Nonhomogeneous Wave Equation ❱

6.2. Nonhomogeneous Wave Equation ❱

6.2.1. Problems in Cartesian Coordinates

6.2.1-1. Domain: − ❴ < ▲ < ❴ ,− ❴ < ▼ < ❴ ,− ❴ < ❵ < ❴ . Cauchy problem. Initial conditions are prescribed:

where the integration is performed over the surface of the sphere ( ♥ = ❤ ❝ ) and the volume of the

✇❚❙ sphere ( ≤

) with center at ( ✉ , ✈ , ❵ ).

Reference : N. S. Koshlyakov, E. B. Glizer, and M. M. Smirnov (1970).

6.2.1-2. Domain: 0 ≤ ✉ ≤ ① 1 ,0≤ ✈ ≤ ① 2 ,0≤ ❵ ≤ ① 3 . Different boundary value problems.

1 ② . The solution of the first boundary value problem for a parallelepiped is given by the formula from Paragraph 6.1.1-3 with the additional term

which allows for the equation’s nonhomogeneity; this term is the solution of the nonhomogeneous equation with homogeneous initial and boundary conditions.

2 ② . The solution of the second boundary value problem for a parallelepiped is given by the formula from Paragraph 6.1.1-4 with the additional term specified in Paragraph 6.2.1-2, Item 1 ② ; the Green’s

function is taken from Paragraph 6.1.1-4.

3 ② . The solution of the third boundary value problem for a parallelepiped is the sum of the so- lution of the homogeneous equation with nonhomogeneous initial and boundary conditions (see

Paragraph 6.1.1-5) and the solution of the nonhomogeneous equation with homogeneous initial and boundary conditions. The latter solution is given by the formula from Paragraph 6.2.1-2, Item 1 ② , in which one should substitute the Green’s function from Paragraph 6.1.1-5.

4 ② . The solutions of mixed boundary value problems for a parallelepiped are given by the formulas from Paragraph 6.1.1-6 to which one should add the term specified in Paragraph 6.2.1-2, Item 1 ② .

6.2.2. Problems in Cylindrical Coordinates

A three-dimensional nonhomogeneous wave equation in the cylindrical coordinate system is written ❛ ❛ ❛ ❛

as

6.2.2-1. Domain: 0 ≤ ⑨ ≤

,0≤ ≤2 ➂ ,0≤ ➃ ≤ ① . Different boundary value problems.

1 ② . The solution of the first boundary value problem for a circular cylinder of finite length is given by the formula from Paragraph 6.1.2-1 with the additional term

which allows for the equation’s nonhomogeneity.

2 ② . The solution of the second boundary value problem for a circular cylinder of finite length is given by the formula from Paragraph 6.1.2-2 with the additional term (1).

3 ② . The solution of the third boundary value problem for a circular cylinder of finite length is the sum of the solution specified in Paragraph 6.1.2-3 and expression (1).

4 ② . The solutions of mixed boundary value problems for a circular cylinder of finite length are given by the formulas from Paragraph 6.1.2-4 with additional terms of the form (1).

6.2.2-2. Domain: ➁ 1 ≤ ♥

2 ,0≤ ≤2 ➂ ,0≤ ➃ ≤ ① . Different boundary value problems.

1 ② . The solution of the first boundary value problem for a hollow cylinder of finite dimensions is given by the formula from Paragraph 6.1.2-5 with the additional term

which allows for the equation’s nonhomogeneity. ➆

2 ② . The solution of the second boundary value problem for a hollow cylinder of finite dimensions is given by the formula from Paragraph 6.1.2-6 with the additional term (2).

3 ② . The solution of the third boundary value problem for a hollow cylinder of finite dimensions is the sum of the solution specified in Paragraph 6.1.2-7 and expression (2).

4 ② . The solutions of mixed boundary value problems for a hollow cylinder of finite dimensions are given by the formulas from Paragraph 6.1.2-8 with additional terms of the form (2).

6.2.2-3. Domain: 0 ≤ ⑨ ≤

,0≤ ≤ 0 ,0≤ ➃ ≤ ① . Different boundary value problems.

1 ② . The solution of the first boundary value problem for a cylindrical sector of finite thickness is given by the formula from Paragraph 6.1.2-9 with the additional term

which allows for the equation’s nonhomogeneity.

2 ② . The solution of a mixed boundary value problem for a cylindrical sector of finite thickness is given by the formula from Paragraph 6.1.2-10 with the additional term (2).

