LETTERS TO THE EDITOR

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Comments on the deep six sound channel P.A. Ramos and D. R. Palmer

National Oceanic and Atmospheric Administration, Atlantic Oceanographic and Meteorological Laboratory, 4301 Rickenbacker Causeway, Miami, Florida 33149

(Received 10 June 1988; accepted for publication 30 November 1988)

Some features of the deep six sound channel [J. C. Miller, J. Acoust. Soc. Am. 71, 859-862 (1982) ], which may be important to those who wish to use the model in various applications, are noted and discussed.

PACS numbers: 43.30.Pc

In an article • (hereafter referred to as M82), Miller pre- sented the "deep six sound channel" for which the sound speed at depth z is given by the expression

c(z) -- •( 1/2)6o• [•//( 1 -- •//6) ]2•, ( 1 )

where •/-- 2(z- •)/B, • is the sound speed at the depth of the SOFAR axis •, 6o is a small parameter related to the fractional sound-speed gradient in an adiabatic isohaline ocean, and B is the ocean stratification parameter. Antici- pating the value of Hamiltonian dynamics in analyzing problems in ray theory, 2 Miller noted the likely utility of Eq. ( 1 ) as a zero-order approximation in perturbation theory to an actual oceanic sound-speed profile.

The realism of Eq. (1) was discussed in terms of its agreement with the Munk 3 reference sound-speed profile,

c(z) =•[1 +6o(e '•-- 1 -- r/)]; (2)

that is, the first three terms in the power series expansion of the difference between the Munk and deep six profiles van- ish, leaving • 6o •/5/1080 for the lowest-order difference. In the following, we make comparisons with the Munk profile from the viewpoint that it is a useful and convenient refer- ence and not from the viewpoint that it represents, in any sense, "truth."

Curiously, it is not the deep six profile that was the sub- ject of the analysis in M82 but another profile,

c(z) =•/{1 -- 6o[r//(1 -- ,//6)]2}1/2, (3) which we call the EQ 13 profile because it is defined by Eq. (13) of M82. Equations ( 1 ) and (3) agree to lowest order in 60. Figure 1 is a comparison of the deep six and EQ 13 profiles with the Munk profile. Below about 700 m, the EQ 13 profile is a better representation of the Munk profile than is the deep six profile. Also plotted in the figure is the lowest-order dif- ference between the Munk and deep six profiles. As can be seen, it is not a good measure of the closeness of the deep six profile to the Munk profile.

The extent that the differences shown in Fig. 1 influence sound propagation modeling depends on the particular ap- plication. We have considered convergence zone structure by tracing rays. For a shallow 300 m source, we traced rays having launch angles varying from -- 10 ø to + 10 ø in incre- ments of 2 ø for the three profiles. We used the values for the parameters •, 2, B, and % listed in M82; namely 1.49 km/s, -- 1.2 km, 1.44 km, and 0.0082, respectively. The results are shown in Fig. 2. The dashed vertical lines were included to suggest the location and width of the convergence zones. Results obtained for the three profiles are in agreement for the first two convergence zones. Beyond this, the EQ 13 pro- file and the Munk profile give results in rough agreement while the results obtained with the deep six profile show differences in the location and width of the convergence zones.

0.0

-1.0

-3.0

-4.0

-5.0 -0.05

1

O.bO 0.•)5 0.•0 0.15 FRACTIONAL DIFFERENCE (%)

FIG. 1. Comparisons between the Munk, deep six, and EQ 13 sound-speed profiles. Plot 1 is the fractional difference between the Munk and deep six profiles as a function of ocean depth. Plot 2 is the fractional difference between the Munk and EQ13 profiles. Plot 3 is % •/5/1080, the lowest-order term in the power series expansion of the difference between the Munk and deep six profiles, as a function of depth, expressed as a fraction of •.

