Parabolic equations are among the most popular numerical techniques in many fields of physics. This article considers extra-wide-angle parabolic equations, wide-angle parabolic equations, and narrow-angle parabolic equations (EWAPEs, WAPEs, and NAPEs, respectively) for sound propagation in moving inhomogeneous media with arbitrarily large variations in the sound speed and Mach number of the (subsonic) wind speed. Within their ranges of applicability, these parabolic equations exactly describe the phase of the sound waves and are, thus, termed the phase-preserving EWAPE, WAPE, and NAPE. Although variations in the sound speed and Mach number are often relatively small, omitting the second-order terms pertinent to these quantities can result in large cumulative phase errors for long propagation ranges. Therefore, the phase-preserving EWAPE, WAPE, and NAPE can be preferable in applications. Numerical implementation of the latter two equations can be performed with minimal modifications to existing codes and is computationally efficient. Numerical results demonstrate that the phase-preserving WAPE and NAPE provide more accurate results than the WAPE and NAPE based on the effective sound speed approximation.
I. INTRODUCTION
The narrow-angle parabolic equation (NAPE) and wide-angle parabolic equation (WAPE) are among the most popular computational techniques in atmospheric acoustics,1–5 ocean acoustics,6–8 nonlinear acoustics,9,10 biomedical acoustics,11 and electromagnetic propagation.12 The NAPE and WAPE are well suited to relatively small computers, large domains, and high frequencies and can handle many phenomena such as atmospheric and ocean stratification and refraction; scattering by turbulence, internal waves, and other inhomogeneities; ground impedance and ocean bottom interactions; and propagation over slowly varying terrain and ocean bathymetry.
In atmospheric acoustics, the NAPE and WAPE should account for the effect of the wind velocity on the sound field. The corresponding equations have been formulated in a number of references, e.g., Ref. 2 and references therein. The WAPEs used in atmospheric acoustics are rather involved because they include the derivatives of the sound, density, and wind velocity, e.g., Refs. 13 and 14. These NAPEs and WAPEs are usually formulated assuming that the variations in the sound speed are small and the wind-speed Mach numbers are low.
A recent article15 suggests considering EWAPEs (extra-wide-angle parabolic equations or one-way equations), NAPEs, and WAPEs in the high-frequency approximation, in which the derivatives of the ambient quantities can be omitted. (Also, see Sec. 11.2 in Ref. 2.) The resulting PEs become much simpler than the previous formulations and still can be used in many applications. Specifically, Ref. 15 considers, in detail, the EWAPE given by Eq. (39) and the NAPE and WAPE derived from that equation. The WAPE from Ref. 15 has already been used in the literature, e.g., Refs. 16–19.
Reference 15 also suggests the other EWAPE, which is given by Eq. (B1), but considers the latter equation very briefly. One of the goals of the present article is to study this EWAPE in detail and determine when it is preferable compared to the EWAPE given by Eq. (39). To this end, starting with the EWAPE given by Eq. (B1) in Ref. 15, the NAPE and WAPE are derived. The newly formulated NAPE is remarkably simple and valid for arbitrary variations in the sound speed and Mach numbers. The new NAPE and the WAPE in the Padé (1,1) approximation can be implemented numerically with minimal modifications to existing codes and are computationally efficient.
In the literature, EWAPEs, NAPEs, and WAPEs are derived by omitting some terms in the Helmholtz-type equations. The ranges of applicability of the resulting equations are often difficult to assess. Hence, another goal of the present article is to study the ranges of applicability of these equations, which is examined by two approaches. In the first approach, starting with these equations, the dispersion relations (which are equivalent to the eikonal equations) are derived and compared to geometrical acoustics. When these dispersion relations agree with geometrical acoustics, the corresponding parabolic equations exactly describe the phase of a sound wave. The present article formulates the EWAPE, NAPE, and WAPE exactly describing the phase of a sound wave within the ranges of their applicability for arbitrary variations in the sound speed and Mach numbers. Therefore, these equations are termed the phase-preserving parabolic equations. On the other hand, many NAPEs and WAPEs from the literature preserve the phase of a sound wave only for low Mach numbers and/or small variations in the sound speed when the second-order terms can be omitted. Although the variations in the sound speed and Mach numbers are often relatively small, omitting the second-order terms might result in large cumulative phase errors for long ranges. For example, Ref. 15 argues that the second-order small terms cannot be omitted for the wind speed greater than or about 15 m/s; in this case, equations for arbitrary Mach numbers should be used. Thus, in many applications (such as sound propagation in the near-ground atmosphere and infrasound propagation in the upper atmosphere), it is desirable to use the phase-preserving EWAPE, NAPE, and WAPE.
