JP2945959B2 - Wave propagation prediction method - Google Patents
Wave propagation prediction methodInfo
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- JP2945959B2 JP2945959B2 JP562497A JP562497A JP2945959B2 JP 2945959 B2 JP2945959 B2 JP 2945959B2 JP 562497 A JP562497 A JP 562497A JP 562497 A JP562497 A JP 562497A JP 2945959 B2 JP2945959 B2 JP 2945959B2
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- equation
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- dimensional
- wave
- prediction
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- Measurement Of Mechanical Vibrations Or Ultrasonic Waves (AREA)
Description
【０００１】[0001]
【発明の属する技術分野】この発明は、電波、音波、弾
性波等について、２次元空間における波動伝搬の振る舞
い及び３次元空間における円筒対称な波動伝搬の遠距離
における振る舞いを高速、高効率、高忠実に予測する方
式に関する。BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to high-speed, high-efficiency, high-speed, It relates to a method of faithfully predicting.
【０００２】[0002]
【従来の技術】境界値問題を初期値問題に帰着すること
で、波動伝搬の予測を効率良く実行する方法としては、
放物型近似が良く知られている。2. Description of the Related Art As a method of efficiently performing wave propagation prediction by reducing a boundary value problem to an initial value problem,
The parabolic approximation is well known.
【０００３】参考文献を以下に示す。[0003] References are listed below.
【０００４】D.Lee and A.D.Pierce:"Parabolic Equati
on Development in Recent Decade,"Journal of Comput
ational Acoustics,Vol.3,No.2(1995)95-173.D. Lee and ADPierce: "Parabolic Equati
on Development in Recent Decade, "Journal of Comput
ational Acoustics, Vol. 3, No. 2 (1995) 95-173.
【０００５】上記参考文献は音波伝搬における放物型近
似についての総合報告である。本文献には、放物型近似
の導出、境界値問題を初期値問題に帰着することで計算
に必要となる記憶容量と計算の手間を減少できる理由、
３次元円筒対称波動伝搬への適用、放物型近似において
大きな角度の波動伝搬の計算に現れる誤差とその減少
法、一様でない媒質中の伝搬への適用、平らでない境界
への適用等について解説してある。The above reference is a comprehensive report on parabolic approximation in sound wave propagation. This paper describes the derivation of the parabolic approximation, the reason that the boundary value problem can be reduced to the initial value problem to reduce the storage capacity and computational effort required for the calculation,
Explains the application to 3D cylindrical symmetric wave propagation, the error that appears in the calculation of large angle wave propagation in parabolic approximation and its reduction method, the application to uneven medium propagation, the application to uneven boundaries, etc. I have.
【０００６】[0006]
【発明が解決しようとする課題】従来の放物型近似で
は、水平方向に対し大きな角度を持つ伝搬モードについ
ては誤差が大きくなる欠点がある。本発明はこのような
問題を解決するためになされたもので、大きな角度の伝
搬モードについても誤差のない予測ができるようにし、
高い精度かつ高い効率で波動伝搬の予測を実行できる波
動伝搬予測方式を提供することを目的とする。In the conventional parabolic approximation, there is a disadvantage that an error becomes large in a propagation mode having a large angle with respect to the horizontal direction. The present invention has been made in order to solve such a problem, and enables prediction without error even for a propagation mode having a large angle,
It is an object of the present invention to provide a wave propagation prediction method capable of executing wave propagation prediction with high accuracy and high efficiency.
【０００７】[0007]
【課題を解決するための手段】上記課題を解決するため
に、本発明の波動伝搬予測方式は、２次元のスカラーHe
lmholtz 方程式In order to solve the above-mentioned problems, a wave propagation prediction system according to the present invention employs a two-dimensional scalar He.