1767 J. Acoust. Soc. Am. 85(4), April 1989 1767

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We turn now to the analytic solution found in M82 for the ray paths. Beginning with Eq. (23) of M82,

A f dv X 2 -- X 1 = -•-•-• ( 1 -- eoA 2),/2 , (4) where we let •' = •/6, a = A/6, and define

R•I --2•'+ (1 -- 1/a2)• '2,

to obtain

A •/2 1 f d•'(1 -- •') x•--x•=•(1--36eo a•) • . (5) a dR Using Eq. 2.264.2 on p. 83 of ReL 4, we have

x2 -- x• = 1 -- 36eo a2) 1/2 2• 1 -- a 2

There are two expressions for the integral in Eq. (6) depend- ing on the value of a (ReL 4, p. 81, Eq. 2.261 ):

x 2 -- x• = (A/2•) ( 1 -- 36eo a2) •/2( 1/I 1 -- a213/2) T, (7)

where

T= --ad[p •-- 1] + sin-'•, (8) for [a[ < 1, and

r= - 41/- - - ], (9)

for la[ > 1. Equation (8) is the solution found in M82, where the

notation 4 -- •/2 = sin- • p, and a = sin y was used. It was shown that Eq. (8) describes the motion of a point on a wheel that is rolling without slipping in a straight line. Be- cause the parametrization a = sin y was made early in the

0 I I I I I

2

4

2

4

I , ,I, I ,I , , , o •o :2o :•o eao 3oo

•.• (•)

•IG. 2. Ray tracings for the three profiles. The rays have launch angles varying from -- 10 ø to + 10 ø in increments of 2 ø. The source depth is 3• m. Plot A is the ray tracing for the Munk profile. Plots B and C are the tracings for the deep six and EQ 13 profiles, respectively. Vertical dashed lines are included to guide the eye.

o

-1

-2

-3

-4 -40 -•0

ANGLE (DECREES)

FIG. 3. Phase space orbits corresponding to A = 6. The orbit on the left was obtained using the parameters listed in M82; the orbit on the right differs from the left one only in that the value of 2 was set equal to -- 2.3. Conse- quently, for this case, [2[/B = 1.6.

analysis in M82, the second solution, Eq. (9), was not found. Equation (9) corresponds to nonoscillatory ray paths that dive into the ocean bottom at large angles, i.e., they have no lower turning point. Figure 3 is a plot of two orbits, obtained by using Eq. (16) of M82 for the limiting case with A = 6 (a -- 1 ), which illustrate this behavior. For the values of B and 2 listed in M82, the second solution leads to orbits hav- ing large angles at all depths (left-hand side of Fig. 3 ) and is irrelevant for long-range, low-frequency propagation. How- ever, if

[2[>•3AB/(A + 6), (10)

it is easy to show that ray paths corresponding to this second solution turn over in the upper portion of the ocean (right- hand side of Fig. 3). Consequently, for a shallow source and small launch angles, it is possible to have ray paths that cor- respond to this unphysical second solution. Do ocean condi- tions forbid condition (10) ? Combining Eqs. ( 11 ) and ( 13 ) of Ref. 3, one has

2--•-[1.78 + ln( 1 + 0'049 Tu)] (11) , . ,

1 -- Tu

where Tu is the Turner number. As A increases from 6 to 8, condition (10) implies a lower bound on Tu that increases from 0.69 to 0.80. We have not been able to rule out this

condition on Tu using ocea. nographic arguments? It is instructive to consider how one might go about

avoiding the unphysical solution in a practical application. By curve fitting or other means, estimates of the parameters 2, c, B, and eo would first be obtained from available data. These quantities are sufficient to define any of the three pro- files Eqs. (1)-(3). Then, with knowledge of the source depth and the launch angles for the rays, Ho can be deter- mined from Eq. ( 11 ) of M82. Given Ho, A then can be de- termined using Eq. (15) of M82. With 2, B, andA available, condition (10) can be considered ifA/6 is close to unity. 6 If this condition is satisfied, EQ13 cannot be used to model propagation, for it will give unphysical results. An alterna- tive, perhaps the Munk profile, should be used.

1768 J. Acoust. Soc. Am., Vol. 85, No. 4, April 1989 Letters to the Editor 1768

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In summary, while it was stated in M82 that the deep six profile is mathematically tractable in the sense that ray paths, travel times, and other quantities of interest can be calculated analytically, this was not demonstrated. It is the profile EQ 13 that was shown to be tractable. In fact, we see no reason for the deep six profile to have been introduced in M82. In addition, even though the deep six profile is equiva- lent to the EQ 13 profile to order •, the differences between the two can lead to significant differences in the calculated acoustic field. Because the Euler transform that Miller used

to derive the deep six profile is singular at r/-- 6, there are two types of solutions for ray paths with the boundary between them defined by the condition A = 6. For A greater than 6, ray paths have no lower turning points. In M82, A was always taken, by fiat, to be less than 6, guaranteeing the existence of lower turning points. We have not been able to find any general argument that restricts A to being less than 6, however, and suggest that, in using the deep six profile (actually the EQ13 profile), one should make efforts to avoid the undesirable second solution.

