Problem Statement
Find the speed of the sound wave in air at sea-level where the pressure and temperature are 9.81 N/cm² (abs.) and 20°C respectively. Take R = 287 J/kg K and k = 1.4.
Given Data & Constants
- Absolute Pressure, \(P = 9.81 \, \text{N/cm}^2\) (Note: Not required for calculation)
- Temperature, \(T = 20^\circ\text{C}\)
- Gas constant, \(R = 287 \, \text{J/kg K}\)
- Adiabatic index, \(k = 1.4\)
Solution
1. Convert Temperature to Absolute Scale (Kelvin)
The formula for the speed of sound requires the temperature to be in Kelvin.
$$ T \, (\text{K}) = T \, (^\circ\text{C}) + 273.15 $$
$$ T = 20 + 273.15 = 293.15 \, \text{K} $$
2. Calculate the Speed of Sound (c)
The speed of sound in an ideal gas is a function of its properties and temperature.
$$ c = \sqrt{k R T} $$
$$ c = \sqrt{1.4 \times 287 \times 293.15} $$
$$ c = \sqrt{117753.87} \approx 343.15 \, \text{m/s} $$
Final Result:
The speed of the sound wave is approximately 343.15 m/s.
Explanation of the Speed of Sound
The speed of sound is the speed at which a small pressure disturbance (a sound wave) propagates through a medium. For an ideal gas like air, this speed is not dependent on the overall pressure of the gas, but rather on its temperature and its inherent properties.
- Temperature (T): Higher temperature means the gas molecules are moving faster and have more energy, allowing them to transmit the pressure wave more quickly.
- Adiabatic Index (k) and Gas Constant (R): These values relate to the gas's compressibility and molar mass, which also affect how quickly a wave can travel through it.
