Question

If fundamental frequency of an open pipe is 200 Hz, then find the fundamental frequency if one end is closed.

Hint:

A highly useful shortcut to memorize is that closing one end of an open pipe immediately halves its fundamental frequency!

Answer 1

Step 1: Understanding the Concept:
A pipe perfectly open at both ends supports a standing wave pattern with antinodes at both the boundaries.
If one end of the exact same pipe is firmly closed, the wave pattern fundamentally shifts to having a node at the closed boundary and an antinode at the open boundary, effectively altering the supported resonant wavelengths.
Step 2: Key Formula or Approach:
The fundamental frequency $f_{\text{open}}$ of a pipe open at both ends is given by $f_{\text{open}} = \frac{v}{2L}$.
The fundamental frequency $f_{\text{closed}}$ of a pipe closed at one single end is given by $f_{\text{closed}} = \frac{v}{4L}$.
Step 3: Detailed Explanation:
By strictly observing the formulas, we can easily establish a direct relationship between the two frequencies.
Since $f_{\text{closed}} = \frac{v}{4L}$ and $f_{\text{open}} = \frac{v}{2L}$, we can write:
\[ f_{\text{closed}} = \frac{1}{2} \left( \frac{v}{2L} \right) = \frac{1}{2} f_{\text{open}} \] The problem states that the initial open pipe fundamental frequency is $f_{\text{open}} = 200\text{ Hz}$.
Substituting this into our derived relationship gives:
\[ f_{\text{closed}} = \frac{200}{2} = 100\text{ Hz} \] Step 4: Final Answer:
The fundamental frequency of the closed pipe is $100\text{ Hz}$.