Question

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.

Short answer

The vibrational frequency of the string is \(347\;{\rm{Hz}}\).

Step-by-step solution

Determination of the beat frequency

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 information

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 .

Evaluation of the vibrational frequency of the string

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 \(347\;{\rm{Hz}}\).