- Email: [email protected]
Solution of the problem of the diffraction of an acoustic pulse by a wedge using wave potentials
Short communications
249
In the case 8o = ?Tthere follows from this the well-known expression for the cross-section of scattering of a soft sphere o=4naa ’
1 3 Pala +O((kaN~
i--
I 1
{
and for f30% 1 that for a circular disk of radius a&-~.For arbitrary parameters ku and 80 system (2.5) was solved by the method of reduction on the M-222 computer. The figure shows the complete scattering cross-section u as a function of the relative wave number ka for a fared aperture angle of the sphere 0 1. l’kanslated by J. Berry. REFERENCES 1.
VINOGRADOV, S. S. and SHESTOPALOV, V. P. The rigorous theory of the scattering of waves by a sphere with an aperture (the Helmholtz resonator). Ninth All-Union acoustics conference, Moscow, 1977 (IX VW. akustich. konf., Moskva, 1977), Subjects ofreports ofsection A (Tezisy dokl. Sekts. A), 79-81, Akistich. in-t, Moscow.
2.
RADIN, A. M. and SHESTOPALOV, V. P. Diffraction of waves by a sphere with an aperture. Dokl. Akad. Nauk SSSR, 212,4,838-841,1973.
3.
SHESTOPALOV, V. P. et aL Diffraction of wavesby lattices (Difraktsiya voln na reshetkakh), Izd-vo Khar’kovsk. un-ta, Khar’kov, 1973.
4.
KANTOROVICH, L. V. and AKILOV, G. P. Functional analysisin normalized spaces (Funktsional’nyi analiz v normirovannykh prostranstvakh), Flzmatgiz, Moscow, 1962.
5.
THOMAS, D. P. Diffraction by a spherical cap. Proc. timbridge Philos. Sot., 59,197-209,1963.
U.S.S.R. Comput. Maths Math. Phys Vol. 18, pp. 249- 254 0 Pergamon Press Ltd. 1979. Printed in Great Britain.
0041-5553/78/1001-0249$07.58/o
SOLUTION OF THE PROBLEM OF THE DIFFRACTION OF AN ACOUSTIC PULSE BY A WEDGE USING WAVE POTENTIALS* A. A. GLADKOV
Moscow (Received 19 October 1977; revised 19 December 1977)
THE FORMULATION and solution of the problem of the diffraction of an acoustic pulse by a wedge using integral equations for the wave surface potentials is considered. The numerical and analytic solutions are compared. 1. The solution of the problem of the diffraction of an acoustic pulse by a wedge has attracted many investigators from the beginning of the century [ 1,2] up to the present time. Especially valuable is [3], which determined the direction of investigations for many years. Below, using the example of the symmetric flow past a wedge we consider the numerical solution of the pulse diffraction problem obtained by using integral equations for the surface wave potentials [4].
*Zh. vjihisl. Mat. mat. Fiz., 18,5, 1324-1329.1978.
250
A. A. Gladkov
2. The wave surface potentials in three-dimensional space for the equation describing, for example, the velocity potential, can be obtained by starting from Kirchhoff s equation
(here n is the outward normal to S, and r is the geometrical distance between PO and P), which connects the value of the velocity potential at the points of the region of space considered with the boundary values of the potential and its normal derivative on the side of the boundary surface internal with respect to the region. For motions beginning from the zero instant, the area of integration S is defined as the part of the boundary surface inside a sphere of radius t with centre at PO. Kirchhoff s formula cannot be used directly to obtain the solution, since it is possible to specify independently the boundary values of either the potential, or its derivative, but not both together. To obtain the wave potentials we proceed by analogy with the derivation of Newtonian surface potentials. We assume that a function has been defined satisfying the wave equation on the outer side of the boundary surface. Then at the internal point PO we obtain 1 Z
1
Js( 7
8
acp-(P, t-r) an
+--
1
dr
q-(P,t-r)
r an
at
(2)
1 ar +---q-(P,t-r)
>
dS==O.