6.2.3. Problems in Spherical Coordinates

A three-dimensional nonhomogeneous wave equation in the spherical coordinate system is repre- sented as

sin ➌ ➉ ⑧ + ♥ 2 2 ➉ ➌ ⑨ ➌ ➌ ➌ 2 ⑩ + q ( , ➌ , , ➇ ).

sin

sin

6.2.3-1. Domain: 0 ≤ ⑨ ≤

,0≤ ➌ ≤ ➂ ,0≤ ≤2 ➂ . Boundary value problem.

1 ② . The solution of the first boundary value problem for a sphere is given by the formula from Paragraph 6.1.3-1 with the additional term

( , , , ) ( , ➌ , , ❦ , ❧ , ♠ , ➇ − ④ ) ❦ sin ❧ ♦ ❦ ♦ ❧ ♦ ♠ ♦ ④ , (1)

which allows for the equation’s nonhomogeneity.

2 ② . The solution of the second boundary value problem for a sphere is given by the formula from Paragraph 6.1.3-2 with the additional term (1).

3 ② . The solution of the third boundary value problem for a sphere is the sum of the solution specified in Paragraph 6.1.3-3 and expression (1).

6.2.3-2. Domain: ⑨ 1 ≤

2 ,0≤ ➌ ≤ ➂ ,0≤ ≤2 ➂ . Boundary value problems.

1 ② . The solution of the first boundary value problem for a spherical layer is given by the formula from Paragraph 6.1.3-4 with the additional term

( ❦ , ❧ , ♠ , ④ ) ⑤ ♥ ( , ➌ , , ❦ , ❧ ♠ ➇ − ✐ ④ ➄ ✐ ✐ ✐ ➆ , , ) ❦ 2 sin ❧ ♦ ❦ ♦ ❧ ♦ ♠ ♦ ④ , (2)

which allows for the equation’s nonhomogeneity. ➆

2 ② . The solution of the second boundary value problem for a spherical layer is given by the formula from Paragraph 6.1.3-5 with the additional term (2).

3 ② . The solution of the third boundary value problem for a spherical layer is the sum of the solution specified in Paragraph 6.1.3-6 and expression (2).

6.3. Equations of the Form ❱ ➎ = ➏ 2 3 – ➑ + ➒ ( ➓ , ➔ , → , )

6.3.1. Problems in Cartesian Coordinates

A three-dimensional nonhomogeneous Klein–Gordon equation in the rectangular Cartesian system of coordinates has the form

6.3.1-1. Fundamental solutions.

1 ② . For ➣ =− ↕ 2 < 0,

where ➜ = ➠ ➤ 2 + ➥ 2 + ➃ 2 , ➛ ( ➦ ) is the Dirac delta function, ➢ ( ➦ ) is the Heaviside unit step function, and 1 ( ➃ ) is the modified Bessel function.

❶✄❸

2 ➨ . For ➣ = ↕ 2 > 0,

, where ➭❚➯ 1 ( ➃ ) is the Bessel function.

( , , , )=

Reference ➫ : V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974).

6.3.1-2. Domain: − ➲ < ➤ < ➲ ,− ➲ < ➥ < ➲ ,− ➲ < ➃ < ➲ . Cauchy problem. Initial conditions are prescribed:

1 ➨ . Solution for ➣ =− ↕ 2 < 0:

, 2 ➃ , ➇ )= ➉

2 − ➜ 2 ➡✼➼ ➽ ➾ ➳ ( ➤ , ➥ , ➚ ) ➪ ➶ ➜ ➹

Here, 0 ( ➚ ) is the modified Bessel function and ➼ ➽ ➾❚➘ ( ➤ , ➥ , ➚ ) ➪ is the average of ➘ ( ➤ , ➥ , ➚ ) over the spherical surface with center at ( ➞ ➤ , ➥ , ➚ ) and radius ➜ :

2 ➨ . Solution for ➮ = ➱ 2 > 0:

( ➤ , ➥ , ➚ , ➇ )= ➉

➭❚➯ where 0 (

) is the Bessel function.

Reference ➫ : V. I. Smirnov (1974, Vol. 2).

6.3.1-3. Domain: 0 ≤ ➤ ≤ ❒ 1 ,0≤ ➥ ≤ ❒ 2 ,0≤ ➚ ≤ ❒ 3 . First boundary value problem.

