A Thermometer. Suppose you have a tube of length L...

A Thermometer. Suppose you have a tube of length $L$ containing a gas whose temperature you want to take, but you cannot get inside the tube. One end is closed, and the other end is open but a small speaker producing sound of variable frequency is at that end. You gradually increase the frequency of the speaker until the sound from the tube first becomes very loud. With further increase of the frequency, the loudness decreases but then gets very loud again at still higher frequencies. Call $f_{0}$ the lowest frequency at which the sound is very loud. (a) Show that the absolute temperature of this gas is given by $T=16 M L^{2} f_{0}^{2} / \gamma R,$ where $M$ is the molar mass of the gas, $\gamma$ is the ratio of its heat capacities, and $R$ is the ideal gas constant. (b) At what frequency above $f_{0}$ will the sound from the tube next reach a maximum in loudness? (c) How could you determine the speed of sound in this tube at temperature $T ?$

Key Concepts

Acoustic Thermometry

Acoustic thermometry is a technique of determining the temperature of a gaseous medium by measuring the speed of sound within it. Since the speed of sound depends on temperature, resonance frequencies in a controlled system (like a tube where standing waves are set up) can be used to calculate the temperature. This approach leverages the interplay between the resonant conditions in the acoustic system and the thermodynamic properties of the gas.

Speed of Sound in Gases

The speed of sound in a gas is fundamentally linked to the medium's properties, such as temperature, molar mass, and heat capacity ratios. It is typically expressed by the relation v = ?(?RT/M), where ? is the ratio of specific heat capacities, R is the ideal gas constant, T is the absolute temperature, and M is the molar mass. This relationship allows quantification of how sound propagation changes with thermodynamic conditions.

Resonance in Acoustic Systems

Resonance refers to the phenomenon where a system oscillates with maximum amplitude at specific frequencies called resonant frequencies. In acoustic systems, resonance is achieved when the sound waves reflected at the boundaries interfere constructively with the incoming waves, leading to large amplitude vibrations. This concept allows the determination of intrinsic properties of the system by identifying these resonant frequencies.

Standing Waves

Standing waves occur when two waves of the same frequency and amplitude travel in opposite directions and interfere, creating fixed nodes (points of zero amplitude) and antinodes (points of maximum amplitude). In an enclosed tube, the formation of standing waves depends on the boundary conditions, such as whether the ends are open or closed, and determines the resonant frequencies. This explains why certain frequencies produce significantly louder sounds than others.

Boundary Conditions in Waveguides

The boundary conditions of a tube, which dictate whether the ends are open or closed, affect how waves are reflected and thereby set up standing wave patterns. For instance, a closed end forces a node, while an open end typically corresponds to an antinode. Understanding these conditions is essential for predicting the wavelengths and frequencies at which resonance will occur in the system.

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