Question

When played in a certain manner, the lowest resonant frequency of a certain violin string is concert A (440 Hz). What is the frequency of the (a) second and (b) third harmonic of the string?

Short answer

1. Frequency of second harmonic is 880 Hz
2. Frequency of third harmonic is 1320 Hz

Step-by-step solution

Given data 

Lowest frequency is$f_1=440\mathrm{Hz}$

Understanding the concept of fundamental frequency

We have to usetheformula for nth harmonic frequency which is n timesthefundamental frequency.

Formula:

The frequency of oscillation of n-loops, $f_n=n\times \frac{v}{2L}........\left(1\right)$

Step 3(a): Calculation for second harmonic frequency

The lowest frequency is given. It means that the first harmonic frequency is 440 Hz , and using equation (1) and the frequency, it is given as:

$440=\frac{v}{2L}......\left(2\right)$

Now the second harmonic frequency using equations (1) and (2) is given as follows:

role="math" localid="1660993667219" $$\begin{gathered} f_2=2\times \frac{v}{2L} \\ =2\times 440 \\ =880\mathrm{Hz} \end{gathered}$$

Hence, the value of second-harmonic frequency is 880 Hz

Step 4(b): Calculation of third harmonic frequency

Third harmonic using equations (1) and (2) is given as follows:

$$\begin{gathered} f_3=3\times \frac{v}{2L} \\ =3\times 440 \\ =1320\mathrm{Hz} \end{gathered}$$

Hence, the value of third-harmonic frequency is 1320 Hz