, the frequency of the other string can be calculated as:

\(\begin{array}{c}{f_B} = \left| {{f_2} \pm {f_1}} \right|\\\left( {3\;{\rm{beats/s}}} \right) = \left| {350\;{\rm{Hz}} \pm {f_1}} \right|\\{f_1} = 347\;{\rm{Hz}}\;{\rm{or}}\;353\;{\rm{Hz}}\end{array}\)

Similarly, when the guitar string produces \(8\;{\rm{beats/s}}\) at \(355\;{\rm{Hz}}\) tuning fork frequency, the frequency of the other string can be calculated as:

\(\begin{array}{c}{f_B} = \left| {{f_2} \pm {f_1}} \right|\\\left( {8\;{\rm{beats/s}}} \right) = \left| {355\;{\rm{Hz}} \pm {f_1}} \right|\\{f_1} = 347\;{\rm{Hz}}\;{\rm{or}}\;363\;{\rm{Hz}}\end{array}\)

From the above-calculated frequencies, the frequency \(347\;{\rm{Hz}}\) works in both scenarios. Hence, the vibrational frequency of the string is \(347\;{\rm{Hz}}\).