Anil K. Chopra - Earthquake Engineering for Concrete Dams

(A2.13)

which is independent of the excitation frequency.

The wave reflection coefficient α is a more physically meaningful description than is the damping coefficient ξ of the behavior of hydrodynamic pressure waves at the reservoir bottom. Although the wave reflection coefficient depends on a pressure wave's angle of incidence at the reservoir bottom, the value α for vertically incident waves, as given in Eq. (A2.13), is used here as a metric to characterize the absorptiveness of the reservoir bottom materials. The wave reflection coefficient, α , may range within the limiting values of 1 and −1. For rigid reservoir bottom materials, C r= ∞ and ξ = 0, resulting in α = 1, i.e. full reflection or no absorption of hydrodynamic pressures waves. For very soft reservoir bottom materials, C rapproaches zero and ξ tends to ∞, resulting in α = − 1. It is believed that α values from 1 to 0 would cover the wide range of materials encountered at the bottom of actual reservoirs (Ghanaat and Redpath 1995); α = 0 indicates no reflection of hydrodynamic pressure waves.

The preceding theory leading up to Eqs. (A2.10)and (A2.13)is also applicable to modeling of wave refraction in the underlying foundation. Such a derivation was presented even earlier by Hall and Chopra (1982).

The wave‐reflection coefficient, α , at the water–foundation boundary can be computed from Eq. (A2.13), wherein ξ = ρ / ρ r C rand ; both C rand ξ are determined from known values of Young's modulus, E r, and mass density, ρ r, for rock.

Wave absorption – or, alternatively, wave refraction – at the reservoir bottom is represented only approximately by the boundary condition of Eq. (A2.10). A hydrodynamic pressure wave impinging on the reservoir bottom results in a reflected hydrodynamic pressure wave in the water and two refracted waves, dilatational and rotational, in the sediments or underlying rock. The angle of reflection is equal to the angle of incidence and the angles of refraction of the two refracted waves are given by Snell's law. Although the boundary condition given by Eq. (A2.10)allows for proper reflection of hydrodynamic pressure waves for any angle of incidence, the only refracted waves allowed in the foundation or in the sediments are downward, vertically propagating waves.

1 †“Reservoir” is the place of storage, not the fluid itself.

2 †The word “foundation” denotes the rock that supports the dam.

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