A rectangular parallelepiped is considered. The following conditions are prescribed:

= ➳ 0 ( ➤ , ➥ , ➚ ) at ➇ =0

(initial condition),

= ➳ 1 ( ➤ , ➥ , ➚ ) at ➇ =0

(initial condition),

= ➵ 1 ( ➥ , ➚ , ➇ ) at ➤ =0 (boundary condition),

= ➵ 2 ( ➥ , ➚ , ➇ ) at ➤ = ❒ 1 (boundary condition),

= ➵ 3 ( ➤ , ➚ , ➇ ) at ➥ =0 (boundary condition),

= ➵ 4 ( ➤ , ➚ , ➇ ) at ➥ = ❒ 2 (boundary condition),

= ➵ 5 ( ➤ , ➥ , ➇ ) at ➚ =0 (boundary condition),

= ➵ 6 ( ➤ , ➥ , ➇ ) at ➚ = ❒ 3 (boundary condition).

) sin( ß Ú à ) sin( á ➚ )

) sin( ß Ú ❰ ) sin( á Ï ) sin ã❍➇✼ä Ü Ú

)+ .

6.3.1-4. Domain: 0 ≤ ≤ ❒ 1 ,0≤ à ≤ ❒ 2 ,0≤ ➚ ≤ ❒ 3 . Second boundary value problem.

A rectangular parallelepiped is considered. The following conditions are prescribed: Þ

0 ( , à , ➚ ) at é =0

(initial condition),

1 ( , ë à , ➚ ) at é =0 (initial condition),

, Þ é ) at =0 (boundary condition),

= ì Þ 2 ( à , ➚ , é ) at

= ❒ 1 (boundary condition),

êí

= ì Þ 3 ( , ➚ , é ) at à =0 (boundary condition),

êí

= ì Þ 4 ( , ➚ , é ) at à = ❒ 2 (boundary condition),

= ì Þ ( , à , é ) at ➚ =0 (boundary condition),

= ì 6 ( , à , é ) at ➚ = ❒ 3 (boundary condition).

) cos( ✌ Ú ✍ ) cos( ✎ )

) cos( Û ✌ Ú ý ) cos( Ù ✎ þ ) sin ✑✓✒✕✔

= ✙ for ✖ = 0,

1 2 3 2 for ✖ > 0. The summation is performed over the indices satisfying the condition +

+ > 0; the term corresponding to =

= 0 is singled out.

6.3.1-5. Domain: 0 ≤ ≤ ✆ 1 ,0≤ ✍ ≤ ✆ 2 ,0≤ ≤ ✆ 3 . Third boundary value problem.

A rectangular parallelepiped is considered. The following conditions are prescribed: ☞

0 ( , ✍ ,) at ✒ =0

(initial condition),

1 ( , ✍ ,) at ✒ =0

(initial condition),

) ☞ at =0

(boundary condition),

2 = ✤ 2 ( ✍ ,, ✒ ) at

= ✆ 1 (boundary condition),

− ✣ ✚ = ✤ ☞ ( ,, ✒ ) at ✍ =0 (boundary condition),

4 = ✤ ✦ ☞ 4 ( ,, ✒ ) at ✍ = ✆ 2 (boundary condition),

− ✣ ✚ 5 = ✤ ☞ 5 ( , ✍ , ✒ ) at

=0 (boundary condition),

= ✆ 3 (boundary condition). The solution ✚

+ ✣ ✚ 6 = ✤ 6 ( , ✍ , ✒ ) at

( , ✍ ,, ✒ ) is determined by the formula in Paragraph 6.3.1-4 where Ù ☞ Ù Û

+ ★ ) sin( ✌ Ú ý + ✩ Ú ) sin( ✎ þ + ✪ ) sin ✑✓✒✕✔

= arctan

= arctan ✆ , ✪ = arctan

4 ) Here, the ☛ , ✌ Ú , and ✎ are positive roots of the transcendental equations

1 ✣ 2 =( ✣ 1 + ✣ 2 ) ☛ cot( ✆ 1 ☛ ), ✌ 2 − ✣ 3 ✣ 4 =( ✣ 3 + ✣ 4 ) ✌ cot( ✆ 2 ✌ ), ✎ 2 − ✣ 5 ✣ 6 =( ✣ 5 + ✣ 6 ) ✎ cot( ✆ 3 ✎ ).

6.3.1-6. Domain: 0 ≤ ≤ ✆ 1 ,0≤ ✍ ≤ ✆ 2 ,0≤ ≤ ✆ 3 . Mixed boundary value problems.

