Proposed Approximations

To derive a more symmetric form, we return to the four-parameter expressions (equations 17 and 18) and propose to modify them as follows:

$\displaystyle v^2_{phase} \approx e(n_1,n_3)(1-\hat{s}) + \hat{s}\sqrt{e^2(n_1,n_3) + \frac{2(\hat{q}-1)w_1w_3n^2_{1}n^2_{3}}{\hat{s}}}~,\\$

(23)

$\displaystyle \frac{1}{v^2_{group}} \approx E(N_1,N_3)(1-\hat{S}) + \hat{S}\sqrt{E^2(N_1,N_3) + \frac{2(\hat{Q}-1)W_1W_3N^2_{1}N^2_{3}}{\hat{S}}}~,$

(24)

$\displaystyle \hat{q}$	$\displaystyle = ~~\frac{q_1 w_1 n^2_1 + q_3 w_3 n^2_3}{w_1 n^2_1 + w_3 n^2_3},~\hat{Q}$	$\displaystyle =~~ \frac{Q_1 W_1 N^2_1 + Q_3 W_3 N^2_3}{W_1 N^2_1 + W_3 N^2_3}~,$	(25)
$\displaystyle \hat{s}$	$\displaystyle = ~~\frac{s_1 w_1 n^2_1 + s_3 w_3 n^2_3}{w_1 n^2_1 + w_3 n^2_3},~\hat{S}$	$\displaystyle =~~ \frac{S_1 W_1 N^2_1 + S_3 W_3 N^2_3}{W_1 N^2_1 + W_3 N^2_3}~.$	(26)

The modifications in equation 25 are equivalent to the second anelliptic approximations by Dellinger et al. (1993). Again, parameters

and

can be found by fitting the velocity profile curvatures at the vertical ( $\theta = 0$ ) and horizontal ( $\theta = \pi/2$ ) axis, respectively and are defined in equations 1 and 2. Analogously,

and

can be found by fitting the fourth-order derivative ( $d^4v_{phase}/d\theta^4$ ) at the same angle. A similar strategy applies to fitting parameters for the group-velocity approximation. Note that expressions for

and

are different from equations 19 and 20.

$\displaystyle s_1$	$\displaystyle =$	$\displaystyle a_{1}/b_{1}~,$	(27)
$\displaystyle a_{1}$	$\displaystyle =$	$\displaystyle (w_3-w_1)(q_1-1)^2(q_3-1)~,$
$\displaystyle b_{1}$	$\displaystyle =$	$\displaystyle 2[w_3(q_1(q_1(q_1-2)+3)-2q_1q_3+q_3^2-1) - w_1(q_3(q_1(q_1-4)+q_3+1) +2q_1 -1 )]~,$

$\displaystyle s_3$	$\displaystyle =$	$\displaystyle a_{3}/b_{3}~,$	(28)
$\displaystyle a_{3}$	$\displaystyle =$	$\displaystyle (w_1-w_3)(q_1-1)(q_3-1)^2~,$
$\displaystyle b_{3}$	$\displaystyle =$	$\displaystyle 2[w_1(q_3(q_3(q_3-2)+3)-2q_1q_3+q_1^2-1) - w_3(q_1(q_3(q_3-4)+q_1+1) +2q_3 -1 )]~,$

$\displaystyle S_1$	$\displaystyle =$	$\displaystyle A_{1}/B_{1}~,$	(29)
$\displaystyle A_{1}$	$\displaystyle =$	$\displaystyle (W_1-W_3)(Q_1-1)^2(Q_3-1)~,$
$\displaystyle B_{1}$	$\displaystyle =$	$\displaystyle 2[W_1(Q_3^2 +2Q_1 +Q_1Q_3(Q_1(Q_1-2)-1)-1)$
		$\displaystyle - W_3(Q_3^2-2Q_1Q_3+Q_1(Q_1(Q_1-1)^2+2)-1)]~,$

$\displaystyle S_3$	$\displaystyle =$	$\displaystyle A_{3}/B_{3}~,$	(30)
$\displaystyle A_{3}$	$\displaystyle =$	$\displaystyle (W_3-W_1)(Q_1-1)(Q_3-1)^2~,$
$\displaystyle B_{3}$	$\displaystyle =$	$\displaystyle 2[W_3(Q_1^2 +2Q_3 +Q_1Q_3(Q_3(Q_3-2)-1)-1)$
		$\displaystyle - W_1(Q_1^2-2Q_1Q_3+Q_3(Q_3(Q_3-1)^2+2)-1)]~.$

Note that equations 23 and 24 introduce three more parameters generating six parameters in total, namely

, and

for equation 23 or

, and

for equation 24. However, expressing

and

in terms of

and

in equations 27-30, we effectively reduce the dependency to four parameters. This reduction leads to four-parameter anelliptic approximations, which fit up to the fourth-order accuracy along both axes. The exact phase- and group-velocity expressions also require the total of four independent parameters. However, the advantage of the proposed approximations lies in the existence of the group-velocity expression (equation 24) with analogous functional form as the phase-velocity expression (equation 23). To reduce the number of parameters to three, we utilize the linear relationships between

and

given in Figure 1. The required

and

parameters for the group-velocity approximations can be found from the reciprocals of

and

for phase-velocity approximations, as mentioned above. Therefore, both phase- and group-velocity approximations derived on the basis of this approach require the same number of parameters.