Subtracting (2) from (1) and regarding the function as continuous on passing through the surface, we obtain the wave potential of the simple layer 1
ui (PO,t) - -g
ss
P (P, t-4 dS
8
t
r
is the density of the wave potential of the simple layer, and where p=acp+/h-&p-/an regarding the normal derivative as continuous, we obtain the wave potential of the double layer (see also [5 ] ) 1 R2 (PO,t)
= --g
i ar JJ[
-gxvP,t-4+--G
#
where
V-cp+--cp-
1
ar avp,t-r) at
I
Qx
is the density of the wave potential of the double layer.
The surface potentials obtained may be used directly to solve boundary value problems. If the normal derivative is specified on the boundary (Neumann’s problem), the equation for the density of the potential of the simple layer is obtained by differentiating ~1 along the normal to the boundary surface as POapproaches the surface and using the connection between the limiting and direct value of the derivative of the potential u 1. If the function is specified on the boundary (Dirichlet’s problem), the equation for the density of the potential of the double layer is obtained from ~2 as PO approaches the surface and by using the connection between the direct and limiting value of the potential ~2 :
Short communications
v PO,t) = -2q PO,t) +
i ar $ JJ[ ~yy(P,
251
1
6
t-4 + ---
ar
r an
avp,t-r)
at
1as.
This form of the equation assumes that a tangent plane exists at the point PO of the boundary surface. Kirchhoff s equation regarded as an integral (more precisely, as an integro-differential) equation for the potential on the boundary surface for a specified normal derivative, is close in idea and in the technique of solution to equations for the density of wave surface layers. In this case it is essentially a matter of finding a combination of wave potentials of a simple and a double layer not creating a field from the outer side of the boundary surface. The equations obtained assume a number of constraints connected with the standard derivation of Kirchhoff s formula. More general possibilities of the application of the equations follow from the approach using Green’s formula in a four-dimensional space including time:
J( r
av ‘cp
aN
acp
)
do=0,
-v-,
aN
where a/aN is the derivative along the normal, v is the solution of the equation conjugate to the wave equation, and 2 is a closed surface in four-dimensional space. It follows from this formula that it is possible to combine regions of space separated by surfaces on which the fust derivative of cpis discontinuous, but the conformal derivative is continuous. Applying Hadamard’s general method, in particular choosing a solution v with a singularity at the point PO, considering a surface Z: consisting of a characteristic hypercone with vertex at PO, of part of the boundary surface with the given boundary condition cut off by the characteristic hypercone, and of the hypersurface t = 0 with zero initial conditions, and isolating the singularity of Y,it is possible again to arrive at Kirchhoff s formula for three-dimensional space [6] . This treatment permits us to consider with the use of the surface wave potential equations, for example, problems of the interaction of incident waves with the surface [7]. In this case by the domain of integration’S in the integral equations for the surface potentials we mean the part of the boundary surface within the sphere of radius I with centre at PO and behind the incident wave. In the case where the initial conditions are non-zero (mixed problem), to the solution obtained must be added the solution of the Cauchy problem, which is obtained in the form of integrals of the given initial values, whereupon correspondingly known additional terms appear in the boundary conditions. These equations may find their principal application in the numerical solution of mixed problems. The approach to the solution of the mixed problem described permits the solution to be divided into two stages: the determination of the density of the surface potentials and the calculation of the perturbed field by it. The advantage of the equations for the surface potentials in comparison, for example, with the method of finite differences is the reduction of the number of measurements of the region considered at the first stage of the solution of the problem, which appears to be especially favourable when the calculation of the whole of the perturbed field is not required. An additional merit of the surface potential equations is the fact that the difficulties connected with the specification of the boundary conditions are to a considerable extent removed. The use of the equations for the density of the simple layer potentials can cause difficulties in some cases because of the differentiation of the limits of integration over $ in the expression for ul (see [8]). In this respect the equations containing the potential of the double layer are preferable.
2.52
A. A. Gladkov
The problem of the diffraction of a wave by a wedge permits us to demonstrate the possibilities of the method and compare the numerical and exact solutions. For the numerical solution the simplest possible formulation of the problem and scheme of solution were taken. We also note that the direct value of the double layer potential, situated on a plane, on the same plane equals zero. This simplifies the evaluation of integrals over S. 3. We consider a wedge with semi-apex angle a, situated in a homogeneous medium symmetrically with respect to an incident acoustic wave (Fig. I). Let the potential of the incident where t is the time multiplied by the speed of wave be of the form cpo(z, t) =(t-s)H(tz), sound, which is measured from the instant when the wave touches the vertex of the wedge, and His a unit step function. The wedge is regarded as absolutely rigid and impenetrable. The perturbed field is described by the wave equation @q@ta=A~.
