- Email: [email protected]
THE INTEGRATION OF δ′(f) IN A MULTI-DIMENSIONAL SPACE
Journal of Sound and
THE INTEGRATION OF d@ ( f ) IN A MULTI-DIMENSIONAL SPACE F. FARASSAT NASA ¸angley Research Center, Hampton,
1. INTRODUCTION
In the study of noise from high-speed surfaces, one needs to evaluate integrals involving d@ ( f ) where f (x)"0 or f (x, t)"0 is a stationary or moving surface on which acoustic sources lie. For example, consider the thickness noise term of the Ffowcs Williams}Hawkings (FW}H) equation when we take the time derivative explicitly: L L [o v d( f )]" [o v d ( f )], Lt 0 J n Lt 0 n (1) "o l5 d ( f )!o v v D + f Dd@ ( f ). 08n 0J n n It is assumed that the function f (x, t) is so de"ned that we have D + f D"1 on this surface. The local normal velocity of the surface is denoted v given by the relation n v "!Lf/Lt and a tilde under a symbol indicates restriction of the variable to the n surface f"0 [1]. This relation shows how the generalized function d@ ( f ) appears in wave propagation problems as the source term of the linear wave equation. We point out two features of equation (1). First, we note that only one of the two normal derivatives multiplying d@ ( f ) is restricted to the surface f"0. Second, the unrestricted function D + f D also appears as a factor of d@ ( f ) which cannot and should not be set equal to 1. In this paper, we will "rst give the interpretation of the integral I": /(x)d@( f ) dx,
(2)
where we will initially assume that D + f DO1 on the surface f"0. We next give two more useful results by "rst assuming that /(x)"Q(x)D ' + f D D+ f D
(3)
C
for some arbitrary C1 function Q(x) and then set D + f D"1 on the surface f"0. 2. DERIVATION OF THE MAIN RESULT
We use Gaussian co-ordinates (u1, u2) on the surface f"0 and then extend these co-ordinates along local normals to the space near the surface. Next, we take u3"f. 0022-460X/00/070460#03 $35.00/0
( 2000 Academic Press
461
LETTERS TO THE EDITOR
We have Jg@ dx"Jg du1 du2 du3" (2) du1 du2 du3. (3) D+ f D
(4)
Here Jg "Jg@ /D + f D2 is the determinant of the coe$cients of the "rst (3) (2) fundamental form in the new variables. The symbol g@ stands for the determinant (2) of the coe$cients of the "rst fundamental form of the surface f"u3"constant in variables (u1, u2, u3). Note that in this case, the surface f"constant is parametrized by the surface variables (u1, u2) but the function g@ is dependent on the variables (2) (u1, u2, u3). The variable u3 enters this function because it is the value of the constant. In accordance with the notation introduced in reference [1], the prime on g@ means that it is a function of u3. (2) Using equation (4) on the right-hand side of equation (2) gives the following result after integrating with respect to variable u3:
P G C DH du1 du2, 1 L / Jg@ "! P GD + f D Ln C D + f D(2)DH du1 du2.
I"!
L / Jg@ (2) Lu3 D+ f D
u3"0
(5)
u3"0
Taking the normal derivative in the integrand and using the following result from di!erential geometry [2]: L Jg@ "!2H Jg@ , (2) f (2) Ln
(6)
we get
P
Pf/0 G
/(x)d@( f ) dx"
C D
H
1 L /(x) 2H /(x) ! # f dS, D + f D Ln D + f D D + f D2
(7)
where H is the local mean curvature of the surface f"constant in equation (6) and f of the surface f"0 in equation (7). Here we have used the following result for element of the surface area of f"0: dS"[Jg@ ]u3"0 du1 du2"Jg du1 du2, (8) (2) (2) where Jg is simply the determinant of the coe$cients of the "rst fundamental (2) form of the surface f"0 in variables (u1, u2). Equation (7) is the main result of this paper. It was "rst explicitly given by the author in reference [1]. Another way of writing equation (7) is
P /(x)d@( f ) dx"Pf/0 G!D + f D2 Ln # 1
L/
H
/(x) n ) + D + f D 2H /(x) # f dS. D + f D2 D + f D3
(9)
Note that n ) + D + f D"L2f /Ln2 so that the integral on the right-hand side of equations (7) and (9) depend on both Lf/Ln"D + f D and L2f/Ln2.
462 2.1.