1 ✮ . A rectangular parallelepiped is considered. The following conditions are prescribed: ☞

= ✛ ☞ 0 ( , ✍ ,) at ✒ =0

(initial condition),

1 ☞ ( , ✍ ,) at ✒ =0

(initial condition),

1 ☞ ( ✍ ,, ✒ ) at =0 (boundary condition),

= ✤ 2 ( ✍ ☞ ,, ✒ ) at

= ✆ 1 (boundary condition),

= ✤ ☞ 3 ( ,, ✒ ) at ✍ =0

(boundary condition),

4 ( ,, ✒ ) at ✍ = ✆ 2 (boundary condition),

5 ( , ✍ , ✒ ) at

=0 (boundary condition),

= ✆ 3 (boundary condition). Solution: ☞

) cos( ✌ Ú ý ) cos( ✎ þ ) sin ✑✓✒✕✔

. A rectangular parallelepiped is considered. The following conditions are prescribed: ☞

= ✛ ☞ 0 ( , ✍ ,) at ✒ =0

(initial condition),

1 ☞ ( , ,) at =0

(initial condition),

) ☞ at =0 (boundary condition),

= ✤ 2 ( ✍ ☞ ,, ✒ ) at

= ✆ 1 (boundary condition),

= ✤ ☞ 3 ( ,, ✒ ) at ✍ =0 (boundary condition),

= ✤ ☞ 4 ( ,, ✒ ) at ✍ = ✆ 2 (boundary condition),

= ✤ ☞ 5 ( , ✍ , ✒ ) at

=0 (boundary condition),

= ✆ 3 (boundary condition). Solution: ☞

) sin( ✌ Ú ✍ ) sin( ✎ ❈ ❃

) sin( ✌ Ú ❀ ) sin( ✎ ❁ ) sin ✑✓✒✕✔

6.3.2. Problems in Cylindrical Coordinates

A nonhomogeneous Klein–Gordon equation in the cylindrical coordinate system is written as ☞

= One-dimensional problems with axial symmetry that have solutions ✚ ( , ✒ ) are treated in ❏

Subsection 4.2.5. Two-dimensional problems whose solutions have the form ▼ = ✚ ( , ,

) or

= ✚ ( , ❃ , ✒ ) are considered in Subsections 5.3.2 and 5.3.3.

6.3.2-1. Domain: 0 ≤ ▼ ≤

,0≤ ≤2 ✗ ,0≤ ❃ ≤ ❈ . First boundary value problem.

A circular cylinder of finite length is considered. The following conditions are prescribed: ❏

0 ( ❏ , , ❃ ) at ✒ =0

(initial condition),

1 ( , , ❃ ) at ✒ ❏ =0

(initial condition),

= ✤ 1 ( ❏ , ❃ , ✒ ) at

(boundary condition),

, ✒ ) at ❃ =0 (boundary condition),

(boundary condition). Solution:

( ❲ ) are the Bessel functions (the prime denotes the derivative with respect to the argument), and the ❯ Ú are positive roots of the transcendental equation ❙ ( ❯ ❴ ) = 0.

6.3.2-2. Domain: 0 ≤ ✌ ≤ ✍ ,0≤ ✎ ≤2 ✏ ,0≤ ✑ ≤ ✒ . Second boundary value problem.

A circular cylinder of finite length is considered. The following conditions are prescribed:

= ✔ 0 ( ✌ , ✎ , ✑ ) at ✕ =0

(initial condition),

= ✔ ✘ 1 ( ✌ , ✎ , ✑ ) at ✕ =0

(initial condition),

= ✙ 1 ( ✖ ✎ ✚ , ✑ , ✕ ) at ✌ = ✍ (boundary condition),

= ( , , ) at

=0 (boundary condition),

(boundary condition). Solution:

cos[ ❄ ( ✎ − ✥ )] cos ✷ ✸

where the ❀ ❃ ( ) are the Bessel functions and the ✿ are positive roots of the transcendental equation

6.3.2-3. Domain: 0 ≤ ✌ ≤ ✍ ,0≤ ✎ ≤2 ✏ ,0≤ ✑ ≤ ✒ . Third boundary value problem.

A circular cylinder of finite length is considered. The following conditions are prescribed:

= ✔ 0 ( ✌ , ✎ , ✑ ) at ✕ =0

(initial condition),

= ✔ 1 ( ✌ , ✎ , ✑ ) at ✕ =0

(initial condition),

at ✌ = ✍

(boundary condition),

( , ✎ , ✕ ) at ✚ ✑ =0 (boundary condition),

(boundary condition). The solution ✓ ( ✌ , ✎ , ✑ , ✕ ) is determined by the formula in Paragraph 6.3.2-2 where ✽ ❁ ✽ ✽ ❁ ✽ ✽ ❁ ✽ ❉ ❉

3 = ✙ 3 ( ✌ , ✎ , ✕ ) at ✑ = ✒

) ❃ ( ✿ ) cos[ ❄ ( ✎ − ✥ )] ❋ ( ✑ ) ❋ ( ✦ ) sin( ❅ ✿ ✧ ✕ ) ( ✌ , ✎ , ✑ , , ✥ , ✦ , ✕ )=

the ❃ ( ) are the Bessel functions, and the

6.3.2-4. Domain: 0 ≤ ✌ ≤ ✍ ,0≤ ✎ ≤2 ✏ ,0≤ ✑ ≤ ✒ . Mixed boundary value problems.

