Abstract
Full Text
THEORY OF ELASTICITY
I. M. YAVORSKAYA
DIFFRACTION OF A PLANE LONGITUDINAL WAVE BY A CIRCULAR CYLINDER
(Presented by Academician L. I. Sedov on 27 IV 1964)
We consider the incidence of a stationary plane longitudinal wave with potential
(\varphi_0(r,t)=e^{i(k_1 r\cos\vartheta-\omega t)}) on a stress-free cylinder of radius (a), situated in an elastic medium (Fig. 1). The solution of analogous problems in the form of series was given in ((^1)), but these series converge well only when (a \ll 2\pi/k), i.e., when the cylinder diameter is small in comparison with the wavelength. In ((^{2,3})) a fairly general method was proposed for solving problems of scattering of elastic waves by curvilinear surfaces. However, the phenomena that arise in this case (in accordance with Kirchhoff’s principle) are considered only in the illuminated region. In the present note we use the method proposed by Watson ((^4)) and further developed in ((^{5-7})), etc., for problems of acoustics and electrodynamics—the method of transforming sums of series into contour integrals. We then investigate the displacements arising on the cylinder, expressed through contour integrals in the regions of “light,” “shadow,” and “penumbra” for the case (k_1a) and (k_2a \gg 1), where (k_1) and (k_2) are the wave numbers of longitudinal and transverse waves. The solution for the potentials of longitudinal and transverse waves outside the cylinder is written in the form ((^1))
Fig. 1
[
\varphi_0+\varphi_1
=
\sum_0^\infty
\varepsilon_n (i)^n
\left[
-\frac{\Delta_{J_n}}{\Delta_n}H_n^{(1)}(k_1 r)
+
J_n(k_1 r)
\right]\cos n\vartheta,
]
[
\psi_1
=
\frac{4i}{\pi}
\sum_0^\infty
\varepsilon_n (i)^n
\frac{n(y^2+2-2n^2)}{\Delta_n}
H_n^{(1)}(k_2 r)\sin n\vartheta,
\tag{1}
]
where (H_n^{(1)}(\rho)) are Hankel functions of the first kind; (J_n(\rho)) are Bessel functions;
[
\varepsilon_0=1;\qquad
\varepsilon_n=2\ (n=1,2,\ldots);\qquad
k_1a=x;\quad k_2a=y;
]
[
\Delta_n
=
\left[4n^2-(2n^2-y^2)^2\right]
H_n^{(1)}(x)H_n^{(1)}(y)
+
4xy(n^2-1)H_n^{(1)\prime}(x)H_n^{(1)\prime}(y)
-
]
[
-2y^2\left[xH_n^{(1)}(y)H_n^{(1)\prime}(x)
+
yH_n^{(1)}(x)H_n^{(1)\prime}(y)\right];
\tag{2}
]
(\Delta_{J_n}) is analogous to (\Delta_n), with (H_n^{(1)}(x)) replaced by (J_n(x)).
Let us study the displacements caused by the incidence of the wave at the boundary of the cylinder:
[
u
=
\frac{2y^2 i}{\pi a}
\sum_0^\infty
\varepsilon_n
\frac{(y^2-2n^2)H_n^{(1)}(y)+2yH_n^{(1)\prime}(y)}
{\Delta_n}
\cos n\vartheta\, e^{in\pi/2},
]
[
v
=
-\frac{4y^2 i}{\pi a}
\sum_0^\infty
\varepsilon_n
\frac{n\left[yH_n^{(1)\prime}(y)-H_n^{(1)}(y)\right]}
{\Delta_n}
\sin n\vartheta\, e^{in\pi/2}.
\tag{3}
]
It is known that if the quantities (k_1a) and (k_2a \gg 1), then these series converge very slowly and are unsuitable for practical use. Therefore we use Watson’s method of transforming the sums of the series (3) into integrals over contours of the complex (\nu)-plane. To this end, consider the functions of the complex variable
[
f_1(\nu)
=
\frac{(y^2-2\nu^2)H_\nu^{(1)}(y)+2yH_\nu^{(1)\prime}(y)}
{\Delta_\nu}
\frac{\cos\nu\vartheta}{\sin\nu\pi}
e^{-i\nu\pi/2},
\tag{4a}
]
[
f_2(\nu)=\frac{\nu\left[yH_\nu^{(1)\prime}(y)-H_\nu^{(1)}(y)\right]}{\Delta_\nu}\,
\frac{\sin \nu\vartheta}{\sin \nu\pi}\,e^{-i\nu\pi/2}.
