Chapter 16: Problem 9
An oscillator vibrating at 1250 Hz produces a sound wave that travels through an ideal gas at 325 m/s when the gas temperature is 22.0\(^\circ\)C. For a certain experiment, you need to have the same oscillator produce sound of wavelength 28.5 cm in this gas. What should the gas temperature be to achieve this wavelength?
Short Answer
Step by step solution
Understand the Problem
Recall the Wave Equation
Convert and Set Known Values
Calculate the Required Speed for Desired Wavelength
Use the Speed of Sound Equation in Gases
Determine the Temperature Change
Convert to Celsius
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Wave Equation
- \( v \) stands for the speed of the wave.
- \( f \) represents the frequency of the wave, which is how often the wave oscillates.
- \( \lambda \) is the wavelength, the distance between successive crests or troughs of a wave.
The equation not only applies to sound waves but also to all types of waves, including light waves and water waves, making it an especially versatile equation in the study of wave phenomena.
Speed of Sound
For gases, the speed of sound is governed by the equation:\[ v = \sqrt{\gamma \frac{R T}{M}} \]where
- \( \gamma \) is the adiabatic index, a constant specific to the gas.
- \( R \) is the universal gas constant.
- \( T \) is the temperature in Kelvin.
- \( M \) is the molar mass of the gas.
Gas Temperature Effect
To quantify this, the relationship between speed and temperature can be expressed as:\[ \frac{v_1}{v_2} = \sqrt{\frac{T_1}{T_2}} \]Here, we see that if the initial speed of sound \( v_1 \) and the final speed \( v_2 \) are known, along with the initial temperature \( T_1 \), one can solve for the final temperature \( T_2 \) in Kelvin.
This relationship is vital in many scientific and industrial processes, where controlling the temperature can precisely alter how fast sound waves travel through a medium. It's also crucial in environmental studies and meteorology, as temperature variations can affect sound propagation in the atmosphere.