A crude approximation of voice production is to consider the...

A crude approximation of voice production is to consider the breathing passages and mouth to be a resonating tube closed at one end. (See Figure $17.32 . )($ a) What is the fundamental frequency if the tube is $0.240-m$ long, by taking air temperature to be $37.0^{\circ} C?$ (b) What would this frequency become if the person replaced the air with helium? Assume the same temperature dependence for helium as for air.

Key Concepts

Resonance in Closed-Ended Tubes

In a tube closed at one end and open at the other, standing waves form with a displacement node at the closed end and an antinode at the open end. The fundamental mode corresponds to a quarter of the wavelength fitting into the tube. This principle underpins how the physical dimensions of the tube determine its resonant frequencies.

Fundamental Frequency Calculation

The fundamental frequency of a resonating tube is calculated using the relation that the speed of sound is equal to four times the product of the frequency and the tube's length. This equation arises because the lowest mode of vibration in a closed tube has a wavelength that is four times the length of the tube, linking the geometric properties directly to the vibrational frequency.

Speed of Sound and Temperature Dependence

The speed of sound in a gas increases with temperature because the kinetic energy of the gas molecules increases, allowing sound waves to travel faster. The dependence of the speed of sound on temperature is typically expressed through formulas that incorporate the absolute temperature, which is crucial for accurate resonance frequency predictions in different thermal conditions.

Effect of Gas Composition on Sound Propagation

The properties of a gas, such as its density and molecular weight, significantly affect the speed of sound within it. When the gas is changed—such as replacing air with helium—the speed of sound alters accordingly, resulting in a shift in the resonant frequencies. Understanding how different gases influence sound propagation is key to accurately predicting changes in acoustic behavior.

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