The second approach for studying the ranges of applicability of parabolic equations is to compare them with the exact equation for sound propagation in stratified moving media. This article shows that the phase-preserving EWAPE, NAPE, and WAPE correctly describe sound propagation in stratified moving media within the ranges of their applicability.
This article is organized as follows. Section II briefly outlines results from Ref. 15 and presents new results such as the phase-preserving NAPE. In Sec. III, the ranges of applicability of EWAPEs, NAPEs, and WAPEs are studied by comparing them to the geometrical acoustics equations. In Sec. IV, these parabolic equations are compared to the exact equation in a stratified moving medium. Section V provides numerical algorithms for the phase-preserving NAPE and WAPE. Numerical results are presented in Sec. VI. The results are summarized in Sec. VII.
II. PARABOLIC EQUATIONS
This section considers parabolic equations for sound propagation in moving media.
A. EWAPE
Next, consider the WAPE obtainable from Eq. (3) by approximating the square-root operator with the Padé series expansion. The Padé series converges for all values of the operators and and, hence, for all values of σ.22 However, if and are small compared to one, fewer terms in this series are needed for a good approximation of the square root than in the case of large and . (This result is illustrated in Table I of Ref. 23.) This, again, dictates the choice of σ given by Eq. (8), at least for small θ.
Comparison between the NAPEs and WAPEs. The first two columns specify the parabolic equation and the function which satisfies this equation. The coefficients pertinent to these equations are in the last four columns.
Equation . | Function . | . | . | . | . |
---|---|---|---|---|---|
NAPE | 1 | 0 | |||
WAPE | 1 | ||||
WAPEM | |||||
NAPEeff | 1 | 0 | |||
WAPEeff |
Equation . | Function . | . | . | . | . |
---|---|---|---|---|---|
NAPE | 1 | 0 | |||
WAPE | 1 | ||||
WAPEM | |||||
NAPEeff | 1 | 0 | |||
WAPEeff |
Equations (9) and (10) coincide with Eqs. (B1) and (B2) in Ref. 15. However, Ref. 15 considers the EWAPE given by Eq. (9) only briefly in Appendix B. One of the main goals of this article is to study in detail this EWAPE and the NAPE and WAPE derived from this equation.
In the literature, the factor σ differs from that given by Eq. (8). In a motionless medium, σ is often set to one, e.g., Ref. 7. In this case, and the Taylor series expansion of the square-root operator in Eq. (3) is valid only for small ε. In a moving medium, the bulk of Ref. 15 considers the EWAPE, NAPE, and WAPE pertinent to . With this choice of σ, the NAPE is valid only for small ε, whereas the WAPE requires more terms in the Padé series.
Thus, the EWAPE given by Eq. (9) might be preferable compared to other equations used in the literature because the terms containing the operators and are small compared to one for arbitrary ε and M (for small θ). This feature of Eq. (9) is achieved by the factor premultiplying the square-root operator, whereas other EWAPEs do not have this factor. The EWAPE given by Eq. (9) does not contain the reference wavenumber k0. It is worthwhile to point out that Eqs. (9) and (10) are simpler (i.e., have fewer terms) than the corresponding Eqs. (39) and (40) in Ref. 15, which pertain to . As expected, these equations coincide if c is constant.
In the literature, EWAPEs, NAPEs, and WAPEs are derived by omitting some terms in starting equations such as Eq. (2). The ranges of applicability of the resulting equations are often difficult to assess. Another goal of this article is to study the ranges of applicability of these equations in detail. To this end, two approaches are used. First, starting with the EWAPE, NAPE, and WAPE, the dispersion relations (which are equivalent to the eikonal equations) are derived and compared to geometrical acoustics. Section III shows that within the ranges of applicability of the EWAPEs given by Eqs. (3) and (9) ( ), the dispersion relations following from these equations coincide with geometrical acoustics. Therefore, Eqs. (3) and (9) are termed the phase-preserving EWAPEs.