FIG. 2
FIG. 1
Here we confine ourselves to obtaining the solution on the surface of the wedge. For this the first stage of the problem alone is essentially sufficient. We will seek the solution in the plane z = 0. For the numerical solution we use two forms of the equations. a. The equation for the density of the double layer potential:
v (2, t) = 29l(z, t) -
2ztga n
8
v(E, r-4 1 *(&t-r) +P P di!
1GG
where
r= -
co9 cc
[cz+&a-2sE co9 2a+c2 co9 a]‘/*.
The axis of z (or 5) is perpendicular to the plane of the diagram. When this equation is used, instead of the external problem the internal Dirichlet problem is posed: the potential on the internal boundary must equal the potential cpowith the opposite sign.
(3)
253
Short communications
b. Kirchhoff s equation, regarded as an integral equation for the potential on the outer boundary of the wedge for the velocity Uo=- sin tl, defined on the surface, is determined from the n~penetration condition:
JJ[
+x
cp(%,t-4
+ /_
r3
9
a
4ce, t-4 a% 4 at 1
(4) >
-
For the numerical solution the surface of the wedge is subdivided into cells with sides Ax/cus a and APAX, the intensity of the singularities was regarded as constant in each cell and equal to its value at the centre of the ceil. The integral equations (3) and (4) were replaced by algebraic equations, which were solved successively for the unknown intensities with time step At. When evaluating the integrals the cells were considered (or not considered) as a whole according to the position of their centre. The integral over a cell was equated to the integrand taken at the centre of the cell multiplied by its area. Linear interpolation in time was carried out between very close values at the same point of space, as they arrived in the c~culation of the time derivative, the general scheme of calculation of the derivative being changed at the ends of the interval. The number of unknowns in the algebraic system was gradually increased as the incident wave advanced; however at each point in time and space only one equation with one unknown had to be solved. The results of lactations for a wedge with semi-apex angle o!= nf4 are given below. The solution of Eq. (3) was recalculated by the potential of the external problem, using the relation for the potential jump on passing through the double layer. Before considering the results of the calculation, we will consider the analytic solution of the problem with which a compa~son is made. 4. For the analytic solution we use the formula given in [9]. The use for the wedge of Green’s function satisfying the wave equation, enables a solution to be obtained for the total field when the plane wave
1
arc ch(i/r)
J
‘5
E) + Q(S+% %IIf&rch
[Q(-+60,
(5)
%)d%,
0
is incident on the wedge, where if
1
a(*)=
\*I+
0 if
p=a(n-a),
I@-s)==(t-z)R(t-4,
sin x(x-9)
sin x(%-k*)
Qf+, %)--:-
&=rc-a,
n
ch x%-cos x(lt+*) - ch x%-cos x(n-9) ’
az
%==-. B
We will also consider the value of I&, t) corresponding to 19= 0. Transforming expression (S), we can reduce it to the form n
cp(r,Q= 9i= $
$
([sin
(n-e,),
+-sin *2Z2(*2)]r
gtf2(q,)
$2
=
$
(n+*of,
-
fsin $JI(*~) --sin %zz*(92)lt)~
A. A. Gbzdkov
254 where
zi(g)-
2 1 exp[x areeh(t/r) I- eos up l-CO8 lp -arctg - arctgx sin 9 [ sin Ip sin 9 810 cb(t/r)
Zz(9)=
s 0
1 (
a ch xE -cos 9
'
We will call the result of the calculation of 9 the “exact” solution. We note that the numerical solution of Eqs. (3), (4) gives the perturbation potential, while the exact solution gives the total field including the incident wave. 5. The results of the solution of Eqs. (3) and (4) are shown in Fig. 2. The step along the x-axis was 0.02, the time step 0.05. This solution gives the potential of the perturbations on the surface for the whole interval x behind the wave, and not only for the segment [0, t] along the side of the wedge, like the exact solution. On the segment along the side of the wedge t - t/cosa the solution is obtained linearly dependent on the distance. A comparison of the solutions of (3) and (4) shows the practical agreement of the results themselves and their agreement with the exact solution, the error increasing somewhat at the ends of the interval. Similar results are obtained for smaller apex angles of the wedge also, although here a tendency for the accuracy of the numerical solution to faB may be observed. Therefore, the results obtained for the numerical example show the practical validity of the computing schemes used and permit them to be used to solve a wide class of diffraction problems. In conclusion the author thanks A. A. Dorodnitsyn for his interest. i’Yan&tedby J. Berry. REFERENCES
1.
