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Boundary

In order for the wavelengths to match at the interface between the two zones there must be a change in direction such that sin 9. X.

sin 92 X2

where 9 is the angle between the wavefront and the horizontal. This relationship is known as Snell's law, and the ratio of the sine terms is called the index of refraction, when the waves are light rather than sound. Snell's law can also be expressed in terms of the velocities of sound in the two media as sin 9 sin 9

ci c2

When the sound velocity is a linear function of the height above the ground we can calculate the shape of the sound path. If we assume sound velocity is a function of height, in Fig. 5.23, which follows the linear relationship, c(y) = Ay + B, where A and B are constants. A is the slope of the line and B is the y axis intercept, which is taken to be zero at c(y) = 0. Substituting into Eq. 5.25 we obtain y cos 0 (y) = — and c(y) = Ay (5.26)

This is the formula for a circle having radius rc , where at the top of the circle 6 (y) = 1 the sound ray cos 0 (y) = 1 and c(y) = c0. At this point we can solve for the radius rc of curvature of cn rc = f (5-27)

Figure 5.23 Relation Between a Velocity Gradient and the Radius of Curvature of a Ray (Kinsler et al., 1982)

Figure 5.23 Relation Between a Velocity Gradient and the Radius of Curvature of a Ray (Kinsler et al., 1982)

where the origin (y = 0) is measured from the point where the sound velocity extrapolates to zero, which is usually below ground level for a positive gradient. Note that the term A in Eq. 5.27 could be due to changes in both wind and thermal gradients. The radius of curvature is (Gutenberg, 1942)

where rc = radius of curvature (m)

c0 = velocity of sound where the ray is horizontal (m/s) 1 cos 0

-is the velocity at angle 0 (Snell's Law)

dc dy dv c0

= rate of change of sound velocity with height (1/s)

, = rate of change of wind velocity with height (1/s)

0 = angle that the wavefront makes with the y axis (rad) When rc is positive (sound velocity increasing with height) the sound ray bends down and when it is negative (sound velocity decreasing with height) it bends up.

If the wind velocity changes from 0 km/hr to 20 km/hr (5.6 m/s) in 5 meters, A would be 4 km/m hr (1.1 s-1). If the velocity at the top of the circle is 40 km/hr (11.2 m/s), then the radius of curvature is the quiescent speed of sound 344 + 11.2 = 355.2 m/s divided by A, yielding a radius of about 322 m. If we assume that a high school band (Lw = 100 dB) is playing and we are located 150 m away, the arc of the sound ray at its highest point is about 10 m above the ground. This path could comfortably clear intervening barriers resulting in a clearly audible sound (45 dB). Without the wind the same source could be inaudible at this distance due to ground effects and shielding. This type of calculation is based on a worst-case scenario without focusing. The most likely scenario would probably include barrier shielding and ground effects if appropriate.

Beranek and Ver (1992) have cited unpublished data by G. S. Anderson giving the approximate attenuations due to refraction over soft ground with and without barrier shielding. Without a barrier the predicted refraction attenuation is

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