The surface subsidence at a horizontal distance \( X \) from the centerline of a cylindrical tunnel is given by \( S = S_{\text{max}} \times \exp\left(\frac{-X^2}{0.5H^2}\right) \), where \( S_{\text{max}} \) is the maximum subsidence above the tunnel and \( H = 50 \) m is the depth of the tunnel centerline. The ratio of surface subsidence at \( X = 10 \) m to that at \( X = 20 \) m is .............
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When dealing with exponential decay, the ratio of subsidence can be found by taking the exponential of the difference in the terms of the equation.
Given the formula for surface subsidence:
\[
S = S_{\text{max}} \times \exp\left(\frac{-X^2}{0.5H^2}\right)
\]
We need to find the ratio of surface subsidence at \( X = 10 \) m to that at \( X = 20 \) m. Step 1: Write the formula for subsidence at \( X = 10 \) m and \( X = 20 \) m:
For \( X = 10 \) m:
\[
S_1 = S_{\text{max}} \times \exp\left(\frac{-(10)^2}{0.5 \times 50^2}\right)
= S_{\text{max}} \times \exp\left(\frac{-100}{0.5 \times 2500}\right)
= S_{\text{max}} \times \exp\left(\frac{-100}{1250}\right)
= S_{\text{max}} \times \exp\left(-0.08\right)
\]
For \( X = 20 \) m:
\[
S_2 = S_{\text{max}} \times \exp\left(\frac{-(20)^2}{0.5 \times 50^2}\right)
= S_{\text{max}} \times \exp\left(\frac{-400}{1250}\right)
= S_{\text{max}} \times \exp\left(-0.32\right)
\]
Step 3: Calculate the exponential term:
\[
\exp\left(0.24\right) \approx 1.271
\]
Thus, the ratio of surface subsidence at \( X = 10 \) m to that at \( X = 20 \) m is approximately 0.7 after rounding to one decimal place.
\[
\boxed{0.7}
\]
