calculation of horizontal ranges and sound intensities by use of numerical integration techniques

13.2 Received 6 June 1968

Calculation of Horizontal Ranges and Sound Intensities by Use of Numerical Integration Techniques*

MAx K. M•LL•

Science Services Division, Texas Instruments Incorporated, Dallas, Texas 75222

The purpose of this paper is to point out some advantages in the calculation of horizontal ranges, travel times, and sound intensities by means of numerical techniques that seem to have been overlooked in the past. The integrals for horizontal range, travel time, and sound intensity are evaluated directly for reflected rays, while appropriate modifications are made for refracted rays. This approach to the problem of calculating sound intensities is discussed, and two numerical examples are given that point out the accuracy and advantages of the method.

INTRODUCTION

HE purpose of this paper is to show how horizontal ranges, travel times, and sound intensities can be calculated for ray paths by a direct method. It is assumed that the sound velocity is a function of depth only and is specified at discrete depth points. This is the practical case for profiles measured by a velocimeter or by the older Nansen cast method. This method was origi- nally devised to calculate horizontal ranges for bottom reflected rays; refractions were a later consideration.

Calculations of the horizontal range as a function of phase velocity (or source angle) for given ray paths and sound intensities are the main considerations here.

Travel times and distance traveled along the curved ray paths may be obtained at the same time with few addi- tional calculations. After horizontal distances are calcu-

lated, they are used to calculate the intensity of the sound field as a function of horizontal range.

The problem of fitting a known function through a set of given points on a velocity profile, and then being able to perform the necessary integration in order to calculate the horizontal range and/or sound intensity for a given ray path is known and has been discussed. •-4 However, one is not always fortunate enough to be able to do this with much success. The problem of false

* A portion of this paper was presented at the annual meeting of the Acoustical Society of America in November 1967 at Miami Beach, Fla. [-J. Acoust. Soc. Amer. 42, 1155(A) (1967)'].

• P. Hirsch and A. H. Carter, J. Acoust. Soc. Amer. 37, 90-94 (1965).

•' K. R. Stewart, J. Acoust. Soc. Amer. 38, 339-347 (1965). 3 M. A. Pedersen, J. Acoust. Soc. Amer. 33, 465--474 (1961). 4 M. A. Pedersen and D. F. Gordon, J. Acoust. Soc. Amer. 41,

419438 (1967).

1690 Volume 44 Number 6 1968

caustics, or infinities in the intensity calculation that re- suit from piecewise linear approximations to the velocity function, has been overcome somewhat by more sophisticated curve-fitting methods. For example, Pedersen and Gordon 4,5 treat this problem in detail.

Instead of expending effort to approximate the velocity function, emphasis is placed on approximating the integrals involved in the calculation of range, travel time, or sound intensity.

I. RAY-THEORY CALCULATIONS

The coordinate system used here is shown in Fig. 1. Note that the source angle is measured from the vertical to the tangent to the ray path. For a velocity function that depends only on depth, the horizontal range X for a ray path is determined by

.• Zr 7•dZ where v is understood to be a function of z, and z8 and zr are, respectively, the source and receiver depths. The phase velocity c is given by Snell's law and is determined from sin0= v/c. A sinfilar expression for the travel time T is given by

zr ½dz T-- • v(c•--v•) • . (2) For such a ray reaching a receiver at coordinates

(zr, X), the relative intensity of the sound field is given

* M. A. Pedersen, J. Acoust. Soc. Amer. 43, 619-634 (1968).

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 141.209.100.60 On: Sat, 20 Dec 2014

22:18:01

CALCULATION OF HORIZONTAL SOUND RANGES

by I sin0,

)•=X cosOr(ax/ao.,)' (3) 18 and I are the intensities at the source and receiver, while 08 and 0r are, respectively, the angles formed by the rays with the vertical at the source and receiver.

From Eq. 3, it is readily seen that if OX/00,= O, there is an infinity in the intensity calculation. In practice, it is customary to replace OX/008 with dX/d08, and this is assumed throughout the remainder of this paper. Furthermore, differentiation under the integral sign is valid in Eq. 1 and it follows that

dX 6 2 f z,. vdz - (c •--v?)l . (4) dO• v• • (c"-v •)• This equation must be modified for refracted rays.