In the second approach, EWAPEs, NAPEs, and WAPEs are compared with the exact equation for sound propagation in stratified moving media. Section IV shows that for , the EWAPEs given by Eqs. (3) and (9) coincide with this exact equation.
Equations presented in this section are the starting point of the subsequent analysis.
B. NAPE
The NAPE [Eq. (13)] is valid for relatively small propagation angles θ. By studying the phase errors of a sound wave in motionless and moving media, Refs. 7 and 15 conclude that NAPEs are valid for the propagation angles . However, this conclusion depends on an acceptable phase error and the problem considered. For near-ground sound propagation, there is generally a consensus2,13 that a NAPE [including Eq. (13)] can be used if .
Section III shows that the dispersion relation derived from Eq. (13) coincides with geometrical acoustics within the ranges of applicability of Eq. (13); therefore, this equation is termed the phase-preserving NAPE. Section IV shows that within the ranges of its applicability, Eq. (13) coincides with the exact equation in a stratified moving medium. Given the wide range of applicability of the phase-preserving NAPE [Eq. (13)], it is remarkably simple. Numerical implementation of Eq. (13) is provided in Sec. V A.
Other NAPEs appearing in the literature are valid either for low Mach numbers and/or small variations in the sound speed. They preserve the phase of a sound wave only approximately and do not exactly coincide with the exact equation in a stratified moving medium. For example, the NAPE given by Eq. (46) in Ref. 15 is valid only for ; that equation is also much more involved than the phase-preserving NAPE [Eq. (13)].
C. WAPE
The Padé (1,1) approximation is a more accurate approximation of the square-root operator in Eq. (9) than the first two terms in the Taylor series. (For example, see Table I in Ref. 23.) Therefore, a WAPE in the Padé (1,1) approximation is valid for larger propagation angles than a NAPE. By studying the phase errors pertinent to WAPEs in the Padé (1,1) approximation, Refs. 7 and 15 conclude that these equations can be used if the propagation angles . As with NAPEs, this conclusion depends on an acceptable accuracy and the problem at hand.
Section III shows that the dispersion relation obtained from Eq. (17) coincides with geometrical acoustics within the ranges of applicability of Eq. (17). Therefore, Eq. (17) is termed the phase-preserving WAPE. In a stratified moving medium, Eq. (17) coincides with the exact equation, again, within the ranges of its applicability (See Sec. IV.).
As the Padé series converges for all values of the operators and in Eq. (3), the WAPE given by Eq. (44) in Ref. 15, which is obtained from Eq. (3) with σ = 1, is also phase preserving. However, as explained in Sec. II, more terms in the Padé series might be needed in Eq. (44) from Ref. 15 than in Eq. (17). Furthermore, Eq. (17) is simpler (has fewer terms) than Eq. (44) from Ref. 15. (See also Sec. V B.) Other WAPEs in moving media, such as those in Refs. 2, 13, and 14, are valid for small ε and M.
III. GEOMETRICAL ACOUSTICS
In this section, the ranges of applicability of EWAPEs, NAPEs, and WAPEs are studied by comparing them with the geometrical acoustics equations.
A. Dispersion relation and eikonal equation
B. EWAPE
The EWAPE given by Eq. (3) is derived from the convective Helmholtz equation [Eq. (2)]. The dispersion relation can also be formulated starting with Eq. (2). To this end, Eq. (24) is substituted into the convective Helmholtz equation. Similar to the analysis above, it can be shown that the operator transforms to and the operator Δ transforms to . With these transformations, the dispersion relation following from Eq. (2) exactly coincides with that used in geometrical acoustics [Eq. (18)].
C. NAPE
Other NAPEs in moving media correctly describe the phase of a sound wave only approximately when M and/or ε are small. For example, starting with Eq. (14), the dispersion relation can be derived. It can be shown that it coincides with the dispersion relation given by Eq. (34) only for small ε and M and the NAPE given by Eq. (46) from Ref. 15 preserves the phase of a sound wave only for small ε.