SOMMERFIELD, A. Theoretisches iiber die Beugung der RBntgenstrahlen. Z. Moth. Pl?ys, 46,11,11-97, 1901.
2.
LAMB, H. On the diffraction of a solitary wave. Proc. Lond. Moth. Sot., 8,422-427,191O.
3.
SOBOLEV, S. L. Theory of the diffraction of plane waves. n Seirmologich. in-ta, No. 41, 1934.
4.
GLADKOV, A. A. On the solution of the wave equation. In “Lecture Notes Phyr” No. 35,187-190, Springer, Berlin, 1975.
5.
FULKS, W. and GUENTHER, R. B. Hyperbolic potential theory. Arch. Ratior, Mech. Analysis, 49,2, 79-88,1972.
6.
HADAMARD, J. Lecture on Gzuchy’s problem in linear partialdifferent&d equations Dover Publ., New York, 1953.
7.
FRIEDMAN, hi. B. and SHOW, R. P. Diffraction of pulses by cylindrical obstacles of arbitrary cross sections. mm. ASME, 29E, 1,40-46,1962.
8.
GLADKOV, A. A. Solution of the mixed problem for the wave equation by means of the wave potential of the simple layer. Uch. zup. TSAGZ,8,1,105-107,1977.
9.
FRIEDLANDER, F. G. Soundpulses (Zvukovye impul’sy), Jzd-vo in. lit., Moscow, 1962.
249
In the case 8o = ?Tthere follows from this the well-known expression for the cross-section of scattering of a soft sphere o=4naa ’
1 3 Pala +O((kaN~
i--
I 1
{
and for f30% 1 that for a circular disk of radius a&-~.For arbitrary parameters ku and 80 system (2.5) was solved by the method of reduction on the M-222 computer. The figure shows the complete scattering cross-section u as a function of the relative wave number ka for a fared aperture angle of the sphere 0 1. l’kanslated by J. Berry. REFERENCES 1.
VINOGRADOV, S. S. and SHESTOPALOV, V. P. The rigorous theory of the scattering of waves by a sphere with an aperture (the Helmholtz resonator). Ninth All-Union acoustics conference, Moscow, 1977 (IX VW. akustich. konf., Moskva, 1977), Subjects ofreports ofsection A (Tezisy dokl. Sekts. A), 79-81, Akistich. in-t, Moscow.
2.
RADIN, A. M. and SHESTOPALOV, V. P. Diffraction of waves by a sphere with an aperture. Dokl. Akad. Nauk SSSR, 212,4,838-841,1973.
3.
SHESTOPALOV, V. P. et aL Diffraction of wavesby lattices (Difraktsiya voln na reshetkakh), Izd-vo Khar’kovsk. un-ta, Khar’kov, 1973.
4.
KANTOROVICH, L. V. and AKILOV, G. P. Functional analysisin normalized spaces (Funktsional’nyi analiz v normirovannykh prostranstvakh), Flzmatgiz, Moscow, 1962.
5.
THOMAS, D. P. Diffraction by a spherical cap. Proc. timbridge Philos. Sot., 59,197-209,1963.
U.S.S.R. Comput. Maths Math. Phys Vol. 18, pp. 249- 254 0 Pergamon Press Ltd. 1979. Printed in Great Britain.
