In a space of width d , the absolute temperature of air linearly increases from T₁ to T₂ . Determine the time taken by a sound wave to travel through this region in terms of T₁ , T₂ , d , and the speed v of sound at 273 K . Evaluate this time for T₁ = 280 K , T₂ = 310 K , d = 33 m , and v = 330 m s ⁻¹ .

Step-by-step solution

The speed of sound at an absolute temperature T is given by: v_T = v T 273 Since the temperature increases linearly with distance x from 0 to d , the temperature at a distance x is: T(x) = T₁ + T₂ - T₁ d x Differentiating with respect to x gives: dT = T₂ - T₁ d dx dx = d T₂ - T₁ dT The time taken by the sound wave to travel a small distance dx is dt = dx v_T . Integrating from x = 0 to x = d (which corresponds to T = T₁ to T = T₂ ): t = _ T₁ ^ T₂ d T₂ - T₁ 273 v T dT t = d 273 v (T₂ - T₁) _ T₁ ^ T₂ T^ -1/2 dT t = d