Consider a seismic source located at the Earth's surface. A vertical well is located at a distance of 1 km away from the seismic source, and a geophone is suspended inside the well at a depth of 800 m, as shown in the figure below. The travel time of the primary reflected wave from a horizontal reflector located at a depth of 2 km is ___________ s (rounded off to three decimal places).
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Use the image-source (mirror) method for a horizontal reflector: reflect the source (or receiver) across the reflector depth and take a straight-line path to the actual receiver (or source), then divide by velocity.
Alternative — find the actual reflection point on the interface using the similar-triangles (unfolded-path) construction, instead of the image-source shortcut.
The source S is a vertical distance \(h_1=2\) km above the reflector, and the receiver R is a vertical distance \(h_2=2-0.8=1.2\) km above the reflector, with a total horizontal separation of \(1\) km between S and R. For a flat reflector, the reflection point divides that horizontal separation in the same ratio as \(h_1:h_2\) (this follows from unfolding the reflected ray into a straight line and using similar triangles):
Now sum the two ray segments directly: \(d_1=\sqrt{x_0^2+h_1^2}=\sqrt{0.625^2+2^2}=\sqrt{0.3906+4}=\sqrt{4.3906}=2.0954\ \text{km}\) \(d_2=\sqrt{(1-x_0)^2+h_2^2}=\sqrt{0.375^2+1.2^2}=\sqrt{0.1406+1.44}=\sqrt{1.5806}=1.2572\ \text{km}\)
Total path length \(=d_1+d_2=2.0954+1.2572=3.3526\) km — identical to the image-source result.
\[ t=\dfrac{3.3526}{3}\approx1.118\ \text{s} \]
Both independent methods agree exactly, confirming the answer sits inside 1.116–1.120 s.