For example, if a ray travels initially downward, passing through some interface zi, and if it becomes horizontal at zh, then

dXi c 2 « f zh vdz .=--(c2-- v8 2 ) clO• v • • • ( c"- v •) •

A similar expression holds for rays that become horizontal after passing through an interface traveling in an upward direction. One must also be aware of the similar problem that arises for horizontal rays in the calculation of X in Eq. 1.

In order to calculate the horizontal range X and the relative sound intensity 1/18, it is necessary to know the velocity function and then be able to perform the integration indicated in Eqs. 1 and 4. Herein lies the difficulty of finding a suitable function that fits the profile and yet allows the integrals in Eqs. 1, 2, and 4 to be evaluated.

II. NUMERICAL CALCULATION OF RANGE AND INTENSITY

Basically, the scheme here is to approximate the integral for X in Eq. 1 (and similarly for T and dX/d08) by a finite series. Thus,

ß = 62_ Vi 2) «

where the W, are weights used in the particular numerical integration scheme used.

Two types of numerical integration methods are considered. One type is a Gaussian-type quadrature that requires unevenly spaced data points. The other type is one that requires evenly spaced data. Simpson's one- third rule is used to illustrate this kind of quadrature formula, while a Chebyshev quadrature is used as an example of the former type.

HORIZONTAL RANGE

ANGLE • RAY PATH

Z

FIG. 1. Coordinate system used. Source angle is measured from the vertical.

In either case, it is possible that one must use for quadrature points values of the velocity at depths that do not always coincide with measured values--certainly a disadvantage. However, velocimeter measurements can provide data sufficiently dense so that the velocity can be interpolated between adjacent points in the profile. In practice, linear interpolation between two adjacent points is within the accuracy of the measuring system, and therefore the spacing of the data usually is not a problem.

When one works with measured velocities, it may be advisable to "smooth" the velocity profile prior to performing any calculations. Smoothing, in effect, is also accomplished by fitting a curve to the measured points.

If the velocities are given at equally spaced depths, then some method that uses equally spaced data such as Simpson's one-third rule has an obvious advantage.

Consider the evaluation of the integral in Eq. 1. For source angles such that rays leaving the source are reflected from either the surface or bottom, c>v. As the ray strikes the surface at angles sufficiently close to vertical, c>>vm, where vm is the maximum velocity in the profile.

Hence, for reflected rays, the integral in Eq. 1 is well behaved and lends itself to numerical integration methods. Only when c approaches v 'do diff, culties arise. This occurs as the ray becomes horizontal. At a turning point where the ray becomes horizontal, c=v and the denominator of the integrand in F q. 1 becomes zero. At the same time, there is a singularity in the integrand in Eq. 4, which is of a higher degree than the one in Fq. 1. The problem of performing these integrations may be approached as Pedersen a does so that the equations for X and dX/dO, have a form similar to Eqs. 5 and 6 in Ref. 3. That approach, however, is not followed here, and the problem concerning the singularities are discussed in a later section. Thus, one has two cases to consider: refracted rays and reflected rays (from either the surface or bottom).

In practice, the velocity profiles are divided into layers so that the integration limits are actually from

The Journal of the Acoustical Society of America 1691


22:18:01

M. K. MILLER

½2 _ v2) •/2

60

40

o I I 1.0 1. 005 1.01

FIG. 2. Integrand for range function versus c/v. For ½=v, rays are refracted and for c >% rays are reflected.

c/v

interface to interface. Then,

and

N N f Zi+l 7)d• X=EXi--E «, dX N dXi

dO• '= dO•

where N is the number of layers, and appropriate modifications must be made in these equations for refracted rays as they become horizontal.

III. REFLECTED RAYS

Consider rays that exit from the source at angles such that they are reflected from either the surface or the bottom. From Eq. 1, one observes that the problem of calculating the horizontal range X is equivalent to that of finding the area under the curve in Fig. 2. (The curve in Fig. 2 is actually taken from the second example in Sec. VI). For reflections c>v, and as c/v increases, the numerical integration becomes more accurate.

Values of c "close" to v correspond to larger horizontal ranges, while at short ranges c>>v and c --• • as x --• 0.

Zh-5 z

Z - h

v•(•.)

v(•.)

(a)

ß v

Zh-6 z

zh I

• x

(b)

Fro. 3. Horizontal ray. (a) Velocity function v•(z) is tangent to v(z) at zh--•z and v2(z) is the straight-line distance between points on the ray- path curve at depths zh--•z and zh. (b) Ray path for ray horizontal at depth zh.