LETTERS TO THE EDITOR SPECIAL CASES
It can be seen that the integral I of equation (2) is not invariant under the change of description of the surface f"0. The same surface S de"ned by two di!erent functions f "0 and f "0 will, in general, give two di!erent values of I. This means 1 2 that the integrand of this equation, in the form written above, does not usually appear in applications. We will next give an integral involving d@( f ) which is invariant under the change of description of the surface f"0. In "nding this integral, we have been guided by examples similar to equation (1). If we assume that the function /(x) is given by equation (3), then, equation (7) becomes
Pf/0 G
C
H
LQ ! #2H Q dS. f Ln
: Q(x, t) D ' + f D D + f D d@ ( f ) dx"
(10)
This is the invariant result that we were seeking as seen from the right-hand side since the mean curvature is a quantity that does not depend on the representation + f D D + f D always of the surface f"0. In applications, we do get the function D ' C multiplying d@ ( f ). For example, in equation (1), by assumption D ' + f D"1 and we see C that D ' + f D D + f D"D + f D does multiply d@( f ). C In case, we have de"ned D ' + f D"1, i.e., D + f D"1 on f"0, then equation (10) C becomes
P
Pf/0 G
Q(x, t) D + f D d@ ( f ) dx"
H
LQ ! #2H Q dS. f Ln
(11)
+ f D"1 can considerably simplify This is a very useful result because the assumption D ' C algebraic manipulations involving generalized functions [1]. The readers may "nd the applications of this equation in wave propagation problems in references [1, 2]. Note that in some of the publications of the author, the function D + f D was left out of the integrand on the left-hand side of equation (11) [1, 2]. However, since this function was also left out in the manipulation of the source terms of a wave equation, mistaken for D ' + f D which was assumed equal to 1, the correct "nal results C were obtained. ACKNOWLEDGMENT
The author is deeply indebted to Dr Alexandre I. Saichev of the Radiophysical Department of Nizhni Novgorod University in Russia, for bringing the mistake referred to above to his notice [3].
REFERENCES 1. F. FARASSAT 1994 NASA ¹echnical Paper 3428 Introduction to generalized functions with applications in aerodynamics and aeroacoustics, Corrected copy (April 1996) (Available at ftp://techreports.larc.nasa.gov/pub/techreports/larc/94/tp3428.ps.Z). 2. F. FARASSAT 1996 NASA ¹echnical Memorandum 110285 The Kirchho! formulas for moving surfaces in aeroacoustics * the subsonic and supersonic cases (Available at ftp://techreports.larc.nasa.gov/pub/techreports/larc/96/NASA-96-tm110285.ps.Z). 3. A. I. SAICHEV 1999 Private communication.
THE INTEGRATION OF d@ ( f ) IN A MULTI-DIMENSIONAL SPACE F. FARASSAT NASA ¸angley Research Center, Hampton,
1. INTRODUCTION
In the study of noise from high-speed surfaces, one needs to evaluate integrals involving d@ ( f ) where f (x)"0 or f (x, t)"0 is a stationary or moving surface on which acoustic sources lie. For example, consider the thickness noise term of the Ffowcs Williams}Hawkings (FW}H) equation when we take the time derivative explicitly: L L [o v d( f )]" [o v d ( f )], Lt 0 J n Lt 0 n (1) "o l5 d ( f )!o v v D + f Dd@ ( f ). 08n 0J n n It is assumed that the function f (x, t) is so de"ned that we have D + f D"1 on this surface. The local normal velocity of the surface is denoted v given by the relation n v "!Lf/Lt and a tilde under a symbol indicates restriction of the variable to the n surface f"0 [1]. This relation shows how the generalized function d@ ( f ) appears in wave propagation problems as the source term of the linear wave equation. We point out two features of equation (1). First, we note that only one of the two normal derivatives multiplying d@ ( f ) is restricted to the surface f"0. Second, the unrestricted function D + f D also appears as a factor of d@ ( f ) which cannot and should not be set equal to 1. In this paper, we will "rst give the interpretation of the integral I": /(x)d@( f ) dx,
(2)
where we will initially assume that D + f DO1 on the surface f"0. We next give two more useful results by "rst assuming that /(x)"Q(x)D ' + f D D+ f D
(3)
C
for some arbitrary C1 function Q(x) and then set D + f D"1 on the surface f"0. 2. DERIVATION OF THE MAIN RESULT
We use Gaussian co-ordinates (u1, u2) on the surface f"0 and then extend these co-ordinates along local normals to the space near the surface. Next, we take u3"f. 0022-460X/00/070460#03 $35.00/0
( 2000 Academic Press
461
LETTERS TO THE EDITOR
We have Jg@ dx"Jg du1 du2 du3" (2) du1 du2 du3. (3) D+ f D
(4)
Here Jg "Jg@ /D + f D2 is the determinant of the coe$cients of the "rst (3) (2) fundamental form in the new variables. The symbol g@ stands for the determinant (2) of the coe$cients of the "rst fundamental form of the surface f"u3"constant in variables (u1, u2, u3). Note that in this case, the surface f"constant is parametrized by the surface variables (u1, u2) but the function g@ is dependent on the variables (2) (u1, u2, u3). The variable u3 enters this function because it is the value of the constant. In accordance with the notation introduced in reference [1], the prime on g@ means that it is a function of u3. (2) Using equation (4) on the right-hand side of equation (2) gives the following result after integrating with respect to variable u3:
P G C DH du1 du2, 1 L / Jg@ "! P GD + f D Ln C D + f D(2)DH du1 du2.
