Question

One of the harmonic frequencies of tube Awith two open ends is 325Hz. The next-highest harmonic frequency is 390Hz. (a) What harmonic frequency is next highest after the harmonic frequency 195Hz? (b) What is the number of this next-highest harmonic? One of the harmonic frequencies of tube Bwith only one open end is 1080Hz. The next-highest harmonic frequency is 1320Hz. (c) What harmonic frequency is next highest after the harmonic frequency 600Hz? (d) What is the number of this next-highest harmonic?

Short answer

1. The harmonic frequency next highest after the harmonic frequency 195Hz is 260Hz.
2. The number of this next highest harmonics is 4.
3. The harmonic frequency next highest after the harmonic frequency 600Hz is 840Hz.
4. The number of this next highest harmonics is 7.

Step-by-step solution

Identification of given data

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

Significance of frequency

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,$f=\frac{f_n-f_{n-1}}{2}$

(a) Determining the harmonic frequency

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.

(b) Determining the number of higher harmonics

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.

(c) Determining the harmonic frequency of B

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:

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

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

d) Determining the number of harmonics of B

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.