0041-5553/78/1001-0249$07.58/o
SOLUTION OF THE PROBLEM OF THE DIFFRACTION OF AN ACOUSTIC PULSE BY A WEDGE USING WAVE POTENTIALS* A. A. GLADKOV
Moscow (Received 19 October 1977; revised 19 December 1977)
THE FORMULATION and solution of the problem of the diffraction of an acoustic pulse by a wedge using integral equations for the wave surface potentials is considered. The numerical and analytic solutions are compared. 1. The solution of the problem of the diffraction of an acoustic pulse by a wedge has attracted many investigators from the beginning of the century [ 1,2] up to the present time. Especially valuable is [3], which determined the direction of investigations for many years. Below, using the example of the symmetric flow past a wedge we consider the numerical solution of the pulse diffraction problem obtained by using integral equations for the surface wave potentials [4].
*Zh. vjihisl. Mat. mat. Fiz., 18,5, 1324-1329.1978.
250
A. A. Gladkov
2. The wave surface potentials in three-dimensional space for the equation describing, for example, the velocity potential, can be obtained by starting from Kirchhoff s equation
(here n is the outward normal to S, and r is the geometrical distance between PO and P), which connects the value of the velocity potential at the points of the region of space considered with the boundary values of the potential and its normal derivative on the side of the boundary surface internal with respect to the region. For motions beginning from the zero instant, the area of integration S is defined as the part of the boundary surface inside a sphere of radius t with centre at PO. Kirchhoff s formula cannot be used directly to obtain the solution, since it is possible to specify independently the boundary values of either the potential, or its derivative, but not both together. To obtain the wave potentials we proceed by analogy with the derivation of Newtonian surface potentials. We assume that a function has been defined satisfying the wave equation on the outer side of the boundary surface. Then at the internal point PO we obtain 1 Z
1
Js( 7
8
acp-(P, t-r) an
+--
1
dr
q-(P,t-r)
r an
at
(2)
1 ar +---q-(P,t-r)
>
dS==O.
Subtracting (2) from (1) and regarding the function as continuous on passing through the surface, we obtain the wave potential of the simple layer 1
ui (PO,t) - -g
ss
P (P, t-4 dS
8
t
r
is the density of the wave potential of the simple layer, and where p=acp+/h-&p-/an regarding the normal derivative as continuous, we obtain the wave potential of the double layer (see also [5 ] ) 1 R2 (PO,t)
= --g
i ar JJ[
-gxvP,t-4+--G
#
where
V-cp+--cp-
1
ar avp,t-r) at
I
Qx
is the density of the wave potential of the double layer.
The surface potentials obtained may be used directly to solve boundary value problems. If the normal derivative is specified on the boundary (Neumann’s problem), the equation for the density of the potential of the simple layer is obtained by differentiating ~1 along the normal to the boundary surface as POapproaches the surface and using the connection between the limiting and direct value of the derivative of the potential u 1. If the function is specified on the boundary (Dirichlet’s problem), the equation for the density of the potential of the double layer is obtained from ~2 as PO approaches the surface and by using the connection between the direct and limiting value of the potential ~2 :
Short communications
v PO,t) = -2q PO,t) +
i ar $ JJ[ ~yy(P,
251
1
6
t-4 + ---
ar
r an
avp,t-r)
at
1as.
