Answer:
Density change, Δρ = 2.4 × 10⁻⁴ kg/m³
Temperature Change, ΔT = 0.0258 K
Velocity Change, Δc = 0.0148 m/s
Explanation:
For sound waves moving through the air,
Pressure and Temperature varies thus
(P₀/P) = (T₀/T)^(k/(k-1))
Where P₀ = initial pressure of air = 101KPa = 101000 Pa
P = final pressure of air due to the change brought about by the moving sound wave = 101000+30 = 101030 Pa
T₀ = initial temperature of air = 30°C = 303.15 K
T = final temperature of air = ?
k = ratio of specific heats = Cp/Cv = 1.4
(101000/101030) = (303.15/T)^(1.4/(1.4-1))
0.9990703 =(303.15/T)^(3.5)
Solving This,
T = 303.1758 K
ΔT = T - T₀ = 303.1758 - 303.15 = 0.0258 K
Density can be calculate in two ways,
First method
Δρ = ρ - ρ₀
P₀ = ρ₀RT₀
ρ₀ = P₀/RT₀
R = gas constant for air = 287 J/kg.k
where all of these are values for air before the wave propagates
P₀ = 101000 Pa, R = 287 J/kg.K, T₀ = 303.15K
ρ₀ = 101000/(287 × 303.15) = 1.1608655 kg/m³
ρ = P/RT
P = 101030 Pa, T = 303.1758K
ρ = 101030/(287×303.1758) = 1.1611115 kg/m³
Δρ = ρ - ρ₀ = 1.1611115 - 1.1608655 = 0.00024 kg/m³ = 2.4 × 10⁻⁴ kg/m³
Second method
(ρ₀/ρ) = (T₀/T)^(1/(k-1))
Where ρ₀ is initially calculated from ρ₀ = P₀/RT₀, then ρ is then computed and the diff taken.
Velocity Change
c₀ = √(kRT₀) = √(1.4 × 287 × 303.15) = 349.00669 m/s
c = √(kRT) = √(1.4 × 287 × 303.1758) = 349.0215415 m/s
Δc = c₀ - c = 349.0215415 - 349.00669 = 0.0148 m/s
QED!
