Fundamental Period of a Soil Site

A layer of soft soil sitting on stiff rock is a resonator. Waves come up from below, reflect off the free surface, come back down, reflect off the impedance contrast at the rock, and return. At one particular period the returning wave arrives in step with the next one and the motion builds. That period is $T = 4H/V_s$, and the 4 is a quarter wavelength: a column fixed at the bottom and free at the top holds exactly a quarter of a shear wave in its fundamental mode, so $H = \lambda/4$ and $T = \lambda/V_s = 4H/V_s$. It is the same standing-wave arithmetic as an organ pipe closed at one end.

The reason this matters is that buildings resonate too. A rough but serviceable rule for a framed building is about 0.1 s of natural period per storey, so a ten-storey block sits near one second and a twenty-storey tower near two. When the site period lands on the building period, the ground hands the structure energy at exactly the rate the structure will accept it, and the response can be several times what the same building would see on rock.

Mexico City on 19 September 1985 is the case every textbook uses, and it deserves the space. The earthquake was Mw 8.0, but its epicentre was 350 km away on the Pacific coast — far enough that the shaking in the surrounding hills was moderate. The city centre, though, sits on the bed of the drained Lake Texcoco: 30 to 50 m of extraordinarily soft clay with $V_s$ around 60 to 80 m/s, giving a site period near two seconds. The clay amplified the long-period motion by a factor of ten or more and rang for over a minute. The buildings that collapsed were overwhelmingly the ten- to twenty-storey ones whose own periods matched. Shorter and taller buildings on the same streets survived. Four years later the Loma Prieta earthquake told the same story in the Marina District of San Francisco, on hydraulic fill.

Two caveats keep this honest. $V_s$ is not constant through a real profile, so the practical form uses a travel-time average, $T = 4\sum(h_i/V_{si})$. And the layer softens as shaking strengthens, which LENGTHENS the period during the event — a site that starts at 0.8 s can drift toward 1.2 s while the earthquake is still going, sweeping across the periods of whatever is standing on it.