of both media and the incident angle are known. If a layer containing a lower modulus

overlies a layer with a higher modulus within the half-space, another surface wave,

known as a Love wave, will occur. This wave is a horizontally oriented S-wave whose

velocity is between the S-wave velocity of the layer and of the underlying medium.

The decay or attenuation of stress waves occurs for two reasons:

geometric or radiation damping, and material or hysteretic damping. An equation

including both types of damping is the following:

(12-18)

r1

C

exp [-α(r2 r1)]

A2 = A1

r2

where

A2 = desired amplitude at distance r2

A1 = known or measured amplitude at radial distance r1 from vibration source

C = constant, which describes geometrical damping

= 1 for body (P- or S-) waves

= 0.5 for surface or R-waves

α = coefficient of attenuation, which describes material damping

12-1.4.3

vibration may sometimes be necessary. In some instances, isolation can be

accomplished by locating the site at a large distance from the vibration source. The

required distance, r2, is calculated from equation (12-18). In other situations, isolation

may be accomplished by wave barriers. The most effective barriers are open or void

zones like trenches or rows of cylindrical holes. Somewhat less effective barriers are

solid or fluid-filled trenches or holes. An effective barrier must be proportioned so that

its depth is at least two-thirds the wavelength of the incoming wave. The thickness of

the barrier in the direction of wave travel can be as thin as practical for construction

considerations. The length of the barrier perpendicular to the direction of wave travel

will depend upon the size of the zone to be isolated but should be no shorter than two

times the maximum plan dimension of the structure or one wavelength, whichever is

greater.

12-1.5

dynamic analysis of a soil-foundation system is the shear modulus, G. The shear

modulus can be determined in the laboratory or estimated by empirical equations. The

value of G can also be computed by the field-measured S-wave velocity and equation

(12-16).

12-1.5.1

machine vibration problems correspond to low shear-strain amplitudes of the order of 1

to 3 x 10-4 percent. These properties may be determined from field measurements of

the seismic wave velocity through soil or from special cyclic laboratory tests.

12-14

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