1 ■ . A circular cylinder of finite length is considered. The following conditions are prescribed:

= ✔ 0 ( ✌ , ✎ , ✑ ) at ✕ =0

(initial condition),

= ✔ 1 ( ✌ , ✎ , ✑ ) at ✕ =0

(initial condition),

= ✙ 1 ( ✎ , ✑ , ✕ ) at ✌ = ✍

(boundary condition),

= ✙ 2 ( ✌ , ✎ , ✕ ) at ✑ =0 (boundary condition),

(boundary condition). Solution:

where the ❁

( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the ✤

are positive roots of the transcendental equation ❃ ( ✍ ) = 0.

✄✂

2 ■ . A circular cylinder of finite length is considered. The following conditions are prescribed:

= ✔ 0 ( ✌ , ✎ , ✑ ) at ✕ =0

(initial condition),

= ( ✌ , ✎ , ✘ ✑ ) at ✕ =0

(initial condition),

= ✙ 1 ( ✎ , ✑ , ✕ ) at ✌ = ✍

(boundary condition),

= ✙ 2 ( ✌ , ✎ , ✕ ) at ✑ =0 (boundary condition),

(boundary condition). Solution:

where the ❀

( ) are the Bessel functions and the

are positive roots of the transcendental equation

( ✍ ) = 0. ✤

6.3.2-5. Domain: ✍ 1 ≤ ✌ ≤ ✍ 2 ,0≤ ✎ ≤2 ✏ ,0≤ ✑ ≤ ✒ . First boundary value problem.

A hollow circular cylinder of finite length is considered. The following conditions are prescribed:

= ✔ 0 ( ✌ , ✎ , ✑ ) at ✕ =0

(initial condition),

= ✔ 1 ( ✌ , ✎ , ✑ ) at ✕ =0

(initial condition),

= ✙ 1 ( ✎ , ✑ , ✕ ) at ✌ = ✍ 1 (boundary condition),

= ✙ 2 ( ✎ , ✑ , ✕ ) at ✌ = ✍ 2 (boundary condition),

= ✙ 3 ( ✌ , ✎ , ✕ ) at ✑ =0

(boundary condition),

= ✙ 4 ( ✌ , ✎ , ✕ ) at ✑ = ✒

(boundary condition).

)= ✽ ❃ ✽ ( ( ✿ ✍ 1 ❖ ) P ( ✿ ✌ )− P ( ✿ ✍ 1 ) ❃ ( ❁ ✿ ✽ ✌ ); the ❃ ( ✌ ) and P ( ✌ ) are the Bessel functions, and the

are positive roots of the transcendental equation

( ✍ 1 ) P ( ✍ 2 )− P ( ✍ 1 ) ❃ ( ✍ 2 ) = 0.

6.3.2-6. Domain: ✍ 1 ≤ ✌ ≤ ✍ 2 ,0≤ ✎ ≤2 ✏ ,0≤ ✑ ≤ ✒ . Second boundary value problem.

A hollow circular cylinder of finite length is considered. The following conditions are prescribed:

= ✔ 0 ( ✌ , ✎ , ✑ ) at ✕ =0

(initial condition),

= ✔ 1 ( ✌ , ✎ , ✑ ) at ✕ =0

(initial condition),

= ✙ 1 ( ✎ , ✑ , ✕ ) at ✌ = ✍ 1 (boundary condition),

= ✙ 2 ( ✎ , ✑ , ✕ ) at ✌ = ✍ 2 (boundary condition),

= ✙ 3 ( ✌ , ✎ , ✕ ) at ✑ =0

(boundary condition),

= ✙ 4 ( ✌ , ✎ , ✕ ) at ✑ = ✒

(boundary condition).

( ❖ ✌ )= ❃ ❈ ( ✿ ✍ 1 ) P ( ✿ ✌ )− ❁ P ✽ ❈ ( ✿ ✍ 1 ) ❃ ( ✿ ✌ ); the ❃ ( ✌ ) and P ( ✌ ) are the Bessel functions, and the

are positive roots of the transcendental ❁

equation

( ✍ 1 ) P ❈ ( ✍ 2 )− P ❈ ( ✍ 1 ) ❃ ❈ ( ✍ 2 ) = 0.