I"!
L / Jg@ (2) Lu3 D+ f D
u3"0
(5)
u3"0
Taking the normal derivative in the integrand and using the following result from di!erential geometry [2]: L Jg@ "!2H Jg@ , (2) f (2) Ln
(6)
we get
P
Pf/0 G
/(x)d@( f ) dx"
C D
H
1 L /(x) 2H /(x) ! # f dS, D + f D Ln D + f D D + f D2
(7)
where H is the local mean curvature of the surface f"constant in equation (6) and f of the surface f"0 in equation (7). Here we have used the following result for element of the surface area of f"0: dS"[Jg@ ]u3"0 du1 du2"Jg du1 du2, (8) (2) (2) where Jg is simply the determinant of the coe$cients of the "rst fundamental (2) form of the surface f"0 in variables (u1, u2). Equation (7) is the main result of this paper. It was "rst explicitly given by the author in reference [1]. Another way of writing equation (7) is
P /(x)d@( f ) dx"Pf/0 G!D + f D2 Ln # 1
L/
H
/(x) n ) + D + f D 2H /(x) # f dS. D + f D2 D + f D3
(9)
Note that n ) + D + f D"L2f /Ln2 so that the integral on the right-hand side of equations (7) and (9) depend on both Lf/Ln"D + f D and L2f/Ln2.
462 2.1.
LETTERS TO THE EDITOR SPECIAL CASES
It can be seen that the integral I of equation (2) is not invariant under the change of description of the surface f"0. The same surface S de"ned by two di!erent functions f "0 and f "0 will, in general, give two di!erent values of I. This means 1 2 that the integrand of this equation, in the form written above, does not usually appear in applications. We will next give an integral involving d@( f ) which is invariant under the change of description of the surface f"0. In "nding this integral, we have been guided by examples similar to equation (1). If we assume that the function /(x) is given by equation (3), then, equation (7) becomes
Pf/0 G
C
H
LQ ! #2H Q dS. f Ln
: Q(x, t) D ' + f D D + f D d@ ( f ) dx"
(10)
This is the invariant result that we were seeking as seen from the right-hand side since the mean curvature is a quantity that does not depend on the representation + f D D + f D always of the surface f"0. In applications, we do get the function D ' C multiplying d@ ( f ). For example, in equation (1), by assumption D ' + f D"1 and we see C that D ' + f D D + f D"D + f D does multiply d@( f ). C In case, we have de"ned D ' + f D"1, i.e., D + f D"1 on f"0, then equation (10) C becomes
P
Pf/0 G
Q(x, t) D + f D d@ ( f ) dx"
H
LQ ! #2H Q dS. f Ln
(11)
+ f D"1 can considerably simplify This is a very useful result because the assumption D ' C algebraic manipulations involving generalized functions [1]. The readers may "nd the applications of this equation in wave propagation problems in references [1, 2]. Note that in some of the publications of the author, the function D + f D was left out of the integrand on the left-hand side of equation (11) [1, 2]. However, since this function was also left out in the manipulation of the source terms of a wave equation, mistaken for D ' + f D which was assumed equal to 1, the correct "nal results C were obtained. ACKNOWLEDGMENT
The author is deeply indebted to Dr Alexandre I. Saichev of the Radiophysical Department of Nizhni Novgorod University in Russia, for bringing the mistake referred to above to his notice [3].
REFERENCES 1. F. FARASSAT 1994 NASA ¹echnical Paper 3428 Introduction to generalized functions with applications in aerodynamics and aeroacoustics, Corrected copy (April 1996) (Available at ftp://techreports.larc.nasa.gov/pub/techreports/larc/94/tp3428.ps.Z). 2. F. FARASSAT 1996 NASA ¹echnical Memorandum 110285 The Kirchho! formulas for moving surfaces in aeroacoustics * the subsonic and supersonic cases (Available at ftp://techreports.larc.nasa.gov/pub/techreports/larc/96/NASA-96-tm110285.ps.Z). 3. A. I. SAICHEV 1999 Private communication.