ACKNOWLEDGMENTS

We thank Professor S. L. Mintz and Dr. H. F. Bezdek

for their encouragement and support. This work was done under the auspices of the Atlantic Oceanographic and Mete-

orological Laboratory/Florida International University Co- operative Education Program.

•J. C. Miller, "Oceanic acoustic rays in the deep six sound channel," J. Acoust. Soc. Am. 71, 859-862 (1982).

2See, for example, D. R. Palmer, M. G. Brown, F. D. Tappert, and H. F. Bezdek, "Classical chaos in nonseparable wave propagation problems," Geophys. Res. Lett. 15, 569-572 (1988); C. Wunsch, "Acoustic tomo- graphy by Hamiltonian methods including the adiabatic approximation," Rev. Geophys. 25, 41-53 (1987); J. C. Miller, "Hamiltonian perturbation theory for acoustic rays in a range-dependent sound channel," J. Acoust. Soc. Am. 79, 338-346 ( 1986); R. Dashen and W. Munk, "Three models of global ocean noise," J. Acoust. Soc. Am. 76, 540-554 (1984).

3W. H. Munk, "Sound channel in an exponentially stratified ocean, with application to SOFAR," J. Acoust. Soc. Am. 55, 220-226 (1974).

4I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

5It should be noted that this type of analysis is based on the phenomenology developed in Ref. 3 and may not always be applicable. For example, a typi- cal value for Tu off Bermuda is 0.8, which implies an unrealistic axial depth of 2.47 km. This problem was recognized in Ref. 3, where it was noted that the s adjustment for the Atlantic is too large. In situations where the analy- sis in Ref. 3 does not apply, Eq. ( 11 ) obviously cannot be used to discuss the implications of Eq. (10).

6Condition A = 6 defines the boundary between solutions that have lower turning points and those that do not. However, the ray paths are continu- ously deformed as this boundary is crossed. As A approaches 6 from below, the radius of curvature at the lower turning point increases until, at A = 6, it is infinite. Consequently, even ifA is less than 6, unphysical results may be obtained. For example, the upper loop distance might be a few tens of kilometers while the lower loop distance might be hundreds or thousands of kilometers.

Length error analysis for impedance tube measurements P. Banks-Leeand H. Peng North Carolina State University, College of Textiles, Raleigh, North Carolina 27695

(Received 27 July 1988; accepted for publication 8 December 1988 )

Length error analysis of the transfer function method and improved standing wave ratio (SWR) method for impedance tube measurements is presented in this letter. The error of measuring the complex reflection coefficients R is caused by errors in measuring the positions of the two microphones (or the two pressure points for the improved SWR method). Results agree with Abom and Boden's [J. Acoust. Soc. Am. 83, 2429-2438 (1988) ] conclusions, i.e., errors will be large if s, the separation distance of the two microphones, is about one-half wavelength and will be small if s is about one-quarter wavelength. Results also show that in order to reduce the amplitude error of R, it is better to locate one microphone near a minimum pressure point and the other near the adjacent maximum pressure point. To reduce the phase error of R for the transfer function method, preference of the two microphone locations depends on the combination of length errors AI• and A12. Phase error reduction for the improved SWR method requires an accurate measurement of the minimum pressure point.

PACS numbers: 43.55.Ev, 43.85.Bh

INTRODUCTION

In a recent issue, Chu • published an interesting article wherein, through well-designed systematic measurements, he concluded that, "for accurate measurements, one of the microphone positions must be close to a minimum pressure

point of the standing wave pattern, preferably the first mini- mum point. The choice of the other microphone position does not seem to be critical as long as the separation of the two positions is not close to a half-wavelength." His conclu- sions were mainly related to changing the two microphone locations. It is our feeling that an analysis of the errors en-

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