Ultrasonics, 3rd Edition

2.10.1    ATTENUATION DUE TO BEAM SPREADING

If a spherical wave front of uniform intensity (Figure 2.27) suffers no loss in traveling from its source, the total energy (P_T) at each spherical surface of total area S at distance r from the center of the source is always

$P_{\text{T}}=I_r{S}_r$

where I_r is the intensity at distance r and S_r is the total area of a sphere or segment of the sphere through which the total energy P_T passes.

The area of a sphere is 4πr². Let r₁ and r₂ correspond to the radii of two concentric spheres containing a point source at their centers radiating a total energy P_T. If there are no losses, the total energy passing through S₁ corresponding to r₁ is equal to that through S₂ corresponding to r₂. From Equation 2.104

$P_{\text{T}}=I_1{S}_1=$