Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 119

There are three main modeling approaches in acoustics, which may be termed wave acoustics, ray acoustics, and energy acoustics. So far in this chapter we have mostly used the wave acoustics approach in which the acoustical quantities are completely defined as functions of space and time. This approach is practical in certain cases where the fluid medium is bounded and in cases where the fluid is unbounded as long as the fluid is homogenous. However, if the fluid properties vary in space due to variations in temperature or due to wind gradients, then the wave approach becomes more difficult and other simplified approaches such as the ray acoustics approach described here and in chapter 3 of the Handbook of Acoustics [1] are useful. This approach can also be extended to propagation in fluid‐submerged elastic structures, as described in chapter 4 of the Handbook of Acoustics [1]. The energy approach is described in Section 3.13.

In the ray acoustics approach, rays are obtained that are solutions to the simplified eikonal equation (Eq. (3.68))

(3.68)

The ray solutions can provide good approximations to more exact acoustical solutions. In certain cases they also satisfy the wave equation [14]. The eikonal S(x, y, z) represents a surface of constant phase (or wavefront) that propagates at the speed of sound c. It can be shown that Eq. (3.68) is consistent with the wave equation only in the case when the frequency is very high [7]. However, in practice, it is useful, provided the changes in the speed of sound c are small when measured over distances comparable with the wavelength. In the case where the fluid is homogeneous (constant sound speed c and density ρ throughout), S is a constant and represents a plane surface given by S = (αx + βy + γz)/c, where α, β, and γ are the direction cosines of a straight line (a ray) that is perpendicular to the wavefront (surface S). If the fluid can no longer be assumed to be homogeneous and the speed of sound c(x, y, z) varies with position, the approach becomes approximate only. In this case some paths bend and are no longer straight lines. In cases where the fluid has a mean flow, the rays are no longer quite parallel to the normal to the wavefront. This ray approach is described in more detail in several books [6, 12, 15, 16] and in chapter 3 of the Handbook of Acoustics [1] (where in this chapter the main example is from underwater acoustics).

The ray approach is also useful for the study of propagation in the atmosphere and is a method to obtain the results given in Figures 3.14–3.16. It is observed in these figures that the rays always bend in a direction toward the region where the sound speed is less. The effects of wind gradients are somewhat different since in that case the refraction of the sound rays depends on the relative directions of the sound rays and the wind in each fluid region.