lmholtz equation
【０００８】[0008]
【数４】 (Equation 4)
【０００９】［但し、ｘ、ｙ：２次元ガウス座標系にお
ける座標成分、φ（ｘ，ｙ）：２次元空間における任意
のスカラー場、ｋ0 ：任意パラメータとする。］に帰着
する２成分１階偏微分方程式の形を持つ波動伝搬予測式[Where x and y are coordinate components in a two-dimensional Gaussian coordinate system, φ (x, y) is an arbitrary scalar field in a two-dimensional space, and k 0 is an arbitrary parameter. ] A wave propagation prediction formula having the form of a two-component first-order partial differential equation resulting in
【００１０】[0010]
【数５】 (Equation 5)
【００１１】［但し、σx 、σy ：パウリ行列と呼ばれ
る２行２列の行列、φ（ｘ，ｙ）：２個の成分φ
1 （ｘ，ｙ）、φ2 （ｘ，ｙ）からなり、[Where σ x , σ y : a matrix of 2 rows and 2 columns called a Pauli matrix, φ (x, y): two components φ
1 (x, y), φ 2 (x, y)
【００１２】[0012]
【数６】 (Equation 6)
【００１３】と表すことのできる２成分場とする。］を
用いて２次元波動伝搬及び円筒対称な３次元波動伝搬の
遠距離場の振る舞いを、高速、効率的かつ高忠実度で予
測することを特徴としている。A two-component field that can be expressed as ], The behavior of the far field of two-dimensional wave propagation and cylindrically symmetric three-dimensional wave propagation is predicted at high speed, efficiently and with high fidelity.
【００１４】ここで、本発明で用いる上記波動伝搬予測
式（２）が２次元Helmholtz 方程式（１）に帰着するこ
とを、以下に示す。The following shows that the wave propagation prediction equation (2) used in the present invention results in a two-dimensional Helmholtz equation (1).
【００１５】式（２）に使われているPauli 行列は次の
性質を満たす。The Pauli matrix used in equation (2) satisfies the following properties.
【００１６】 σx 2 ＝σy 2 ＝Ｉ …式（３）及びσx σy ＋σy σx ＝０ …式（４） ［但し、Ｉは下記式（５）による２行２列の単位行列で
ある。］Σ x 2 = σ y 2 = I Equation (3) and σ x σ y + σ y σ x = 0 Equation (4) [where I is a unit of 2 rows and 2 columns by the following equation (5)] It is a matrix. ]
【００１７】[0017]
【数７】 (Equation 7)
【００１８】σx とσy の具体的な例としては、下記式
（６）と式（７）のように与えられる。Specific examples of σ x and σ y are given by the following equations (6) and (7).
【００１９】[0019]
【数８】 (Equation 8)
【００２０】[0020]
【数９】 (Equation 9)
【００２１】この性質を使うと、式（２）より下記式
（８）を得る。Using this property, the following equation (8) is obtained from the equation (2).
【００２２】[0022]
【数１０】 (Equation 10)
【００２３】そして、式（８）は下記式（９）に変形で
きる。The equation (8) can be transformed into the following equation (9).
【００２４】[0024]
【数１１】 [Equation 11]
【００２５】従って、式（２）のφの各成分が２次元ス
カラーHelmholtz 方程式（１）を満たすことがわかる。Therefore, it is understood that each component of φ in the equation (2) satisfies the two-dimensional scalar Helmholtz equation (1).
【００２６】この波動伝搬予測方式において使用する上
記予測式は、近似無しにHelmholtz方程式に等価な１階
偏微分方程式である。このように何らの近似を用いるこ
となくHelmholtz 方程式を１階偏微分方程式に変形して
いる点が、従来の放物型近似と大きく異なる点である。
このため、上記予測式の満たす分散関係は、Helmholtz
方程式の分散関係と同じであり、これが任意角度の伝搬
モードに対しても誤差の無い伝搬の予測を実行でること
を保証する。また、上記予測式は従来の放物型近似と同
じく１階の偏微分方程式の形をしているため、波動伝搬
予測のための積分の計算が初期値問題に帰着され、波動
伝搬の予測を効率良く実行できる。The above-mentioned prediction equation used in this wave propagation prediction method is a first-order partial differential equation equivalent to the Helmholtz equation without approximation. Thus, the point that the Helmholtz equation is transformed into a first-order partial differential equation without using any approximation is significantly different from the conventional parabolic approximation.
For this reason, the dispersion relation satisfied by the above prediction equation is Helmholtz
This is the same as the variance relation of the equation, which guarantees that error-free propagation prediction can be performed even for a propagation mode at an arbitrary angle. In addition, since the above-mentioned prediction formula is in the form of a first-order partial differential equation as in the conventional parabolic approximation, the calculation of the integral for wave propagation prediction is reduced to an initial value problem, and the wave propagation prediction is performed. It can be executed efficiently.