\tag{46}
]
Then the displacements at the boundary of the cylinder are represented in the form
[
u=u_1+u_{1R}
=-\frac{2y^2}{\pi a}\int_C f_1(\nu)\,d\nu
-\frac{4y^2 i}{a}\operatorname{Res}\,[f_1(\nu)]_{\nu^*},
\tag{5}
]
[
v=v_1+v_{1R}
=\frac{4y^2}{\pi a}\int_C f_2(\nu)\,d\nu
+\frac{8y^2 i}{a}\operatorname{Res}\,[f_2(\nu)]_{\nu^*},
]
Fig. 2
where (C) is a loop around the real semiaxis, (\nu^*) is the real positive root of the equation (\Delta_\nu=0) (Fig. 2). (f_1(\nu)) and (f_2(\nu)) are odd analytic functions of (\nu), meromorphic in the right half-plane, having poles only in the first quadrant and on the real axis.
On the basis of what has been said, we transform the path of integration (C) and split it into parts, as shown in Fig. 2, where (A, B, D, E, F) are points of a circle of infinitely large radius; (\nu_1,\ldots,\nu_k) are zeros of the function (\Delta_\nu) lying in the first quadrant.
The integrals over portions of infinite circles, as can readily be shown using the properties of the functions (f_1(\nu)) and (f_2(\nu)), tend to zero. The integrals over (BD), owing to the evenness of the integrand functions, vanish. Thus,
[
u_1=-\frac{2y^2}{\pi a}\int_E^F
\frac{(y^2-2\nu^2)H_\nu^{(1)}(y)+2yH_\nu^{(1)\prime}(y)}
{\Delta_\nu}\,
\frac{\cos \nu\vartheta}{\sin \nu\pi}\,
e^{-i\nu\pi/2}\,d\nu,
]
[
v_1=\frac{4y^2}{\pi a}\int_E^F
\frac{\nu}{\Delta_\nu}
\left[yH_\nu^{(1)\prime}(y)-H_\nu^{(1)}(y)\right]
\frac{\sin \nu\vartheta}{\sin \nu\pi}\,
e^{-i\nu\pi/2}\,d\nu .
\tag{6}
]
The integrals over (EF) can be evaluated through the sum of the residues of the integrand functions at the zeros (\nu_1,\ldots,\nu_k,\ldots), lying in the first quadrant:
[
u_1=-\frac{4y^2 i}{a}\sum_{k=1}^{\infty}
\frac{(y^2-2\nu_k^2)H_{\nu_k}^{(1)}(y)+2yH_{\nu_k}^{(1)\prime}(y)}
{(\partial\Delta_\nu/\partial\nu)_{\nu_k}}\,
\frac{\cos \nu_k\vartheta}{\sin \nu_k\pi}\,
e^{-i\nu_k\pi/2},
\tag{7}
]
[
v_1=\frac{8y^2 i}{a}\sum_{k=1}^{\infty}
\frac{\nu_k}{(\partial\Delta_\nu/\partial\nu){\nu_k}}
\left[yH(y)\right]}^{(1)\prime}(y)-H_{\nu_k}^{(1)
\frac{\sin \nu_k\vartheta}{\sin \nu_k\pi}\,
e^{-i\nu_k\pi/2}.
]
From the subsequent investigation it will be seen that these series converge in the region where (|\vartheta|<\pi/2), i.e., in the shadow.