D. WAPE
The WAPE obtained from Eq. (3) in the Padé approximation also preserves the phase of a sound wave within the ranges of its applicability but might require more terms in the corresponding Padé series. (See Sec. II.) Other WAPEs appearing in the literature2,13,14 preserve the phase of a sound wave only for small ε and M.
IV. STRATIFIED MOVING MEDIUM
In this section, the ranges of applicability of EWAPEs, NAPEs, and WAPEs are studied by comparing them to an exact result for sound propagation in a stratified moving medium.
A. Exact equation
B. EWAPE
C. NAPE
D. WAPE
It can be shown that the WAPE given by Eq. (44) in Ref. 15 also coincides with the exact equation for sound propagation in a stratified medium within the ranges of its applicability. However, this equation might require more terms in the Padé series than in Eq. (53), as discussed in Sec. II. The WAPEs in Refs. 2, 13, and 14 coincide with the exact equation in a stratified moving medium only approximately for small ε and M.
V. NUMERICAL IMPLEMENTATION OF THE PARABOLIC EQUATIONS
A. NAPE
B. WAPE
C. Intercomparison of the parabolic equations
It follows from Secs. V A and V B that the phase-preserving NAPE and WAPE and the corresponding equations based on the effective sound speed approximation, NAPEeff and WAPEeff, reduce to the same equation [Eq. (57)] but with different coefficients . These coefficients are summarized in Table I. The rows “NAPE” and “WAPE” correspond to the phase-preserving NAPE and WAPE [Eqs. (13) and (17)], and the rows “NAPEeff” and “WAPEeff” correspond to the equations based on the effective sound speed approximation. The row “WAPEM” corresponds to Eq. (78) in Ref. 15, which is the 2D WAPE in the Padé (1,1) approximation and can be obtained from Eq. (3) in this article by setting σ = 1 and written in the form given by Eq. (57). As mentioned in the Introduction, Eq. (78) from Ref. 15 has already been used in the literature16–19 and is valid for arbitrary Mach numbers but small variations in the sound speed. In the WAPEM row, .
Interestingly, although the parabolic equations in Table I describe similar sound propagation phenomena, the coefficients in these equations differ significantly. Also, these coefficients are more complex for the WAPEM than for the phase-preserving WAPE. This result elucidates the statements above that the EWAPE, NAPE, and WAPE, which are obtained from Eq. (3) by setting σ equal to Eq. (8), are simpler than those pertinent to σ = 1.
Another difference between these parabolic equations is that the phase-preserving NAPE and WAPE and the WAPEM are formulated for the complex amplitude of the auxiliary function , while the NAPEeff and WAPEeff are formulated for the complex amplitude of the sound pressure .
VI. NUMERICAL RESULTS
In this section, numerical results obtained with the phase-preserving NAPE and WAPE and the NAPEeff and WAPEeff are compared to analytical and reference solutions. We will also briefly consider results obtained with the WAPEM.
In all numerical examples to follow, we consider a point source with the frequency Hz located above a perfectly reflecting, flat surface. The reference sound speed is m/s. The computational domain is 500 m long and 125 m high. At the lower boundary, the vertical derivative of for NAPE and WAPE and the vertical derivative of p for NAPEeff and WAPEeff are zero due to a perfectly reflecting surface. For the upper boundary, a perfectly matched layer (PML) is used.26 The thickness of the PML is 30 grid points. It is implemented as described in Ref. 19. In all numerical examples, for x greater than about 100 m but smaller than 500 m, the propagation angles .
A. Uniformly moving medium
The SPL calculated with the parabolic equations is depicted in Fig. 1, which also presents the SPL corresponding to the exact analytical solution of the problem considered, presented in Sec. VI B of Ref. 15. The SPL has several minima and maxima due to interference between the direct wave from the source to receiver and that reflected from the surface. For m, the SPL calculated with the phase-preserving WAPE is indistinguishable from the exact analytical solution. The phase-preserving NAPE results are also close to the analytical solution but at larger propagation ranges. The results obtained with the WAPEeff and NAPEeff noticeably deviate from the analytical solution.