【００２７】そして、本発明により構成した波動伝搬予
測装置を利用することにより、電波伝搬、音波伝搬、弾
性波伝搬等の予測を効率的かつ高精細に実行できる。こ
れにより、各種電波源、音響源、振動源等の環境へ与え
る影響等を効率よく評価することができる。また、これ
らの波動を利用する各種装置の性能を評価するためにも
有効である。さらに、場の振る舞いから電波源、音響
源、振動源等の振る舞いを予測する装置へのこの方式の
応用も有効である。The use of the wave propagation prediction device constructed according to the present invention enables efficient and high-definition prediction of radio wave propagation, sound wave propagation, elastic wave propagation and the like. This makes it possible to efficiently evaluate the effects of various radio wave sources, sound sources, vibration sources, and the like on the environment. It is also effective for evaluating the performance of various devices using these waves. Further, the application of this method to a device that predicts the behavior of a radio wave source, an acoustic source, a vibration source, and the like from the behavior of a field is also effective.
【００２８】[0028]
【発明の実施の形態】以下、本発明の波動伝搬予測方式
の実施の形態について説明する。DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of the wave propagation prediction system according to the present invention will be described below.
【００２９】図１の長方形領域ＡＢＣＤ内の音波伝搬の
予測を例に考える。辺ＤＡについては音源の分布に対応
した音圧分布が与えられているとする。辺ＡＢと辺ＣＤ
についてはある境界条件が与えられているとする。辺Ｂ
Ｃにおいては＋ｘ方向へ広がる波のみが存在するとす
る。Consider, for example, the prediction of sound wave propagation in the rectangular area ABCD in FIG. It is assumed that a sound pressure distribution corresponding to the distribution of the sound source is given to the side DA. Side AB and Side CD
It is assumed that a certain boundary condition is given for. Side B
In C, it is assumed that only a wave spreading in the + x direction exists.
【００３０】本発明の式（２）の波動伝搬予測式をこの
問題に適用するには、以下のように行う。式（２）はｘ
について１階の偏微分方程式であるから、あるｘにおけ
る場の振る舞いφ（ｘ，ｙ）が与えられれば、そこから
小さな正の量Δｘだけずれた位置ｘ＋Δｘにおける場の
振る舞いφ（ｘ＋Δｘ，ｙ）は、式（２）を＋ｘ方向に
積分することで求めることができる。この操作を繰り返
すことで、あるｘにおける場の振る舞いからそれより＋
ｘ方向の全ての領域における場の振る舞いを求めること
ができる。この手続を模式的に表したのが図２である。
図２は、ｘ＋Δｘにおける場の振る舞いは、ｘにおける
場の振る舞いから計算され、ｘ−Δｘより左の領域の場
の振る舞いの情報を必要としないことを表したものであ
る。これにより予測計算が効率良く実行できる。To apply the wave propagation prediction formula of the formula (2) of the present invention to this problem, the following is performed. Equation (2) is x
Is a first-order partial differential equation, the field behavior φ (x, y) at a position x + Δx shifted by a small positive amount Δx from a given field behavior φ (x, y) at a certain x Can be obtained by integrating equation (2) in the + x direction. By repeating this operation, from the behavior of the field at a certain x,
The field behavior in all regions in the x direction can be obtained. FIG. 2 schematically shows this procedure.
FIG. 2 shows that the field behavior at x + Δx is calculated from the field behavior at x and does not require information on the field behavior in the region to the left of x−Δx. As a result, the prediction calculation can be executed efficiently.
【００３１】この＋ｘ方向への積分計算において、各辺
における境界条件がどのように処理されるかを以下に述
べる。The following describes how the boundary conditions on each side are processed in the integral calculation in the + x direction.
【００３２】式（２）のφ（ｘ，ｙ）の第１成分、φ1
（ｘ，ｙ）、のｘ＝０における値は、図１の辺ＤＡにお
いて与えられている音圧分布に一致させる。これによ
り、上記の積分計算で得られたφ（ｘ，ｙ）のうち、第
１成分φ1 （ｘ，ｙ）を見れば、求めたい予測音圧が得
られることになる。The first component of φ (x, y) in equation (2), φ 1
The value of (x, y) at x = 0 is made to match the sound pressure distribution given on the side DA in FIG. Thus, by looking at the first component φ 1 (x, y) of φ (x, y) obtained by the above-described integral calculation, the expected sound pressure to be obtained can be obtained.