For the illuminated region the integrals (6) should be transformed, and then the displacements in the illuminated region are represented in the form of the sum of three terms:
[
u=-\frac{2y^2}{\pi a}\int_E^F
\frac{(y^2-2\nu^2)H_\nu^{(1)}(y)+2yH_\nu^{(1)\prime}(y)}
{\Delta_\nu}\,
\frac{\cos \nu(\pi-\vartheta)}{\sin \nu\pi}\,
e^{i\nu\pi/2}\,d\nu
+
]
[
+\frac{2y^2 i}{\pi a}\int_E^F
\frac{(y^2-2\nu^2)H_\nu^{(1)}(y)+2yH_\nu^{(1)\prime}(y)}
{\Delta_\nu}\,
e^{i\nu(\pi/2-\vartheta)}\,d\nu+u_{1R},
\tag{8a}
]
[
v=-\frac{4y^2}{\pi a}\int_E^F
\frac{\nu\left[yH_\nu^{(1)\prime}(y)-H_\nu^{(1)}(y)\right]}{\Delta_\nu}
\frac{\sin \nu(\pi-\vartheta)}{\sin \nu\pi}e^{i\nu\pi/2}\,d\nu+
]
[
+\frac{4y^2}{\pi a}\int_E^F
\frac{\nu\left[yH_\nu^{(1)\prime}(y)-H_\nu^{(1)}(y)\right]}{\Delta_\nu}
e^{i\nu(\pi/2-\vartheta)}\,d\nu+v_{1R}.
\tag{86}
]
The first integrals, analogously to the preceding one, are evaluated through the sum of residues at the zeros of (\Delta_\nu); the second can be transformed to an integral over (BD)
[
u=u_2+u_3+u_R=
]
[
=-\frac{4y^2i}{a}\sum_{k=1}^{\infty}
\frac{(y^2-2\nu_k^2)H_{\nu_k}^{(1)}(y)+2yH_{\nu_k}^{(1)\prime}(y)}
{(\partial\Delta_\nu/\partial\nu)_{\nu_k}}
\frac{\cos\nu_k(\pi-\vartheta)}{\sin\nu_k\pi}e^{i\nu_k\pi/2}-
]
[
-\frac{2y^2i}{\pi a}\int_B^D
\frac{(y^2-2\nu^2)H_\nu^{(1)}(y)+2yH_\nu^{(1)\prime}(y)}
{\Delta_\nu}
e^{i\nu(\pi/2-\vartheta)}\,d\nu+u_R;
\tag{9}
]
[
v=v_2+v_3+v_R=-\frac{8y^2i}{a}\sum_{k=1}^{\infty}
\frac{\nu_k\left[yH_{\nu_k}^{(1)\prime}(y)-H_{\nu_k}^{(1)}(y)\right]}
{(\partial\Delta_\nu/\partial\nu)_{\nu_k}}
\frac{\sin\nu_k(\pi-\vartheta)}{\sin\nu_k\pi}e^{i\nu_k\pi/2}-
]
[
-\frac{4y^2}{\pi a}\int_B^D
\frac{\nu\left[yH_\nu^{(1)\prime}(y)-H_\nu^{(1)}(y)\right]}
{\Delta_\nu}
e^{i\nu(\pi/2-\vartheta)}\,d\nu+v_R,
]
where
[
u_R=-\frac{4y^2i}{a}
\frac{(y^2-2\nu^{2})H_{\nu^}^{(1)}(y)+2yH_{\nu^}^{(1)\prime}(y)}
{(\partial\Delta_\nu/\partial\nu)_{\nu^}}
\frac{\cos\nu^(\pi-\vartheta)}{\sin\nu^\pi}e^{i\nu^*\pi/2},
]
[
v_R=-\frac{8y^2i}{a}
\frac{\nu^\left[yH_{\nu^}^{(1)\prime}(y)-H_{\nu^}^{(1)}(y)\right]}
{(\partial\Delta_\nu/\partial\nu)_{\nu^}}
\frac{\sin\nu^(\pi-\vartheta)}{\sin\nu^\pi}e^{i\nu^*\pi/2}.