22:18:01

CALCULATION OF HORIZONTAl. SOUND RANGES

For the case where the rays exit the source with small angles and are reflected from the bottom or the surface, the velocity profile may be approximated with piecewise linear segments and good results still be expected for the range, tra.vel time, and intensity calculations. Only as the range increases and c---) Vm, at which time refraction occurs, does the accuracy of the calculations deteriorate and such things as false caustics occur.

For bottom bounce and surface-reflected rays, the method described here still offers some advantages. One advantage is that no effort (either human or computer) is required to approximate the profile with broken line segments or some other velocity function. This makes for a relatively simple approach to the problem. Another advantage is that the computations required are fast on a computer.

IV. REFleCTED RAYS

From Fig. 2 one sees that numerically evaluating Eq. 1 (and the similar integrals for travel time and dX/dOs) is made difficult due to the singularity as c---) Vm. The following method was used for calculating X in the numerical examples that follow in Sec. VI. The calculation of dX/d08 are discussed in the following section.

Let zh be the depth at which a downgoing ray is refracted and reaches a horizontal position. Then Eq. 1 may be written

X - fz zr vdz 8 (C•'--v2) «

f zh-•z V(]Z f zh VdZ • z• (•-•)• •-• (•-•)•

f z+az vdz Ff•z' vdz + (c*- ½) • +• (c•- •,•) • ß / Zh

•h-•z vdz fz• vdz _ __ nL2 --- • • ½*-v*)• • •_• ½*-v•)•

+ (5) • •+• (c*- v •) •'

where •i, is an arbitrarily small distance in the z direction. However, •i, is such that the first and last integral in Eq. 5 may be evaluated numerically as previously discussed.

Now, consider the middle integral in Eq. 5. As a, ---) 0, v---) c and the denominator term goes to zero. Even though the denominator of the integrand goes to zero, the integral may still exist. This is the case for the piecewise linear profile. For example, if the velocity profile is approximated in the interval (zh--az, zn) by the linear function v(z)=a+bz, where a and b are constants, the integral can be evaluated. For a= va = v(zn-- •,), one has

4800 0 i

2000

4000

60oo

VELOCITY (FT/SEC)

4900 50

Fie,. 4. A typical velocity profile.

for the contributation Xx to the total range X:

fz• vdz •z• (va-]-b•)dz = •./•{ (,•*- •,a')"--- [½*-- •,•(•,,)]"-},

b•O.

(6)

The integral is somewhat simplified for b=0 and

f z zh VdZ Z• at• z x•= = . (7) h,--•z ((72__ ,2) • (C2__ •)a 2) «

For/•z sufficiently small, the velocity function may be approximated by a linear profile over the small range of z values. In practice, this may turn out to be as small as 1 or 2 m, and the approximation is within the accuracy of the measured velocities. The error introduced by this linear approximation affects only X•, which may be a small contribution to the total value X.

An indication of the error that results from this

approximation can be obtained. Consider the velocity function and ray path between two adjacent interfaces as shown in Fig. 3.

Let v be the actual velocity function for values of z that lie between z•-$z and z•. Assume that v(z) is sufficiently "smooth" in this interval. Let vi and v• be velocity functions approximated by piecewise linear segments. That is, v•(z)=v(zh--$z)q-b•z and v•(z) =v(zn--$•)+b•z. Choose b• such that b•=['v(zn) --v(zn--$z)-]/$z. A value for b• may be obtained numerically (as in Eq. 8) or if the slope of the profile is near zero, let b•=0. If b•_<bs, then v•(z)_<v(z)_<w.(z) for z•(zn--$z, zh). For v•(z)_<v(z)_<v,(z),

f ,z, ,•(z)& f'" •_,• {•*-FVl(Z)•*}•- •_,• {•-F,½)•*} •

< .z•



22:18:01

20.0

M. K. MILLER

15.0

5.0 -

8-LAYER PIECEWISE LINEAR PROFILE

CHEBYSHEV INTEGRATION

i I I I I I I

10 20 30 40 50 60 70

SOURCE ANGLE (ø)

FiG. 5. Horizontal range versus source angle for bottom reflected rays. The profile shown in Fig. 3 was used for calculations. Source and

receiver depths are 328 ft.

