Wavefront Curvature Calculation in Graded Media
It is possible to compute the ray intensity by changing Intensity computation to Compute intensity or Compute intensity in graded media in the settings window for the Ray Acoustics interface. The options Compute intensity and power and Compute intensity and power in graded media can also be used to compute intensity, with the only difference being that these options define an additional auxiliary dependent variable for the total power transferred by the ray. The setting Compute intensity is more robust and accurate than Compute intensity in graded media, but is only applicable to homogeneous media. The setting Compute intensity in graded media can be used for both homogeneous and graded media, but it introduces more numerical error than Compute intensity.
Assumptions for Computing Intensity in Homogeneous Media
When Compute intensity is selected, the ray intensity is computed along each ray path using the following assumptions:
1
The wavefront represented by each ray subtends a constant plane angle (2D) or solid angle (3D), which can change only when rays are reflected or refracted at boundaries.
2
Given the radii of curvature at two positions within the same domain and the intensity at one of these two positions, the intensity at the other position can be computed using the relationship
3
Except when rays are reflected or refracted at boundaries, the principal radii of curvature change at a constant rate:
4
The principal curvature directions do not change, except when rays are reflected or refracted at boundaries.
These assumptions are not valid in graded media, in which the speed of sound changes continuously as a function of position. It is possible for the angle between two different rays to change as they propagate through the medium, so the solid angle subtended by the wavefront is no longer constant. As a result, the ray intensity cannot be expressed using the ratio of initial and final principal radii of curvature.
Curvature Tensor Definition in Graded Media
The calculation of ray intensity in graded media is based on the concept of a curvature tensor K, defined in terms of the principal curvatures κ1 and κ2 and the corresponding principal curvature directions e1 and e2:
Because e1 and e2 are orthogonal, it follows that κ1 and κ2 are eigenvalues of K. It also follows that K is singular because there is no contribution that is orthogonal to both e1 and e2.
The signs of the principal curvatures are chosen so that positive curvature indicates that the wavefront is converging, whereas negative curvature indicates that the wavefront is diverging.
The selection of the coordinate system in which the curvature tensor is defined is crucial. It is convenient to describe the coordinate system so that two of the coordinate axes lie in the plane containing e1 and e2, since this reduces the number of nonzero terms in K. Further reduction in the number of nonzero terms can be achieved if K can be defined in a coordinate system in which e1 and e2 are basis vectors. This is possible in 2D because one of the principal curvature directions is always parallel to the out-of-plane direction, but in 3D it is not feasible because the principal curvature directions can change as rays propagate through a graded medium.
In the most general 3D case, the curvature tensor is described using the following symbols, each of which corresponds to a different orthonormal basis:
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KX: curvature tensor defined in the Cartesian coordinate system with basis vectors x, y, and z.
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KW: curvature tensor defined in the coordinate system consisting of the two principal curvature directions e1 and e2 and the direction of propagation t.
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KS: curvature tensor defined in a coordinate system in which one basis vector is the direction of propagation t. The other two basis vectors s1 and s2 can be determined arbitrarily, as long as they are orthogonal to each other and to t, such that . It is more convenient to operate in this coordinate system if the basis vectors can be expressed strictly in terms of x, y, z, and t.
The basis vectors s1 and s2 are defined as follows:
To avoid poles in the definition of the basis vectors, the following alternative definitions are used when the rays propagate nearly parallel to the z-axis:
The relationship between s1, s2, e1, and e2 is given by
where the rotation angle is an auxiliary dependent variable that is stored for each ray.
The relationship between KS and KW is given by
where Q0 is the rotation matrix:
The relationship between KX and KS is given by
where Q is the rotation matrix:
In 2D, one of the principal curvature directions is always known, so it is possible to define s1 and s2 so that they coincide with e1 and e2. The rotation matrix Q0 is the identity, and the auxiliary dependent variable for the angle can be omitted.
Derivatives of the Curvature Tensor
Following Ref. 3, the derivative of the curvature tensor along the ray trajectory in a graded medium is given by the expression
where the terms Mi correspond to the different ways in which the curvature can depend on the medium properties. They are defined using the following expressions:
where Π is the projection matrix:
The gradient operator ∇S consists of derivatives that are taken with respect to the local coordinates with basis vectors s1, s2, and t.
Derivatives of the Principal Curvatures
By application of the chain rule, the derivatives of the nonzero elements of KS can be expressed in terms of the principal curvatures that occupy the diagonal elements of KW and the rotation angle :
Numerical Stabilization
The principal curvatures κ1 and κ2 are not ideal choices for the auxiliary dependent variables stored by each ray because their values can become arbitrarily large as rays approach caustics. Similarly, the principal radii of curvature r1 and r2 are not ideal choices because their values can become arbitrarily large when a diverging wavefront begins to converge while propagating through a graded medium.
Instead, the auxiliary dependent variables stored by each ray are the help variables α1 and α2. In 2D, only one help variable is allocated because the out-of-plane principal radius of curvature is assumed to be infinite. The principal curvatures are expressed in terms of the help variables using the expression
where κ0 = 1 1/m and i ∈ [1, 2]. The derivatives are then related by the expression
Similarly, the intensity may become infinitely large at caustics, and its reciprocal becomes infinitely large as rays undergo attenuation, so a help variable Γ is used to represent the intensity of the ray:
where I0 = 1 W/m2. The relationship between the ray intensity help variable and the principal curvatures is
where α is the attenuation coefficient of the medium.