Step 1: Understanding the Concept:
When an air column inside a pipe closed at one end vibrates in its fundamental mode, it forms a stationary wave.
The simplest standing wave pattern has an antinode at the open end and a node at the closed end.
This pattern encompasses exactly one-quarter of a full wavelength.
Step 2: Key Formula or Approach:
For a pipe of length \(L\) closed at one end, the fundamental wavelength \(\lambda\) is given by:
\[ L = \frac{\lambda}{4} \implies \lambda = 4L \]
The frequency (\(f\)) of the wave is related to wave speed (\(v\)) and wavelength (\(\lambda\)) by the equation:
\[ f = \frac{v}{\lambda} \]
Additionally, the speed of the wave is the distance traveled divided by the time taken. If it takes time \(t\) to travel the length \(L\) of the pipe:
\[ v = \frac{L}{t} \]
Step 3: Detailed Explanation:
The problem states that the sound wave travels from the open end to the closed end in time '\(t\)'.
The distance it covers is the length of the pipe, \(L\).
Thus, we can express the speed of sound as \(v = \frac{L}{t}\).
We also established that for the fundamental mode, the wavelength is \(\lambda = 4L\).
Substitute these expressions for \(v\) and \(\lambda\) into the fundamental frequency formula:
\[ f = \frac{v}{\lambda} = \frac{\left( \frac{L}{t} \right)}{4L} \]
Simplifying the resulting fraction:
\[ f = \frac{L}{t \times 4L} \]
The \(L\) terms cancel out from the numerator and the denominator:
\[ f = \frac{1}{4t} \]
This can be expressed using a negative exponent format as found in the options:
\[ f = (4t)^{-1} \]
Step 4: Final Answer:
The frequency of vibration is \((4t)^{-1}\).