To find the time it takes for sound waves to travel from point A to point B when the air temperature varies linearly from T1 to T2, we'll first need to determine the average velocity of sound in that temperature gradient. Then, we can use this average velocity to calculate the time taken for the journey.
Let's break down the steps:
- Average velocity of sound (v_avg) between points A and B: Since the temperature varies linearly between T1 and T2, we can find the average temperature (T_avg) between the two points: T_avg = (T1 + T2) / 2
Now, we can find the corresponding average velocity (v_avg) using the given velocity-temperature relationship: v_avg = α * T_avg^(1/2)
Distance between points A and B: Assuming points A and B are not at the same location, we need to know the distance between them (d_AB). This will be crucial for calculating the time taken.
Time taken (t) for sound waves to travel from A to B: Using the average velocity (v_avg) and the distance between A and B (d_AB), we can calculate the time (t) taken for the sound waves to travel:
t = d_AB / v_avg
Now that we have all the information, we can proceed with the calculations. Keep in mind that the units used for temperature, velocity, distance, and time must be consistent.
Example: Let's consider an example where T1 = 20°C, T2 = 30°C, α = 0.606 m/s√°C, and the distance between A and B (d_AB) is 500 meters.
Average temperature (T_avg): T_avg = (T1 + T2) / 2 T_avg = (20 + 30) / 2 T_avg = 25°C
Average velocity of sound (v_avg): v_avg = α * T_avg^(1/2) v_avg = 0.606 * √(25) v_avg = 0.606 * 5 v_avg = 3.03 m/s
Time taken (t): t = d_AB / v_avg t = 500 / 3.03 t ≈ 165.02 seconds
So, in this example, it would take approximately 165.02 seconds for sound waves to travel from point A to point B, considering the linear variation in temperature between the two points.