\tag{10}
]
Next, to compute the displacements it is necessary to use the asymptotic form of the Hankel functions for large arguments. For the regions where
[
|x^2-\nu^2|\gg Ax^{4/3},\qquad |y^2-\nu^2|\gg Ay^{4/3},\qquad A\sim 3,
\tag{11}
]
one may use for (H_\nu^{(1)}(x)), (H_\nu^{(1)}(y)) and their derivatives Debye’s asymptotic formula (8). The equation (\bar{\Delta}\nu\equiv \left(H\nu^{(1)}(x)H_\nu^{(1)}(y)\right)^{-1}\Delta_\nu=0) in the Lebedev approximation has the form
[
\left(2\frac{\nu^2}{y^2}-1\right)^2
-4\frac{x}{y}\frac{\nu^2}{y^2}
\sqrt{\frac{\nu^2}{x^2}-1}\sqrt{\frac{\nu^2}{y^2}-1}
-
\tag{12}
]
[
-\frac{2}{y}\left[\frac{x}{y}\sqrt{\frac{\nu^2}{x^2}-1}+
\sqrt{\frac{\nu^2}{y^2}-1}\right]
-\frac{4}{y^2}\left[\frac{\nu^2}{y^2}
-\frac{x}{y}\sqrt{\frac{\nu^2}{x^2}-1}\sqrt{\frac{\nu^2}{y^2}-1}\right]=0.
]
To accuracy up to (y^{-1}), equation (12) is equivalent to the Rayleigh equation, which has a real positive root (\nu^=\chi y), where (\chi>1) and (\chi) depends on (\varepsilon=k_1/k_2<1). Therefore the residues at the point (\nu^), corresponding to the displacement components (u_R, v_R, u_{1R}), and (v_{1R}), give the displacements in the surface wave of Rayleigh-wave type. The displacement components (u_3) and (v_3), represented in (9) by integrals over (BD), can be evaluated by the saddle-point method:
[
u_3=
\frac{2k_1i(2\varepsilon^2\sin^2\vartheta-1)\cos\vartheta}
{(2\varepsilon^2\sin^2\vartheta-1)^2
-4\varepsilon^3\sin^2\vartheta\cos\vartheta\sqrt{1-\varepsilon^2\sin^2\vartheta}}
e^{ik_1a\cos\vartheta},
]
[
v_3=-
\frac{4\varepsilon k_1i\sin\vartheta\cos\vartheta\sqrt{1-\varepsilon^2\sin^2\vartheta}}
{(2\varepsilon^2\sin^2\vartheta-1)^2
-4\varepsilon^3\sin^2\vartheta\cos\vartheta\sqrt{1-\varepsilon^2\sin^2\vartheta}}
e^{ik_1a\cos\vartheta}.
\tag{13}
]
From (14) we see that (u_3) and (v_3) give the geometrical components of the displacement field.
Debye’s asymptotic formula for the function (H_\nu^{(1)}(x)) will be inapplicable: (a) near the boundary of the shadow, since there (|\nu_0^2-x^2|\ll Ax^{4/3}), i.e. (|\cos(\pi-\vartheta)|\sim (k_1a)^{-1/3}), and (b) in the calculation of the smallest roots of the equation (\Delta_\nu=0), since they lie inside the Fock circle. In these cases for the Hankel function
for (H_\nu^{(1)}(y)) one can as before use Debye’s asymptotic formula, and for (H_\nu^{(1)}(x)) the more accurate Fock–Hankel asymptotic formula.
For calculating the diffraction components of the displacements (u_2, v_1) and (u_2, v_2), it is usually sufficient to take several first terms of the series in (\nu_k). For (\nu_k=x+(x/2)^{1/3}t_k), lying inside the Fock region, (t_k (x/2)^{-2/3}\ll 1), where (t_k) is a root of the equation (w'(t)-q_1w(t)=0), and (w(t)) is the Airy function. Then
[
u_1=\frac{k_1 i\sqrt{\pi}}{2}\,
\frac{1-2\varepsilon^2}{\varepsilon^3\sqrt{1-\varepsilon^2}}
\sum_k
\frac{1}{w(t_k)(t_k-q_1^2)}
\frac{\cos \nu_k\vartheta}{\sin \nu_k\pi}
e^{-i\nu_k\pi/2},
]
[
v_1=\frac{k_1\sqrt{\pi}}{\varepsilon^2}
\sum_k
\frac{1}{w(t_k)(t_k-q_1^2)}
\frac{\sin \nu_k\vartheta}{\sin \nu_k\pi}
e^{-i\nu_k\pi/2},
\tag{14}
]
where (q_1) is a constant quantity equal to
(-\,i(x/2)^{1/3}(2\varepsilon^2-1)^2/4\varepsilon^3\sqrt{1-\varepsilon^2}).