This form of the equation assumes that a tangent plane exists at the point PO of the boundary surface. Kirchhoff s equation regarded as an integral (more precisely, as an integro-differential) equation for the potential on the boundary surface for a specified normal derivative, is close in idea and in the technique of solution to equations for the density of wave surface layers. In this case it is essentially a matter of finding a combination of wave potentials of a simple and a double layer not creating a field from the outer side of the boundary surface. The equations obtained assume a number of constraints connected with the standard derivation of Kirchhoff s formula. More general possibilities of the application of the equations follow from the approach using Green’s formula in a four-dimensional space including time:
J( r
av ‘cp
aN
acp
)
do=0,
-v-,
aN
where a/aN is the derivative along the normal, v is the solution of the equation conjugate to the wave equation, and 2 is a closed surface in four-dimensional space. It follows from this formula that it is possible to combine regions of space separated by surfaces on which the fust derivative of cpis discontinuous, but the conformal derivative is continuous. Applying Hadamard’s general method, in particular choosing a solution v with a singularity at the point PO, considering a surface Z: consisting of a characteristic hypercone with vertex at PO, of part of the boundary surface with the given boundary condition cut off by the characteristic hypercone, and of the hypersurface t = 0 with zero initial conditions, and isolating the singularity of Y,it is possible again to arrive at Kirchhoff s formula for three-dimensional space [6] . This treatment permits us to consider with the use of the surface wave potential equations, for example, problems of the interaction of incident waves with the surface [7]. In this case by the domain of integration’S in the integral equations for the surface potentials we mean the part of the boundary surface within the sphere of radius I with centre at PO and behind the incident wave. In the case where the initial conditions are non-zero (mixed problem), to the solution obtained must be added the solution of the Cauchy problem, which is obtained in the form of integrals of the given initial values, whereupon correspondingly known additional terms appear in the boundary conditions. These equations may find their principal application in the numerical solution of mixed problems. The approach to the solution of the mixed problem described permits the solution to be divided into two stages: the determination of the density of the surface potentials and the calculation of the perturbed field by it. The advantage of the equations for the surface potentials in comparison, for example, with the method of finite differences is the reduction of the number of measurements of the region considered at the first stage of the solution of the problem, which appears to be especially favourable when the calculation of the whole of the perturbed field is not required. An additional merit of the surface potential equations is the fact that the difficulties connected with the specification of the boundary conditions are to a considerable extent removed. The use of the equations for the density of the simple layer potentials can cause difficulties in some cases because of the differentiation of the limits of integration over $ in the expression for ul (see [8]). In this respect the equations containing the potential of the double layer are preferable.
2.52
A. A. Gladkov
The problem of the diffraction of a wave by a wedge permits us to demonstrate the possibilities of the method and compare the numerical and exact solutions. For the numerical solution the simplest possible formulation of the problem and scheme of solution were taken. We also note that the direct value of the double layer potential, situated on a plane, on the same plane equals zero. This simplifies the evaluation of integrals over S. 3. We consider a wedge with semi-apex angle a, situated in a homogeneous medium symmetrically with respect to an incident acoustic wave (Fig. I). Let the potential of the incident where t is the time multiplied by the speed of wave be of the form cpo(z, t) =(t-s)H(tz), sound, which is measured from the instant when the wave touches the vertex of the wedge, and His a unit step function. The wedge is regarded as absolutely rigid and impenetrable. The perturbed field is described by the wave equation @q@ta=A~.
FIG. 2
FIG. 1
Here we confine ourselves to obtaining the solution on the surface of the wedge. For this the first stage of the problem alone is essentially sufficient. We will seek the solution in the plane z = 0. For the numerical solution we use two forms of the equations. a. The equation for the density of the double layer potential:
v (2, t) = 29l(z, t) -
2ztga n
8
v(E, r-4 1 *(&t-r) +P P di!
1GG
where
r= -
co9 cc
[cz+&a-2sE co9 2a+c2 co9 a]‘/*.
The axis of z (or 5) is perpendicular to the plane of the diagram. When this equation is used, instead of the external problem the internal Dirichlet problem is posed: the potential on the internal boundary must equal the potential cpowith the opposite sign.