(Color online) SPL in a uniformly moving medium with M = 0.5 produced by a point source located 50 m above a perfectly reflecting, flat surface. The sound frequency is 200 Hz. The subplots correspond to the exact analytical solution and numerical results obtained with the parabolic equations as labeled.
(Color online) SPL in a uniformly moving medium with M = 0.5 produced by a point source located 50 m above a perfectly reflecting, flat surface. The sound frequency is 200 Hz. The subplots correspond to the exact analytical solution and numerical results obtained with the parabolic equations as labeled.
These conclusions are reinforced by Fig. 2, in which the relative level obtained with the analytical solution and parabolic equations, is plotted versus the propagation range x for four heights z above the surface. For m, propagation angles θ can be relatively large such that the ranges of applicability of the NAPEs and WAPEs might not be fulfilled. Therefore, the predictions based on the parabolic equations can significantly deviate from the analytical solution. For m, the results obtained with the phase-preserving WAPE are very close to the analytical solution. The results obtained with the phase-preserving NAPE slightly deviate from the analytical solution in the range m because the NAPE is valid for smaller θ than the WAPE. However, for m (where θ becomes small), the NAPE is close to the analytical solution. On the other hand, Fig. 2 clearly shows that the results obtained with the WAPEeff and NAPEeff significantly deviate from the analytical solution. For the problem considered (but with different source height and frequency), inaccurate predictions of WAPEeff are also reported in Ref. 15.
(Color online) Relative level versus the propagation range x for four heights z above the surface. The geometry of the problem is the same as that for the SPL in Fig. 1. Different curves correspond to the analytical solution and parabolic equations.
(Color online) Relative level versus the propagation range x for four heights z above the surface. The geometry of the problem is the same as that for the SPL in Fig. 1. Different curves correspond to the analytical solution and parabolic equations.
(Color online) Normalized errors ϵ between the analytical solution and results obtained with the parabolic equations from Fig. 2.
(Color online) Normalized errors ϵ between the analytical solution and results obtained with the parabolic equations from Fig. 2.
Cumulative errors pertinent to the normalized errors ϵ shown in Fig. 3.
Equation . | z = 1 m . | z = 2 m . | z = 5 m . | z = 10 m . |
---|---|---|---|---|
NAPE | 0.03 | 0.03 | 0.09 | 0.15 |
WAPE | 0.00 | 0.01 | 0.01 | 0.02 |
NAPEeff | 0.44 | 0.32 | 0.45 | 0.43 |
WAPEeff | 0.46 | 0.51 | 0.59 | 0.76 |
Equation . | z = 1 m . | z = 2 m . | z = 5 m . | z = 10 m . |
---|---|---|---|---|
NAPE | 0.03 | 0.03 | 0.09 | 0.15 |
WAPE | 0.00 | 0.01 | 0.01 | 0.02 |
NAPEeff | 0.44 | 0.32 | 0.45 | 0.43 |
WAPEeff | 0.46 | 0.51 | 0.59 | 0.76 |
B. Linear sound speed profile
The SPL calculated with these parabolic equations is depicted in Fig. 4, which also shows the SPL calculated with the finite-difference time-domain (FDTD) solution of linearized equations of fluid dynamics. (For details, see the Appendix.) For the linear sound speed profile, the FDTD approach can be considered as a reference solution.
(Color online) SPL for a linear profile of the sound speed produced by a point source located 10 m above a perfectly reflecting, flat surface. The sound frequency is 200 Hz. The subplots correspond to the results obtained with the FDTD solution and parabolic equations as labeled.
(Color online) SPL for a linear profile of the sound speed produced by a point source located 10 m above a perfectly reflecting, flat surface. The sound frequency is 200 Hz. The subplots correspond to the results obtained with the FDTD solution and parabolic equations as labeled.
It can be observed from Fig. 4 that the phase-preserving WAPE provides the best agreement with the FDTD solution, followed by the WAPEeff, NAPE, and NAPEeff. This conclusion is further illustrated in Fig. 5, where the relative level calculated with the FDTD approach and parabolic equations is plotted versus the propagation range x for four heights z above the ground. Figure 5 clearly shows that for m, the results obtained with the phase-preserving WAPE are in a very good agreement with those obtained with the FDTD solution.