【００３３】φ（ｘ，ｙ）の第２成分φ2 （ｘ，ｙ）の
ｘ＝０における値は、図１の辺ＢＣにおいて与えられて
いる＋ｘ方向に広がる音波のみが含まれているという条
件を使って定めてやる。式（２）に下記式（１０）を代
入し、式（６）と式（７）のσx とσy を使って固有値
方程式を無限空間について解くと、下記式（１１）によ
る分散関係を得る。The value at x = 0 of the second component φ 2 (x, y) of φ (x, y) is said to include only the sound wave extending in the + x direction given on the side BC in FIG. I'll use conditions to determine it. When the following equation (10) is substituted into the equation (2), and the eigenvalue equation is solved for the infinite space using σ x and σ y of the equations (6) and (7), the dispersion relation by the following equation (11) is obtained. obtain.
【００３４】[0034]
【数１２】 (Equation 12)
【００３５】[0035]
【数１３】 (Equation 13)
【００３６】このうち、＋符号が＋ｘ方向へ広がる波に
対応する。式（１１）の分散関係のうち、＋ｘ方向へ広
がる波に対応する固有関数は、下記式（１２）で与えら
れる。Of these, the + sign corresponds to a wave spreading in the + x direction. The eigenfunction corresponding to the wave spreading in the + x direction in the dispersion relation of Expression (11) is given by Expression (12) below.
【００３７】[0037]
【数１４】 [Equation 14]
【００３８】この固有関数を重ね合わせて下記式（１
３）を作る。The eigenfunctions are superimposed to obtain the following equation (1)
Make 3).
【００３９】[0039]
【数１５】 (Equation 15)
【００４０】式（１３）に現れる重み関数ｗ（ｋ）は、
通常のFourier 変換の手法を使って、式（１３）のφ
（０，ｙ）の第１成分φ1 （０，ｙ）が、辺ＤＡにおい
て与えられている音圧分布に一致するように定めてや
る。このようにして辺ＤＡにおける音圧分布と辺ＢＣに
おける＋ｘ方向へ広がる波のみ含まれるという条件か
ら、２成分場φ（ｘ，ｙ）の辺ＤＡにおける振る舞いを
定めることができる。上記の説明は、ｋ0 が定数で、固
有関数の計算は無限に広い空間の場合を使った。実際の
場合では、音源は局在しており、対応して辺ＤＡにおけ
る音圧分布も局在したものである。また、ｋ0 が空間に
依存している場合も、その依存性は興味の対象である波
長に比べるとゆっくりとした変化である場合が多い。こ
のような状況下では、φ（ｘ，ｙ）の初期値の設定に
は、上記の説明で使った固有関数等を用いた設定法が近
似的に使えるであろう。The weighting function w (k) appearing in equation (13) is
Using the usual Fourier transform technique, φ of equation (13)
The first component φ 1 (0, y) of (0, y) is determined so as to match the sound pressure distribution given on the side DA. In this way, the behavior of the two-component field φ (x, y) on the side DA can be determined from the condition that only the sound pressure distribution on the side DA and the wave extending in the + x direction on the side BC are included. In the above description, the case where k 0 is a constant and the calculation of the eigenfunction is an infinitely large space is used. In an actual case, the sound source is localized, and the sound pressure distribution on the side DA is also localized correspondingly. Also, when k 0 depends on the space, the dependency often changes slowly compared to the wavelength of interest. In such a situation, the setting method using the eigenfunction or the like used in the above description may be approximately used for setting the initial value of φ (x, y).
【００４１】こうして定めた初期場（１３）を使って、
式（２）に基づく＋ｘ方向への積分計算を実行してい
く。辺ＡＢと辺ＣＤにおける境界条件は、この＋ｘ方向
への積分を実施する際に考慮してやればよい。この積分
手続とそれに伴う辺ＡＢと辺ＣＤについての境界条件の
考慮については、従来の放物型近似で用いられてきた方
法等が活用できる。このようにして音波伝搬の予測が実
行できる。なお、ここでは長方形領域における計算の実
施の形態を説明したが、従来の放物型近似で行われてい
るように、辺ＡＢあるいは辺ＣＤが直線でない場合につ
いても本発明の方法を適用することができる。Using the initial field (13) thus determined,
The integral calculation in the + x direction based on Expression (2) is executed. The boundary conditions on the side AB and the side CD may be considered when performing the integration in the + x direction. For the integration procedure and the consideration of the boundary conditions for the side AB and the side CD associated therewith, a method or the like used in the conventional parabolic approximation can be used. In this way, prediction of sound wave propagation can be performed. Although the embodiment of the calculation in the rectangular area has been described here, the method of the present invention can be applied to the case where the side AB or the side CD is not a straight line as in the conventional parabolic approximation. Can be.