6.3.2-7. Domain: ✍ 1 ≤ ✌ ≤ ✍ 2 ,0≤ ✎ ≤2 ✏ ,0≤ ✑ ≤ ✒ . Third boundary value problem.

A hollow circular cylinder of finite length is considered. The following conditions are prescribed:

= ✔ 0 ( ✌ , ✎ , ✑ ) at ✕ =0

(initial condition),

= ✔ 1 ( ✌ , ✎ , ✑ ) at ✕ =0

(initial condition),

− ✓ 1 = ✙ 1 ( ✎ , ✑ , ✕ ) at ✌ = ✍ 1 (boundary condition),

) at ✌ = ✍ 2 (boundary condition),

− ✓ 3 = ✙ 3 ( ✌ , ✎ , ✕ ) at ✑

(boundary condition),

4 = ✙ 4 ( ✌ , ✎ , ✕ ) at ✑ = ✒

(boundary condition).

The solution ✓ ( ✌ , ✎ , ✑ , ✕ ) is determined by the formula in Paragraph 6.3.2-6 where ✽ ❁ ✽

( ❲ ) = cos( ❅ ❲ )+ ✸ sin( ❅ ❲ ), ❋ ❁ ✽ = ✸ 2 2 ✸ 2 + ✸ 3 + ❳ ✷

are positive roots of the transcen- dental equation ❁ ✽ ❁

where the ✸

( ❯ ) and P ( ❯ ) are the Bessel functions, and the

are positive roots of the transcendental equation

6.3.2-8. Domain: ❱ 1 ≤ ❯ ≤ ❱ 2 ,0≤ ❨ ≤2 ❩ ,0≤ ❲ ≤ ❳ . Mixed boundary value problems.

1 ■ . A hollow circular cylinder of finite length is considered. The following conditions are prescribed:

= ❭ 0 ( ❯ , ❨ , ❲ ) at ❫ ❪ ❴ =0

(initial condition),

= ❭ 1 ( ❯ , ❨ , ❲ ) at ❪ =0

(initial condition),

= ❵ 1 ( ❨ , ❲ , ❪ ) at ❯ = ❱ 1 (boundary condition),

= ❵ 2 ( ❫ ❨ ❛ , ❲ , ❪ ) at ❯ = ❱ 2 (boundary condition),

= ❵ 3 ( ❯ , ❨ , ❪ ) at ❫ ❲ ❛ =0 (boundary condition),

(boundary condition). Solution:

= ❵ 4 ( ❯ , ❨ , ❪ ) at ❲ = ❳

( ⑥ ❶ 1 ) ⑩ ( ⑥ ❸ ); the ⑩ ( ❸ ) and ➆ ( ❸ ) are the Bessel functions, and the

are positive roots of the transcendental ⑨

equation

2 ➇ . A hollow circular cylinder of finite length is considered. The following conditions are prescribed:

= ➉ ➊ 0 ➋ ( ❸ , ❺ , ❾ ) at ➁ =0

(initial condition),

= ➉ 1 ( ❸ , ❺ , ❾ ) at ➊ ➁ ➌ =0

(initial condition),

, ) at

= 1 (boundary condition), = ➍ 2 ( ❺ , , ➁ ❾ ) at ❸ = ❶ 2 (boundary condition),

= ➍ 3 ( ❸ , ❺ , ➁ ) at ❾ =0

(boundary condition),

(boundary condition). Solution:

= ➍ 4 ( ❸ , ❺ , ➁ ) at ❾ =

Here, ➃

sin ➀✭➁ ⑦ ✮ ✐ ➒

sin ❻ ❼ ❽ ❿ ❾ ➑ sin ❻ ❼ ❽ ❿ ❾ ➑

× cos[ ⑥

( ❺ − ❣ )] sin ❻ ❼ ❽ ❿ ❾ ➑ sin ❻ ❼ ❽ ❿ ➑

sin

( ❸ )= ⑩ ➓ ( ⑥ ❶ 1 ) ➆ ( ⑥ ❸ )− ⑨ ➆ ⑤ ➓ ( ⑥ ❶ 1 ) ⑩ ( ⑥ ❸ ); the ⑩ ( ❸ ) and ➆ ( ❸ ) are the Bessel functions, and the

are positive roots of the transcendental ⑨

equation

6.3.2-9. Domain: 0 ≤ ❿

≤ ❶ ,0≤ ❺ ≤ ❺ 0 ,0≤ ❾ ≤ . First boundary value problem.