--

or X•_< X•_<X2. As •, becomes smaller, the values of X• and X2 approach the same value and the true value of ,V• is bounded between X• and X, The total value for horizontal range is the sum

iv

X= I] Xi, i=1

where X• is one element of the sum. If b• is determined numerically, the previous bounds are affected by the error in the calculation of b•. For such cases, there is only an indication of the accuracy of X•; otherwise (for bl= 0), there is truly an error bound on X1.

For refracted rays, therefore, the horizontal range X can be calculated by using the integrals in Eq. 5. In this case, the integrals

z, (62-- T2) } and f•zr vdz •+• (c•-v9 •

TABLE I. Eight-layer piecewise linear profile.

Depth Velocity (ft) (ft/sec)

0 4960 328 4940 427 4937 502 4939 765 4942 885 4949

1490 4960 1910 4965 5935 5025

are calculated numerically. The integral

z z• vdz is evaluated by approximating with sufficient accuracy the velocity profile with a linear function and using the exact result for this profile as given in Eqs. 6 and 7.

V. CALCULATION OF dX/d0s

Calculation of the term dX/d08, which is used in Eq. 3 for the intensity calculation, was done in two ways. For reflected rays (see the first numerical example) numerical integration of the integral specified in Eq. 4 was performed. This gives dX/dO8 directly.

An alternate method was used for the calculation of

dX/dO, in the second numerical example where refracted rays are considered. Here values of X are calculated at different source angles; then dX/dO, is determined numerically from the values of X. If the range-versus- source angle curve is relatively "smooth," dX/dO, can be adequately determined. A visual inspection of the range curve is a practical aid for determing the validity of this step.

Several formulas for computing the derivative numerically have been used. Difficulties encountered in computing a derivative numerically are not discussed. However, the relatively simple relation

dXi/dOs-- (-X'i-t-1-- X/__i)/2A0 (8)

seems to work satisfactorily and is used for all the cal-



22:18:01


culations here. In Eq. 8, the X• are calculated at evenly spaced values of 0, A0 units apart.

VI. NUMERICAL EXAMPLES

For the first example, consider the velocity profile shown in Fig. 4. This profile is similar to those published by the U.S. Naval Oceanographic office. ø Source and receiver depths are each 328 ft.

For these calculations, it is known that the velocity profile approximated with piecewise linear segments yields satisfactory results for bottom-reflected rays. The approximation of the velocity function with piecewise linear segments, which is done either by hand or automatically on a computer, is avoided when numerical integration techniques are used. Only the discrete values of the velocity profile and the source and receiver depths are required input into the computer program.

Both Simpson's one-third rule and the Chebyshev quadrature method give essentially the same results for this example. Comparisons are made with results obtained by approximating the velocity profile with a piecewise linear function. The eight-layer profile used for comparison is shown in Table I.

Figure $ shows range as a function of source angle for the profile in Fig. 4. Results show little observable difference between the values calculated using integration techniques and the results obtained using the 8 layer profile. Only at the larger ranges are there differences. The contribution to each X• and dX•/d08 was computed by integrating between points that coincide with interfaces in the eight-layer model. Ea•ch integration interval was subdivided into two segments for Simpson's one-third rule, and two quadrature points were used in the Chebyshev integration routine. Both integration routines gave essentially the same results. Very little improvement in accuracy was obtained by increasing the number of quadrature points or by further subdividing the integration intervals except for large ranges, at which point the phase velocity decreases and approaches the maximum velocity in the profile. By using four quadrature points in the Chebyshev integration routine, the number of integration intervals was reduced to three, with no appreciable loss of accuracy. These interfaces were located at 328, 885, 1910, and 5935 ft.

The relative intensity I/Is-rs-range plot for the same velocity profile is shown in Fig. 6.

This example points out the advantages of calculating the range and intensity in the manner described here. A computer program that will perform the calculations requires source depth, receiver depth, and the values of sound velocities at given depths. With this information, X and 1/18 may be easily computed. The simplicity of the program lies in the fact that at most the four-layer

6 "Oceanographic Atlas of the North Atlantic Ocean, Section VI Sound Velocity," U.S. Naval Oceanographic Office, ?ubl. No. 700, Washington, D.C. (1967).

