To solve this problem, we need to understand the harmonic concepts of sound waves in open and closed pipes.
Open Pipes: In an open-ended pipe, both ends are open. The number of nodes (points of zero amplitude) for any harmonic is equal to the harmonic number. Therefore, for an open pipe:
- \(n = \text{harmonic number}\)
Closed Pipes: In a pipe closed at one end, an antinode (point of maximum amplitude) occurs at the open end and a node at the closed end. Only odd harmonics are possible. For the \(k^{\text{th}}\) harmonic, where \(k\) is odd, the number of nodes is:
- \(m = \frac{k + 1}{2}\)
We are given:
- The \(5^{\text{th}}\) harmonic of an open-ended pipe, thus \(n = 5\).
- The \(9^{\text{th}}\) harmonic of the same pipe with one end closed, meaning this is the 5th odd harmonic, so \(m = \frac{9 + 1}{2} = 5\).
Therefore, the ratio \(\frac{n}{m}\) is:
- \(\frac{n}{m} = \frac{5}{5} = 1\)
Thus, the answer is:
- 1
This matches the correct answer in the options provided.
Similar Questions
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Important Questions from Oscillations and Waves
- Two tuning forks $A$ and $B$ are sounded together giving rise to $8$ beats in $2\text{ s}$. When fork $A$ is loaded with wax, the beat frequency is reduced to $4$ beats in $2\text{ s}$. If the original frequency of tuning fork $B$ is $380\text{ Hz}$ then original frequency of tuning fork $A$ is _________ $\text{Hz}$.
- A simple pendulum of string length 30 cm performs 20 oscillations in 10 s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is ___________ cm. [Assume that the mass of the pendulum remains same.]
- The velocity of sound in air is doubled when the temperature is raised from $0^\circ\text{C}$ to $\alpha^\circ\text{C}$. The value of $\alpha$ is ________.
- The equation of a transverse wave is $y = y_0 \sin 2\pi(ft - \frac{x}{\lambda})$. If the maximum particle velocity be four times that of wave velocity then
- The velocity of a particle executing simple harmonic motion along $x$-axis is described as $v^2 = 50 - x^2$, where $x$ represents displacement. If the time period of motion is $\frac{x}{7}~\text{s}$, the value of $x$ is ________.