(3)
253
Short communications
b. Kirchhoff s equation, regarded as an integral equation for the potential on the outer boundary of the wedge for the velocity Uo=- sin tl, defined on the surface, is determined from the n~penetration condition:
JJ[
+x
cp(%,t-4
+ /_
r3
9
a
4ce, t-4 a% 4 at 1
(4) >
-
For the numerical solution the surface of the wedge is subdivided into cells with sides Ax/cus a and APAX, the intensity of the singularities was regarded as constant in each cell and equal to its value at the centre of the ceil. The integral equations (3) and (4) were replaced by algebraic equations, which were solved successively for the unknown intensities with time step At. When evaluating the integrals the cells were considered (or not considered) as a whole according to the position of their centre. The integral over a cell was equated to the integrand taken at the centre of the cell multiplied by its area. Linear interpolation in time was carried out between very close values at the same point of space, as they arrived in the c~culation of the time derivative, the general scheme of calculation of the derivative being changed at the ends of the interval. The number of unknowns in the algebraic system was gradually increased as the incident wave advanced; however at each point in time and space only one equation with one unknown had to be solved. The results of lactations for a wedge with semi-apex angle o!= nf4 are given below. The solution of Eq. (3) was recalculated by the potential of the external problem, using the relation for the potential jump on passing through the double layer. Before considering the results of the calculation, we will consider the analytic solution of the problem with which a compa~son is made. 4. For the analytic solution we use the formula given in [9]. The use for the wedge of Green’s function satisfying the wave equation, enables a solution to be obtained for the total field when the plane wave
1
arc ch(i/r)
J
‘5
E) + Q(S+% %IIf&rch
[Q(-+60,
(5)
%)d%,
0
is incident on the wedge, where if
1
a(*)=
\*I+
0 if
p=a(n-a),
I@-s)==(t-z)R(t-4,
sin x(x-9)
sin x(%-k*)
Qf+, %)--:-
&=rc-a,
n
ch x%-cos x(lt+*) - ch x%-cos x(n-9) ’
az
%==-. B
We will also consider the value of I&, t) corresponding to 19= 0. Transforming expression (S), we can reduce it to the form n
cp(r,Q= 9i= $
$
([sin
(n-e,),
+-sin *2Z2(*2)]r
gtf2(q,)
$2
=
$
(n+*of,
-
fsin $JI(*~) --sin %zz*(92)lt)~
A. A. Gbzdkov
254 where
zi(g)-
2 1 exp[x areeh(t/r) I- eos up l-CO8 lp -arctg - arctgx sin 9 [ sin Ip sin 9 810 cb(t/r)
Zz(9)=
s 0
1 (
a ch xE -cos 9
'
We will call the result of the calculation of 9 the “exact” solution. We note that the numerical solution of Eqs. (3), (4) gives the perturbation potential, while the exact solution gives the total field including the incident wave. 5. The results of the solution of Eqs. (3) and (4) are shown in Fig. 2. The step along the x-axis was 0.02, the time step 0.05. This solution gives the potential of the perturbations on the surface for the whole interval x behind the wave, and not only for the segment [0, t] along the side of the wedge, like the exact solution. On the segment along the side of the wedge t - t/cosa the solution is obtained linearly dependent on the distance. A comparison of the solutions of (3) and (4) shows the practical agreement of the results themselves and their agreement with the exact solution, the error increasing somewhat at the ends of the interval. Similar results are obtained for smaller apex angles of the wedge also, although here a tendency for the accuracy of the numerical solution to faB may be observed. Therefore, the results obtained for the numerical example show the practical validity of the computing schemes used and permit them to be used to solve a wide class of diffraction problems. In conclusion the author thanks A. A. Dorodnitsyn for his interest. i’Yan&tedby J. Berry. REFERENCES
1.
SOMMERFIELD, A. Theoretisches iiber die Beugung der RBntgenstrahlen. Z. Moth. Pl?ys, 46,11,11-97, 1901.
2.
LAMB, H. On the diffraction of a solitary wave. Proc. Lond. Moth. Sot., 8,422-427,191O.
3.
SOBOLEV, S. L. Theory of the diffraction of plane waves. n Seirmologich. in-ta, No. 41, 1934.
4.
GLADKOV, A. A. On the solution of the wave equation. In “Lecture Notes Phyr” No. 35,187-190, Springer, Berlin, 1975.
5.
FULKS, W. and GUENTHER, R. B. Hyperbolic potential theory. Arch. Ratior, Mech. Analysis, 49,2, 79-88,1972.
6.
HADAMARD, J. Lecture on Gzuchy’s problem in linear partialdifferent&d equations Dover Publ., New York, 1953.
7.
FRIEDMAN, hi. B. and SHOW, R. P. Diffraction of pulses by cylindrical obstacles of arbitrary cross sections. mm. ASME, 29E, 1,40-46,1962.
8.
GLADKOV, A. A. Solution of the mixed problem for the wave equation by means of the wave potential of the simple layer. Uch. zup. TSAGZ,8,1,105-107,1977.
9.
FRIEDLANDER, F. G. Soundpulses (Zvukovye impul’sy), Jzd-vo in. lit., Moscow, 1962.