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Chapter 9: Problem 298

A cylindrical tube, open at both ends has a fundamental frequency, \(f\) in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air-column is now (B) \(\frac{f}{2}\) (C) \(\frac{3 f}{4}\) (A) \(\bar{f}\) (D) \(2 f\)

Short Answer

Expert verified
(D) \(2 f\)

Step by step solution

01

Identify the frequency relationship

The natural frequency, f, of a tube open on both ends (an open tube), is represented by the equation: \(f = \frac{v}{2L}\) where v is the speed of sound, and L is the length of the tube. If we half the length of the tube by submerging half of it in water, the new frequency \(f'\) of the tube becomes: \(f' = \frac{v}{2L'}\), where \(L'\) is the new length of the tube. \ Now, the fraction \( \frac{f'}{f} = \frac{2L}{2L'}\). Since we halved the length of the tube, \(L'\) is \( \frac{L}{2}\). Substituting we get, \( \frac{f'}{f} = \frac{2L}{2 * \frac{L}{2}} = 2 \)
02

Answer

The fundamental frequency doubles when the tube is filled halfway with water. \ Hence, the answer is (D) \(2 f\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Acoustics
Acoustics, the branch of physics concerned with the properties of sound, plays a crucial role in understanding the behavior of sound waves in various environments. The fundamental frequency, or the lowest frequency at which a system naturally resonates, is a key concept in acoustics. Imagine a guitar string; when plucked, it vibrates to produce a sound at a certain frequency, which is its fundamental frequency.

In the context of the textbook exercise, acoustic principles help us analyze the changes in the fundamental frequency of an open cylindrical tube when its length is altered by submerging it in water. Sound waves in the tube are affected by the length of the air column; a shorter column produces a higher pitch, which is why the fundamental frequency increases when the tube is half-filled with water.
Harmonic Waves
Harmonic waves are periodic waves that can be described by sinusoidal functions, and they are found in many physical phenomena, especially in sound and music. In our textbook exercise, harmonic waves are demonstrated by the air vibrations within the tube. These vibrations create standing waves composed of nodes and antinodes, where the nodes represent points of no movement, and antinodes are points of maximum movement.

Understanding harmonics is key to grasping how an instrument like the tube in the exercise can have a fundamental frequency alongside higher frequencies, known as harmonics or overtones. When we change the effective length of the tube by submerging it in water, we alter the positions of these nodes and antinodes, therefore changing the fundamental frequency and harmonic series.
Speed of Sound in Air
The speed of sound in air is a constant value that significantly impacts the way sound travels and is perceived. It is determined by the properties of the air, such as temperature, humidity, and pressure, and is approximately 343 meters per second at room temperature. In our exercise, the speed of sound is denoted by the variable 'v' and remains constant regardless of the length of the tube.

When solving problems related to sound, such as calculating the fundamental frequency of the tube, it's essential to remember this constancy of the speed of sound. It allows us to focus on the relationship between the tube's length and its frequency without having to consider variations in the speed of propagation of the waves.

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Most popular questions from this chapter

A particle vibrates in SHM along a straight line. Its greatest acceleration is \(5 \pi^{2} \mathrm{~cm} \mathrm{~s}^{-2}\) and its distance from the equilibrium position is \(4 \mathrm{~cm}\), the velocity of the particle is \(3 \pi \mathrm{cms}^{-1}\), then (A) The amplitude is \(10 \mathrm{~cm}\) (B) The period of oscillation \(2 \mathrm{~s}\) (C) The amplitude is \(5 \mathrm{~cm}\) (D) The period of oscillation \(4 \mathrm{~s}\)

For a certain stretched string, three consecutive resonance frequencies are observed as \(105,175,245 \mathrm{~Hz}\), respectively. Then select the correct alternatives (A) The string is fixed at both ends. (B) The string is fixed at one end only. (C) The fundamental frequency is \(35 \mathrm{~Hz}\). (D) The fundamental frequency is \(52.5 \mathrm{~Hz}\).

The track followed for two perpendicular SHMs is a perfect ellipse when \(\left(\delta\right.\)-phase difference, \(A_{1}, A_{2}\) amplitudes) (A) \(\delta=\frac{\pi}{4}, A_{1} \neq A_{2}\) (B) \(\delta=\frac{3 \pi}{4}, A_{1} \neq A_{2}\) (C) \(\delta=\frac{\pi}{2}, A_{1} \neq A_{2}\) (D) \(\delta=\pi, A_{1}=A_{2}\)

The equation of a wave on a string of linear mass density \(0.04 \mathrm{~kg} \mathrm{~m}^{-1}\) is given by $$ y=0.02(m) \sin \left[2 \pi\left(\frac{t}{0.04(s)}-\frac{x}{0.50(m)}\right)\right] \text {. The ten- } $$ sion in the string is (A) \(4.0 \mathrm{~N}\) (B) \(12.5 \mathrm{~N}\) (C) \(0.5 \mathrm{~N}\) (D) \(6.25 \mathrm{~N}\)

A pipe open at both ends has a fundamental frequency \(f\) in air. The pipe is dipped vertically in water so that half of it is in water. The fundamental frequency of the air column is now (A) \(\frac{3 f}{4}\) (B) \(2 f\) (C) \(f\) (D) \(\frac{f}{2}\)
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