Question

If we study the vibration of a pipe open at both ends, then the following statement is not true: (1) Fundamental frequency will be inversely proportional to the length of pipe (2) All harmonics of the fundamental frequency will be generated (3) Pressure change will be maximum at both ends (4) Open end will be displacement antinode

Short answer

The incorrect statement is (3) Pressure change will be maximum at both ends.

Step-by-step solution

Understand the fundamental frequency

For a pipe open at both ends, the fundamental frequency is given by the equation: \( f = \frac{v}{2L} \). Here, \( v \) is the speed of sound and \( L \) is the length of the pipe. The fundamental frequency is indeed inversely proportional to the length of the pipe.

Identify harmonics

A pipe open at both ends supports all harmonics. The frequencies of the harmonics are given by: \( f_n = n \cdot f \), where \( n \) is an integer (n=1, 2, 3,...), and \( f \) is the fundamental frequency.

Analyze pressure changes

In a pipe open at both ends, the pressure change is zero at the ends because they are open to the atmosphere. Therefore, pressure change is minimum (not maximum) at both ends.

Understand displacement nodes and antinodes

An open end is a displacement antinode where the air molecules can move freely. This is true for both ends of the pipe since both are open.

Determine the incorrect statement

Based on the analysis: (1), (2), and (4) are correct for a pipe open at both ends, but (3) is not true because the pressure change is minimum at both ends.

Key concepts

Fundamental Frequency

The fundamental frequency of a pipe open at both ends is a crucial concept in understanding the vibration behavior of the pipe. The fundamental frequency can be defined by the equation: \( f = \frac{v}{2L} \), where \( v \) is the speed of sound in the air, and \( L \) is the length of the pipe. This relationship means that the fundamental frequency is inversely proportional to the length of the pipe. In other words, if you make the pipe longer, the fundamental frequency will decrease. Conversely, if the pipe is shorter, the fundamental frequency will increase. This concept is essential for musical instruments like flutes and organ pipes, where changing the length alters the pitch.

Harmonics

Harmonics are the multiples of the fundamental frequency. They are critical for understanding the complex sounds produced by vibrating systems like pipes. In a pipe open at both ends, all harmonics are supported. The frequency of the nth harmonic is given by: \( f_n = n \times f \), where \( n \) is any positive integer (1, 2, 3, ...), and \( f \) is the fundamental frequency. For example, if the fundamental frequency (first harmonic) is 100 Hz, the second harmonic will be 200 Hz, the third will be 300 Hz, and so on. This pattern is why musical instruments produce rich and complex sounds, instead of a single pure tone.

Pressure Change

Understanding pressure change in a vibrating pipe is essential. At the open ends of the pipe, the pressure change is zero because they are exposed to the atmosphere. This means that the pressure variation is minimum at both ends of the pipe. Although it might initially seem counterintuitive, this behavior aligns with the wave dynamics inside the pipe. When a sound wave travels through the pipe, its pressure fluctuations must average out to zero at the openings to join seamlessly with the external atmospheric pressure. Hence, the pressure change is not maximum at the ends—it's actually the opposite.

Displacement Nodes and Antinodes

In the context of a vibrating pipe open at both ends, displacement nodes and antinodes are critical concepts. A displacement antinode is a point where the displacement of air molecules is at its maximum, and in an open pipe, antinodes are located at the open ends. This means that at both open ends, the air molecules can move freely and experience maximum displacement. Conversely, a displacement node is a point where the displacement is zero. In a pipe open at both ends, displacement nodes occur within the pipe, spaced evenly between the antinodes. Understanding where nodes and antinodes are located helps in visualizing and predicting the patterns of sound wave vibrations inside the pipe.