10 -8 8-LAYER PIECEWISE LINEAR PROFILE

_

,

10 -9

z x 1o- 10 I I I 5 10 15

RANGE (KY DS)

FzG. 6. Relative intensity I/Is versus range for bottom reflected rays. The profile shown in Fig. 3 was used for calculations. Source and receiver depths are 328 ft.

interfaces (z-0, z= receiver depth, z-- source depth, and z=water depth) must be read as input into the computer program. The computer program may be written to subdivide these layers further automatically. Of course, for bottom-reflected rays, no integration is done from z=0 to z-min (source depth, receiver depth). Figure 7 depicts the situation for bottom-reflected rays. The integration implies calculation of horizontal range, travel time, or intensity for bottom-reflected rays.

As previously mentioned, the technique described here was developed for use in the calculation of horizontal ranges for reflected rays. However, because of the

FiG. 7. Ray path VELOCITY z= 0 -- for numerical cal-

culation of bottom SOURCE

bounce rays. Inte- ii --- grations for range, REC•EIVER intensity, or travel • time are indicated by • - (.). The integrals • are given by •

ay (')= (') AY PATH • Path '

+2 (.). r Zb-I-



22:18:01

lOO

300

500

700

900

11oo

4800

M. K. MILLER

VELOCITY (FT/SEC) 4850 49OO

FZG. 8. Epstein profile.

good results obtained when refractions were considered, an example is given here to point out the usefulness, especially in the calculation of ranges, of this numerical approach.

For the second example, consider the Epstein profile. This example was chosen because it has been used in the literature 4.7 and therefore, some comparisons can be easily made. In order to make direct comparisons, the same parameters for the Epstein profile and the same source and receiver depths (324 ft) are used as those found in Ref. 4, where the exact solution is given. Only upgoing rays are considered.

The velocity function given below defines the Epstein profile used. The velocity is determined by

v(z)= {7.5X 10 -9 sechE(z-- 71)/76•+ 7.018X 10 -9 X tanhi-(z- 71)/76•+3.797X 10-7}-L

A plot of the velocity function is shown in Fig. 8. Discrete values of velocity profile used were selected

so that the input velocity points used in the program were spaced at 3-yd intervals. Linear interpolation between adjacent points was used to calculate the velocity at the quadrature points required by the numerical integrations.

Figure 9 shows horizontal range as a function of phase velocity. For these calculations, each 3-yd layer was

7 R. L. Davenport, Radio Sci. 1, 709-724 (1966).

used as an integration interval. That is, a contribution Xi to the total horizontal range X was obtained by integrating from one layer interface Zi+l to an adjacent layer interface Zi (a distance of 3 yd, except for the layer in which the ray becomes horizontal). Then X=•X•.

Chebyshev numerical integration was used for calculating the horizontal ranges plotted in Fig. 9. Two quadrature points were used for the evaluation of each X•, except for the layer in which a ray becomes horizontal. In that case, four quadrature points were used. Values of X were calculated by varying the source angle in 0.25 ø increments.

As seen in Fig. 9, the computed values, in general, agree quite well with true values. For source angles near 90 ø , the results are not as good.

The curve in Fig. 9 has a relative maximum and a relative minimum. Thus, there will occur two infinities in the intensity calculations. These occur at source angles near 90.5 ø and near 95 ø . Note also that for source angles near 99.5 ø , the critical distance for refraction occurs and the rays are reflected for larger source angles.

Figure 10 shows relative intensity I/I• as a function horizontal range. The two-point formula in Eq. 8 with A0=0.25 ø was used to determine the slope of the curve in Fig. 9. This value of dX/dO, was then use in Eq. 3 to determine I/I,.



22:18:01


Z.3

Z. 1

•1.9

•1.7

1.5

1.:3

EPSTEIN PROFILE

O CHEBYSHEV INTE

REFLECTED RAYS

REFRACTED • •

1605 1610 1615 1620 1625 1630 1635 PHASE VELOCITY (YD/SEC)

FIG. 9. Range versus phase velocity for upgoing refracted rays. The Epstein profile shown in Fig. 8 was used. Source and receiver depths are 324 ft.