The equation (w'(t)-q_1w(t)=0) is studied in detail in work (5). The investigation shows that the series (16) converge for (|\vartheta|<\pi/2), i.e., they can be used to calculate displacements only in the shadow region. Analogous series are obtained for the displacements (u_2) and (v_2) in the illuminated region:
[
u_2=\frac{k_1 i\sqrt{\pi}}{2}\,
\frac{1-2\varepsilon^2}{\varepsilon^3\sqrt{1-\varepsilon^2}}
\sum_k
\frac{1}{w(t_k)(t_k-q_1^2)}
\frac{\cos \nu_k(\pi-\vartheta)}{\sin \nu_k\pi}
e^{i\nu_k\pi/2},
]
[
v_2=-\frac{k_1\sqrt{\pi}}{\varepsilon^2}
\sum_k
\frac{1}{w(t_k)(t_k-q_1^2)}
\frac{\sin \nu_k(\pi-\vartheta)}{\sin \nu_k\pi}
e^{i\nu_k\pi/2}.
\tag{15}
]
The series (15) converge everywhere, i.e., they are also suitable in the penumbra region. To calculate the total displacement field in the penumbra, it remains to indicate how to calculate the integrals over (BD) for (\vartheta\sim\pi/2), since the saddle-point approximation is not applicable there owing to the closeness of (\nu_0) to (k_1a). Using the Hankel–Fock approximation, we obtain
[
u_3=
\frac{k_1 i\exp{ik_1a(\pi/2-\vartheta)}}{4\varepsilon^3\sqrt{\pi}}
\int_{-\infty}^{\infty}
\frac{1-2\varepsilon^2\left[1+(x/2)^{-2/3}t/2\right]^2}
{\left[1+(x/2)^{-2/3}t/2\right]^2
\sqrt{1-\varepsilon^2\left[1+(x/2)^{-2/3}t/2\right]^2}}
\times
\frac{\exp{i(x/2)^{1/3}t(\pi/2-\vartheta)}}
{w'(t)-qw(t)}\,dt,
]
[
v_3=
\frac{k_1 i\exp{ik_1a(\pi/2-\vartheta)}}{2\varepsilon^2\sqrt{\pi}}
\int_{-\infty}^{\infty}
\frac{\exp{i(x/2)^{1/3}t(\pi/2-\vartheta)}}
{\left[1+(x/2)^{-2/3}t/2\right]}
\frac{dt}{w'(t)-qw(t)}.
\tag{16}
]
The integrals (16) can be calculated by quadrature formulas, replacing, in the (t)-plane, integration over (BD) by integrals over the broken line (\Gamma_1), going from (\infty e^{2i\pi/3}) to 0 along the straight line (\arg t=2\pi/3), and from 0 to (\infty) along the real axis. The resulting integrals with real limits are not difficult to compute by means of the tables given in (5).
In conclusion, the author expresses gratitude to G. A. Skuridin for valuable advice and discussions of the work.
Received
29 IV 1964
REFERENCES
- V. D. Kupradze, Fundamental Problems of the Mathematical Theory of Diffraction, 1934.
- G. A. Skuridin, Izv. AN SSSR, ser. geofiz., No. 2 (1957).
- G. A. Skuridin, Prikl. matem. i mekh., 25, No. 3 (1961).
- G. N. Watson, Proc. Roy. Soc., 95, No. A 666 (1918).
- V. A. Fock, Diffraction of Radio Waves around the Earth’s Surface, Publishing House of the Academy of Sciences of the USSR, 1946.
- W. Franz, Zs. Naturforsch., 9a, No. 9 (1954).
- I. Imai, Zs. Phys., 137, 31 (1954).
- G. I. Petrashen, N. S. Smirnova, G. I. Makarov, Uch. zap. LGU, issue 27, No. 170 (1953).