Step 1: Understanding the Concept:
When two sound sources of slightly different frequencies are sounded together, they produce beats.
The beat frequency is defined as the absolute difference between the two frequencies.
For an organ pipe open at both ends, the fundamental frequency depends on the velocity of sound and the length of the pipe.
Step 2: Key Formula or Approach:
The fundamental frequency $f$ of an open organ pipe of length $l$ is:
\[ f = \frac{v}{2l} \]
where $v$ is the velocity of sound in air.
The beat frequency $n$ is:
\[ n = |f_1 - f_2| \]
Step 3: Detailed Explanation:
Let the length of the first open pipe be $l_1 = L$.
Its fundamental frequency is:
\[ f_1 = \frac{v}{2L} \]
Let the length of the second open pipe be $l_2 = L + L_1$.
Its fundamental frequency is:
\[ f_2 = \frac{v}{2(L + L_1)} \]
Since $L<L + L_1$, the frequency $f_1$ will be greater than $f_2$.
The beat frequency $n$ produced when they are sounded together is:
\[ n = f_1 - f_2 \]
Substitute the expressions for the frequencies:
\[ n = \frac{v}{2L} - \frac{v}{2(L + L_1)} \]
Factor out the common term $\frac{v}{2}$:
\[ n = \frac{v}{2} \left[ \frac{1}{L} - \frac{1}{L + L_1} \right] \]
Find a common denominator to subtract the fractions:
\[ n = \frac{v}{2} \left[ \frac{(L + L_1) - L}{L(L + L_1)} \right] \]
Simplify the numerator:
\[ n = \frac{v}{2} \left[ \frac{L_1}{L(L + L_1)} \right] \]
\[ n = \frac{vL_1}{2L(L + L_1)} \]
Step 4: Final Answer:
The beat frequency will be $\frac{vL_1}{2L(L+L_1)}$.