(Color online) Relative level versus the propagation range x for a linear profile of the sound speed and four heights z above the surface. The geometry of the problem is the same as that for the SPL in Fig. 4 Different curves correspond to the results obtained with the FDTD solution and parabolic equations.
(Color online) Relative level versus the propagation range x for a linear profile of the sound speed and four heights z above the surface. The geometry of the problem is the same as that for the SPL in Fig. 4 Different curves correspond to the results obtained with the FDTD solution and parabolic equations.
The normalized errors ϵ pertinent to the results displayed in Fig. 5 are depicted in Fig. 6. It follows from Fig. 6 that the normalized errors for the phase-preserving WAPE are smaller compared to those for the WAPEeff. Similarly, the phase-preserving NAPE has smaller normalized errors compared to the NAPEeff. The WAPEeff outperforms the NAPE because this section considers sound propagation in a motionless medium for which the effective sound speed approximation is not relevant.
(Color online) Normalized errors ϵ between the FDTD solution and results obtained with the parabolic equations from Fig. 5.
(Color online) Normalized errors ϵ between the FDTD solution and results obtained with the parabolic equations from Fig. 5.
Table III shows the cumulative errors pertinent to the normalized errors ϵ in Fig. 6. It follows from Table III that the phase-preserving WAPE has smaller cumulative errors compared to the WAPEeff, as does the phase-preserving NAPE compared to the NAPEeff.
Cumulative errors pertinent to the normalized errors ϵ depicted in Fig. 6.
Equation . | z = 1 m . | z = 2 m . | z = 5 m . | z = 10 m . |
---|---|---|---|---|
NAPE | 0.34 | 0.51 | 0.68 | 1.01 |
WAPE | 0.08 | 0.09 | 0.11 | 0.17 |
NAPEeff | 0.52 | 1.09 | 1.15 | 1.55 |
WAPEeff | 0.27 | 0.41 | 0.50 | 0.69 |
Equation . | z = 1 m . | z = 2 m . | z = 5 m . | z = 10 m . |
---|---|---|---|---|
NAPE | 0.34 | 0.51 | 0.68 | 1.01 |
WAPE | 0.08 | 0.09 | 0.11 | 0.17 |
NAPEeff | 0.52 | 1.09 | 1.15 | 1.55 |
WAPEeff | 0.27 | 0.41 | 0.50 | 0.69 |
C. Stratified moving medium
Figure 7 shows the relative level versus the propagation range x for four heights z above the surface. The results correspond to the FDTD solution, the phase-preserving NAPE and WAPE, and the NAPEeff and WAPEeff. For the considered case of a stratified moving medium, the FDTD solution is valid for arbitrary Mach numbers (see Sec. IV D in Ref. 27). It follows from Fig. 7 that the results obtained with the phase-preserving WAPE are very close to the FDTD solution and the phase-preserving NAPE outperforms the NAPEeff and WAPEeff. The results obtained with the latter two parabolic equations often deviate from the reference solution not only at the interference minima and maxima but also at the ranges between them.
(Color online) Relative level versus the propagation range x for a stratified moving medium (Sec. V C) and four heights z above a perfectly reflecting, flat surface. The source is located 10 m above the surface and the sound frequency is 200 Hz. Different curves correspond to the results obtained with the FDTD solution and parabolic equations.
(Color online) Relative level versus the propagation range x for a stratified moving medium (Sec. V C) and four heights z above a perfectly reflecting, flat surface. The source is located 10 m above the surface and the sound frequency is 200 Hz. Different curves correspond to the results obtained with the FDTD solution and parabolic equations.
Table IV shows the cumulative errors pertinent to the results in Fig. 7. It follows from the table that the phase-preserving WAPE has the smallest errors, followed by the phase-preserving NAPE, WAPEeff, and NAPEeff. The WAPEM was also used to calculate the relative level for the stratification considered in this section. The results obtained are very close to those for the phase-preserving WAPE with the same cumulative errors. This can be explained by the fact that for the stratification considered, the variation in the sound speed is relatively small.
