91Ó°ÊÓ

1. One of the harmonic frequencies of A, (f_a1) = 325Hz.
2. The next highest frequency of A, (f_a2) = 390Hz
3. One of the frequencies of B, (f_b1) = 1080Hz
4. The next highest frequency of B, (f_b2) = 1320Hz
5. One of the frequencies of A, (f_a1) = 195Hz
6. One of the frequencies of B, (f_b1) = 600Hz

The number of waves passing a fixed location in a unit of time is referred to as frequency in physics.

We can find the fundamental frequency from two given successive frequencies for a pipe open at both ends. Using it, we can easily find the harmonic frequency next highest after the harmonic frequency 195 Hz. The ratio of the given frequency with the first harmonic frequency will give several modes. Similarly, we can answer parts c) and d) using formulae for pipe closed at one end.

Formula:

For a pipe open at both ends,

1. f = f_n - f_n-1
2. The number of harmonics, f_n / f
3. For pipe open at one end,$f=\frac{f_n-f_{n-1}}{2}$

Using equation (i) from the formula, the first harmonic frequency of A for both ends open is given as:

$\left(f_a\right)=f_{a2}-f_{a1}$

Therefore, the first harmonic frequency of A is given as:

$\left(f_a\right)=f_{a2}-f_{a1}$

The next highest frequency after 195 Hz is: 195Hz+65Hz = 260Hz

Hence, the value of the harmonic frequency is 260Hz.

The number of harmonic frequencies of A is given using formula (ii) as follows:

$\begin{array}{rcl}{N}_A& =& \frac{260Hz}{65Hz}\\ & =& 4\end{array}$

Hence, the number of harmonic frequencies is 4.

Using equation (iii) from the formula, the first harmonic frequency of B for one end open is given as:$\left(f_b\right)=\left(f_{b2}-f_{b1}\right)/2$

The first harmonic frequency of B:

$\begin{array}{rcl}{f}_b& =& \frac{1}{2}\left(1320Hz-1080Hz\right)\\ & =& \frac{240Hz}{2}\\ & =& 120Hz\end{array}$

The next highest frequency after 600Hz is given as:

$\begin{array}{rcl}\left(f_{b2}^{\text{'}}\right)& =& f_{b2}^{\text{'}}+f_b\\ & =& 600Hz+2×120Hz\\ & =& 840Hz\end{array}$

Hence, the value of the harmonic frequency is 840Hz.

The number of harmonic frequencies of B is given using formula (ii) as follows:

$\begin{array}{rcl}\left(N_B\right)& =& \frac{840Hz}{120Hz}\\ & =& 7\end{array}$

Hence, the number of harmonics is 7.