Step 1: Understanding the Concept:
The fundamental frequency of an open organ pipe depends on the speed of sound in the air inside it and the length of the pipe.
When two pipes are joined in series, they form a single, longer open pipe whose length is the sum of the individual lengths.
Step 2: Key Formula or Approach:
The fundamental frequency $f$ of an open pipe of length $L$ is given by:
\[ f = \frac{v}{2L} \]
From this, we can express length in terms of frequency:
\[ L = \frac{v}{2f} \]
When joined in series, the new total length is $L_{\text{series}} = L_1 + L_2$.
The new fundamental frequency will be $f_{\text{series}} = \frac{v}{2L_{\text{series}}}$.
Step 3: Detailed Explanation:
Let the lengths of the two pipes be $L_1$ and $L_2$.
Their fundamental frequencies are:
$f_1 = \frac{v}{2L_1} \implies L_1 = \frac{v}{2f_1}$
$f_2 = \frac{v}{2L_2} \implies L_2 = \frac{v}{2f_2}$
When connected in series, the combined length $L'$ is:
\[ L' = L_1 + L_2 \]
Substitute the expressions for $L_1$ and $L_2$:
\[ L' = \frac{v}{2f_1} + \frac{v}{2f_2} \]
Factor out $\frac{v}{2}$:
\[ L' = \frac{v}{2} \left( \frac{1}{f_1} + \frac{1}{f_2} \right) \]
Find a common denominator:
\[ L' = \frac{v}{2} \left( \frac{f_2 + f_1}{f_1 f_2} \right) \]
The fundamental frequency of this combined series pipe $f'$ is:
\[ f' = \frac{v}{2L'} \]
Substitute the expression we found for $L'$:
\[ f' = \frac{v}{2 \left[ \frac{v}{2} \left( \frac{f_1 + f_2}{f_1 f_2} \right) \right]} \]
The $\frac{v}{2}$ terms cancel out:
\[ f' = \frac{1}{\frac{f_1 + f_2}{f_1 f_2}} \]
\[ f' = \frac{f_1 f_2}{f_1 + f_2} \]
Step 4: Final Answer:
The fundamental frequency of the series combination is $\frac{f_1 f_2}{f_1+f_2}$.