Cumulative errors pertinent to the results depicted in Fig. 7.
Equation . | z = 1 m . | z = 2 m . | z = 5 m . | z = 10 m . |
---|---|---|---|---|
NAPE | 0.09 | 0.08 | 0.13 | 0.13 |
WAPE | 0.04 | 0.02 | 0.02 | 0.05 |
NAPEeff | 0.18 | 0.80 | 0.56 | 0.92 |
WAPEeff | 0.05 | 0.14 | 0.17 | 0.21 |
Equation . | z = 1 m . | z = 2 m . | z = 5 m . | z = 10 m . |
---|---|---|---|---|
NAPE | 0.09 | 0.08 | 0.13 | 0.13 |
WAPE | 0.04 | 0.02 | 0.02 | 0.05 |
NAPEeff | 0.18 | 0.80 | 0.56 | 0.92 |
WAPEeff | 0.05 | 0.14 | 0.17 | 0.21 |
VII. CONCLUSIONS
This article considered the phase-preserving EWAPEs, WAPEs, and NAPEs for sound propagation in moving media. These equations [Eqs. (9), (17), and (13), respectively] were derived in the high-frequency approximation while omitting the derivatives of the sound speed and medium velocity. Within the ranges of their applicability, the EWAPE, WAPE, and NAPE exactly describe the phase of sound waves and are valid for arbitrary variations in the sound speed and arbitrary (subsonic) Mach numbers. These equations also correctly describe sound propagation in a stratified moving medium within the ranges of their applicability.
Although variations in sound speed and wind Mach number are often relatively small, omitting the corresponding second-order terms can result in significant phase errors for long propagation ranges. Therefore, it is preferable to use the phase-preserving WAPE and NAPE, at least in some practical applications.
This article further showed that the WAPE given by Eq. (44) in Ref. 15 can also correctly describe the phase of a sound wave but might require more terms in the Padé series expansion. Moreover, that equation involves more terms than the phase-preserving WAPE given by Eq. (17) in this article. The WAPEs from Refs. 2, 13, and 14 preserve the phase of sound waves only approximately for small variations in sound speed and Mach numbers. Previous NAPEs from the literature (e.g., Refs. 2 and 15) correctly describe the phase of sound waves only for small variations in the sound speed and/or low Mach numbers.
Numerical simulations showed that the results obtained with the phase-preserving WAPE and NAPE agree better with the analytical and reference solutions than those with the parabolic equations based on the effective sound speed approximation (the WAPEeff and NAPEeff), which are often used in the literature. Limitations of this approximation have been reported elsewhere, e.g., Refs. 2, 15, and 28. There are, of course, problems for which the effective sound speed approximation might be suitable. However, because the complexity of numerical implementation of the phase-preserving WAPE and NAPE is about the same as that for the WAPEeff and NAPEeff, it makes sense to use the phase-preserving equations.
ACKNOWLEDGMENTS
This research was partially funded by the U.S. Army Engineer Research and Development Center (ERDC) basic research program. Permission to publish was granted by Director, Cold Regions Research and Engineering Laboratory. Any opinions expressed in this article are those of the authors and are not to be construed as official positions of the funding agency or the U.S. Department of the Army unless so designated by other authorized documents. This work was also performed within the framework of the LABEX CeLya (Grant No. ANR-10-LABX-0060) of Université de Lyon within the program Investissements d'Avenir (Grant No. ANR-16-IDEX-0005), operated by the French National Research Agency (ANR).
AUTHOR DECLARATION
Conflict of Interest
The authors have no conflicts to disclose.
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
Appendix
The FDTD solution uses the optimized fourth-order finite-difference technique (see Ref. 29 for details). The mesh grid is uniform with a grid step of 0.1 m, which corresponds to 17 points per wavelength for f = 200 Hz. The Courant-Friedrichs-Lewy number is set to 0.5, yielding a time step s. The FDTD simulation runs for 1.54 s.
The root mean square (rms) sound pressure is determined from the FDTD solution using the last three periods of the sound signal. For comparison with the parabolic equations, the rms pressure is normalized using the factor .