A pipe of length L closed at one end is located along x-axis...

A pipe of length $L$ closed at one end is located along $x$-axis with closed end at origin and open end at $(l,$, 0). The pipe resonates in its $n^{\text {th }}$ overtone with maximum amplitude of air molecules to be equal to $a_{0}$. Calculate the $x$-co-ordinates of those points, where maximum pressure change $\left(\Delta P_{m}\right)$ occurs and calculate $\left(\Delta P_{m}\right) .$ Density of air is equal to $\rho$ and velocity of sound in air is $v$.

A pipe of length $L$ closed at one end is located along $x$-axis with closed end at origin and open end at $(l,$,
0). The pipe resonates in its $n^{\text {th }}$ overtone with maximum amplitude of air molecules to be equal to $a_{0}$. Calculate the $x$-co-ordinates of those points, where maximum pressure change $\left(\Delta P_{m}\right)$ occurs and calculate $\left(\Delta P_{m}\right) .$ Density of air is equal to $\rho$ and velocity of sound in air is $v$.

Key Concepts

Pressure Amplitude Calculation

The maximum pressure change in a standing wave can be calculated by differentiating the displacement amplitude function. For a sinusoidal displacement profile in a resonant air column, this results in an expression for the pressure amplitude that involves the density of the air, the square of the velocity of sound, the maximum displacement amplitude, and the wave number associated with the particular mode. This ties together the physical parameters of the system with the geometry of the standing wave.

Relationship Between Displacement and Pressure

In an acoustic wave, the pressure variation is proportional to the spatial derivative of the displacement of the air molecules. This means that where the displacement changes most rapidly (typically at the nodes of the displacement pattern), the pressure variation reaches a maximum. This relationship is derived from the basic equations of fluid dynamics for sound waves.

Standing Waves

Standing waves form when waves of identical frequency and amplitude traveling in opposite directions interfere, resulting in fixed positions along the medium where the amplitude is always zero (nodes) or maximal (antinodes). This phenomenon is central to understanding acoustic resonance in pipes and other systems.

Boundary Conditions in Air Columns

Air columns, such as those in pipes, have boundary conditions that determine how standing waves develop. In a pipe closed at one end and open at the other, the closed end must be a displacement node (where air molecules cannot move) leading to a pressure antinode (maximum pressure variation), while the open end is a displacement antinode (where the particles move most freely) corresponding to a pressure node (minimal pressure change).

Resonance and Overtones in Pipes

Resonance occurs when the dimensions of the pipe support standing waves at certain frequencies, known as the resonant frequencies. Overtones are the higher-order resonant frequencies beyond the fundamental frequency. In a pipe with one closed end, the allowed resonant modes correspond to wavelengths that satisfy the condition imposed by the boundary conditions, and these modes appear as odd harmonics of the fundamental frequency.

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