【００４２】[0042]
【発明の効果】以上のように、本発明の波動伝搬予測方
式を用いると、式（１１）の分散関係は、式（１）のHe
lmholtz 方程式から得られるものと同じであり、波数ベ
クトル（ｗ，ｋ）で伝わる波に対応する。波数ベクトル
がｘ軸となす角は波の伝わる方向を表すのであるが、本
発明の場合の波数ベクトルはHelmholtz 方程式から得ら
れる波数ベクトルと同じであり、波の伝わる方向によら
ず誤差無しに正しい波数ベクトルを与える。このように
して、本発明の波動伝搬予測方式を用いれば、波の伝わ
る角度によらず、誤差無しに波動伝搬の予測が実行でき
る。また、本発明は従来の方法と同じく境界値問題を初
期値問題に帰着できるため、計算に要する計算機の記憶
容量を小さくでき、計算量も低減され、非常に効率の良
い波動伝搬の予測が実行できる。As described above, when the wave propagation prediction method of the present invention is used, the dispersion relation of the equation (11) is expressed by He of the equation (1).
It is the same as that obtained from the lmholtz equation, and corresponds to the wave transmitted by the wave vector (w, k). The angle formed by the wave vector and the x-axis represents the direction in which the wave travels, but the wave vector in the case of the present invention is the same as the wave vector obtained from the Helmholtz equation, and is correct without error regardless of the direction in which the wave travels. Gives the wave vector. In this way, by using the wave propagation prediction method of the present invention, the wave propagation can be predicted without error regardless of the angle at which the wave propagates. Further, since the present invention can reduce the boundary value problem to the initial value problem as in the conventional method, the storage capacity of the computer required for the calculation can be reduced, the calculation amount is reduced, and very efficient wave propagation prediction is performed. it can.
【図１】本図は本発明に係わる波動伝搬予測方式の実施
の形態において考えた２次元長方形領域の模式図であ
る。FIG. 1 is a schematic diagram of a two-dimensional rectangular area considered in an embodiment of a wave propagation prediction method according to the present invention.
【図２】本図は本発明に係わる波動伝搬予測方式によ
る、予測計算実施手順の模式図である。FIG. 2 is a schematic diagram of a procedure for performing a prediction calculation according to a wave propagation prediction method according to the present invention.
ＡＢＣＤ…長方形領域。 ABCD: rectangular area.
Claims (1)
分、φ（ｘ，ｙ）：２次元空間における任意のスカラー
場、ｋ0 ：任意パラメータとする。］に帰着する２成分
１階偏微分方程式の形を持つ波動伝搬予測式 【数２】 ［但し、σx ，σy ：パウリ行列と呼ばれる２行２列の
行列、φ（ｘ，ｙ）：２個の成分φ1 （ｘ，ｙ）、φ2
（ｘ，ｙ）からなり、 【数３】 と表すことのできる２成分場とする。］を用いて２次元
波動伝搬及び円筒対称な３次元波動伝搬の遠距離場の振
る舞いを予測することを特徴とする波動伝搬予測方式。1. Two-dimensional scalar Helmholtz equation [However, x, y: a coordinate component in a two-dimensional Gaussian coordinate system, φ (x, y): an arbitrary scalar field in a two-dimensional space, k 0 : an arbitrary parameter. A wave propagation prediction equation having the form of a two-component first-order partial differential equation resulting in [However, σ x , σ y : a 2 × 2 matrix called a Pauli matrix, φ (x, y): two components φ 1 (x, y), φ 2
(X, y), And a two-component field that can be expressed as ], Which predicts the behavior of a far field in two-dimensional wave propagation and cylindrically symmetric three-dimensional wave propagation.
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