A cylindrical sector of finite thickness is considered. The following conditions are prescribed:

= ➉ 0 ( ❸ , ❺ , ❾ ) at ➊ ➁ ➋ =0

(initial condition),

= ➉ 1 ( ❸ , ❺ , ❾ ) at ➁ =0

(initial condition),

= ➍ 1 ( ❺ , ❾ , ➁ ) at ❸ = ❶

(boundary condition),

= ➍ 2 ( ❸ , ❾ , ➁ ) at ❺ =0

(boundary condition),

= ➍ 3 ( ❸ , , ➁ ❾ ) at ❺ = ❺ 0 (boundary condition),

= ➍ 4 ( ❸ , ❺ , ➁ ) at

(boundary condition),

(boundary condition). Solution:

where the ➵

0 ( ➨ ) are the Bessel functions and the ➧ ➢ are positive roots of the transcendental equation

6.3.2-10. Domain: 0 ≤ ➨ ≤ ➩

≤ ➲ ,0≤ ➸ ≤ . Mixed boundary value problem.

A cylindrical sector of finite thickness is considered. The following conditions are prescribed:

) at

(initial condition),

, ➸ ) at

(initial condition),

= ❰ 1 ( ➲ , ➸ , ➼ ) at ➨ = ➩

(boundary condition),

= ❰ 2 ( ➨ , ➸ , ➼ ) at ➲ =0

(boundary condition),

( ➨ , ➸ , ➼ ) at

= ➲ 0 (boundary condition),

, ) at ➸ =0 ➺

(boundary condition),

(boundary condition). Solution:

0 = 1 and = 2 for ≥ 1; the ❞ ä 0 ( Ù ) are the Bessel functions; and the å ➵ á are positive roots of the transcendental equation ã ã

6.3.3. Problems in Spherical Coordinates Ò

A nonhomogeneous Klein–Gordon equation in the spherical coordinate system is written as

One-dimensional problems with central symmetry that have solutions of the form è = ( Ù , ➼ ) are treated in Subsection 4.2.6.

6.3.3-1. Domain: 0 ≤ Ù ≤ ç ,0≤ é ≤ ➯ ,0≤ ➲ ≤2 ➯ . First boundary value problem.

A spherical domain is considered. The following conditions are prescribed:

) at

=0 (initial condition),

= ê 1 ( Ù , é , ) at ➼ =0 (initial condition),

(boundary condition). Solution:

(cos é ) î (cos ❣ ) cos[ ( − Ó ➲ )] sin ➻✭➼✼ï Ö 2 í 2 á

+1 ä 2 ( ø ) are the Bessel functions, the î ( å ) are the associated Legendre functions expressed æ

in terms of the Legendre polynomials â â à î ( å ) as

and the í û ü are positive roots of the transcendental equation õ û +1 ý 2 ( í þ ) = 0.

6.3.3-2. Domain: 0 ≤ ø ≤ þ ,0≤ ÿ ≤,0≤ ✁ ≤ 2 . Second boundary value problem.

A spherical domain is considered. The following conditions are prescribed:

= ✄ 0 ( ø , ÿ , ✁ ) at ☎ =0

(initial condition),

= ✄ 1 ( ø , ÿ , ✁ ) at ☎ =0

(initial condition),

(boundary condition). Solution:

at ø = þ

0 ( ✡ , ❣ , ☛ ) ☞ ( ø , ÿ , ✁ , ✡ , ❣ , ☛ , ☎ ) ✡ sin ❣ ù ✡ ù ❣ ù ☎ ☛

1 ( ✡ , ❣ , ☛ ) ☞ ( ø , ÿ , ✁ , ✡ , ❣ , ☛ , ☎ ) ✡ sin ❣ ù ✡ ù ❣ ù ☛

( ❣ , ☛ , ✌ ) ☞ ( ø , ÿ , ✁ , þ , ❣ , ☛ , ☎ − ✌ ) sin ❣ ù ❣ ù ☛ ù ✌

, 2 , ✡ , ❣ , ☛ , ☎ − ✌ ) ✡ sin ❣

(cos ÿ ) ✩ û ú (cos ❣ ) cos[ ó ( ✁ − ☛ )] sin ✛✜☎✼ï ÷ 2 ★ 2 û ü + ð ñ , where

the õ û +1 ý 2 ( ø ) are the Bessel functions, the ✩ û ú ( ✬ ) are the associated Legendre functions (see Paragraph 6.3.3-1), and the ★ û ü are positive roots of the transcendental equation

6.3.3-3. Domain: 0 ≤ ø ≤ þ ,0≤ ÿ ≤,0≤ ✁ ≤ 2 . Third boundary value problem.