From Fig. 9, one can see that there is an infinity in the I/Is curve at approximately 1.76 kyd. The details of this portion of the I/Is curve in Fig. 10 are difficult to calculate accurately because the slope of the range curve in Fig. 9 is small near 1.76 kyd. The "flatness" of the curve causes small errors in computed ranges to cause relatively large errors in dX/dOs. These errors are, in turn, magnified in the calculation of the relative intensity I/Is. By going to a more sophisticated numerical routine for calculating dX/dO•, greater accuracy might be obtained.

The portion of the curve in Fig. 10, which corresponds to the peak at approximately 1.85 kyd for refracted rays (source angle near 90.5 ø in Fig. 9), presents difficulties in calculation of 1/18. This is due to the rapidly changing slope of the range curve in Fig. 9. However, the peak of the range curve is easily determined to be approximately 1.85 kyd, and thus the corresponding infinity in Fig. 10 occurs at this distance.

The portion of the I/Is curve in Fig. 10, for which no calculated values are plotted, corresponds to source angles that lie close to 90 ø (less than 90.5ø). Here the range varies so rapidly as 08 is increased slightly from 90 ø that no meaningful results were obtained for source angles less than 90.5 ø . For rapidly varying curves (such as this one), it is virtually impossible to calculate with any degree of accuracy the slope of the range curve, and one would not expect accurate results for the calculation of dX/dO• in this region.

The same calculations were performed using Simp- son's one-third rule, although they are not shown on

the plots in Figs. 9 and 10. For comparisons, however, Table II shows some of the results for both numerical

schemes. For these calculations, each 3-yd layer was divided into two subintervals for use in Simpson's one- third rule.

Except for the range of values near 90 ø , the results of the intensity calculations obtained are good.

VII. CONCLUSIONS

These results show that horizontal range, travel time, and relative intensity may be calculated by evaluating the integral expressions for these quantities directly. By observing the relative maxima and minima on the range curve, one can infer at what ranges the infinities in the intensity calculations approximately occur (especially useful for refracted rays).

TABLE II. Typical results from two numerical integration techniques.

,

Phase Source Range Range 1/18 I/I, (velocity angle Chebyshev Simpson's « Chebyshev Simpson's «

yd/sec) (deg.) (yd) (yd) (yd-2) (yd-•')

1604.65 92.0 1840.45 1840.50 0.754 X10-* 0.644 X10-* 1605.87 93.0 1781.56 1781.58 0.545 X10-* 0.534 X10-* 1607.58 94.0 1761.33 1761.77 0.964 X10-5 0.959 X10 -• 1609.79 95.0 1759.07 1759.39 0.186 X10 -4• 0.1152 X10 -4• 1612.50 96.0 1783.77 1783.83 0.227 X10-* 0.249 X10-* 1615.71 97.0 1840.18 1840.56 0.978 X10-e 0.978 X10 -6 1619.42 98.0 1946.39 1946.63 0.452 X10-e 0.452 X10 -6 1623.66 99.0 2136.67 2136.77 0.197 X10-• 0.197 X10 -•

This value not shown in Fig. 10.



22:18:01

M.•K. MILLER

<

-5 10

10 -6 --

-7 10 --

_

-8 lO

o

REFRACTED RAYS

REFLECTED RAYS

EPSTEIN PROFILE

o CHEBYSHEV INTEGRATION

I I 1 0.5 1.0 1.5 2.0

RANGE (KYD)

2.5

Fro. 10. Relative intensity I/Is versus range X for upgoing refracted rays. The Epstein profile shown in Fig. 8 was used. Source and receiver depths are 324 ft.

Calculations involving bottom or surface reflections may be done by inputting the velocity profile and source and receiver depths. For refracted rays, more care is required and the calculation of the contribution X• for that layer to the total range X is subject to large error. The interpolation of the basic velocity data to obtain values for quadrature points is a disadvantage; however, this is usually within the accuracy of measurements.

Since the results of the intensity calculation depend on the accuracy of the range calculation, the "smooth-

ness" of the range-vs-08 curve is a critical step in the intensity calculation. There is a disadvantage to this approach for refracted rays if the range curve varies too rapidly or if dX/dO8 cannot be evaluated accurately.

The method introduced here is not intended to deny the good results obtained by curve-fitting techniques such as those found in Ref. 5. It is intended to be used

both as a research tool and as an automated process (reflection cases) for use on large numbers of data. It should be especially useful for calculation of horizontal ranges for either case.



22:18:01

calculation of horizontal ranges and sound intensities by use of numerical integration techniques

Documents