A spherical domain is considered. The following conditions are prescribed:

= 0 ( ø , ÿ , ✯ ✁ ) at ☎ =0

(initial condition),

= ✄ 1 ✮ ø ( , ÿ , ✁ ) at ☎ ✆ =0 (initial condition),

(boundary condition). The solution ✂ ( ø , ÿ , ✁ , ☎ ) is determined by the formula in Paragraph 6.3.3-2 where

× ✩ û (cos ÿ ) ✩ û (cos ❣ ) cos[ ✲ ( ✁ − ☛ )] sin ✛✜☎✴✳ ÷ 2 ★ 2 û ü + ✵ ✶ . Here,

1 2 for = 0, û ü (2 ✸ + 1)( ✸ − ✲ )! = ✷

2 ★ ✰ 2 2 2 ; for ≠ 0, ( ✸ + ✲ )! ✪ þ û ü +( ✹ þ + ✸ )( ✹ þ − ✸ − 1) ✫ ✪✻✺ û +1 ý 2 ( ★ û ü þ ) ✫

the ✺ û +1 ý 2 ( ø ) are the Bessel functions, the ✩ û ( ✬ ) are the associated Legendre functions (see Paragraph 6.3.3-1), and the ★ û ü are positive roots of the transcendental equation

6.3.3-4. Domain: þ 1 ≤ ø ≤ þ 2 ,0≤ ÿ ≤ ,0≤ ✁ ≤ 2 . First boundary value problem.

A spherical layer is considered. The following conditions are prescribed:

= ✄ 0 ( ø , ÿ , ✁ ) at ☎ =0

(initial condition),

= ✄ 1 ( ø , ÿ , ✁ ) at ☎ =0

(initial condition),

= ✝ 1 ( ÿ , ✁ , ☎ ) at ø = þ 1 (boundary condition),

= ✝ 2 ( ÿ , ✁ , ☎ ) at ø = þ 2 (boundary condition).

(cos ❇ ) ✩ ❂ ú (cos ❣ ) cos[ ✹ ( ❈ − ☛ )] sin ✛✜☎✴✳

( ✬ ) are the associated Legendre functions expressed in terms of the Legendre polynomials ✩ ❂ ( ✬ ) as

where the ❂ +1 ❅ ❆ 2 ( ) are the Bessel functions, the ✩ ❂

and the ❉ ★ are positive roots of the transcendental equation +1 ❅ 2 ( ★ 2 ) = 0.

6.3.3-5. Domain: ❉ 1 ≤ ≤ 2 ,0≤ ❇ ≤ ● ,0≤ ❈ ≤2 ● . Second boundary value problem.

A spherical layer is considered. The following conditions are prescribed:

= ■ 0 ( ❆ , ✮ ❇ ✯ , ❈ ) at ❏ =0

(initial condition),

( , ❑ ❇ , ❈ ) at ❏ =0

(initial condition),

1 ❉ ( ❇ , ❈ , ❏ ) at = 1

(boundary condition),

= ❉ 2 ( , , ) at = 2 (boundary condition). Solution:

0 ( P , ◗ , ❘ ) ❙ ( , ❇ , ❈ , P , ◗ , ❘ , ❏ ) P sin ◗ ❋ P ❋ ◗ ❋ ❘

, ) ❙ ( , ❇ , ❈ , P , ◗ , ❘ , ❏ ) P sin ◗ ❋ P ❋ ◗ ❋ ❘

1 ( , , ) ( , , , 1 , , , ❏ − ❱ ) sin ◗ ❋ ◗ ❋ ❘ ❋ ❱

2 ( , , ) ( , , , 2 , , , − ) sin ◗ ❋ ◗ ❋ ❘ ❋ ❱

, ▼ ❘ ❖ , ❏ − ❱ ) P sin ◗ ❋ P ❋ ◗ ❋ ❘ ❋ ❱ ,

(cos ◗ ) cos[ ② ( ❩ − ❘ )]

( ❹ ) are the associated Legendre functions (see Paragraph 6.3.3-4), and the ♥ ✇ ♣ are positive roots of the transcendental equation ♥

where the ♥

+1 ✈ 2 ( ❲ ) and ❷ +1 ✈ 2 ( ❲ ) are the Bessel functions, the ①

6.3.3-6. Domain: ❦ 1 ≤ ❲ ≤ ❦ 2 ,0≤ ❳ ≤ ❥ ,0≤ ❩ ≤2 ❥ . Third boundary value problem.

A spherical layer is considered. The following conditions are prescribed:

= ❻ 0 ( ❼ ❲ , ❳ , ❩ ) at ③ =0

(initial condition),

= ❻ 1 ( ❲ , ❳ , ❩ ) at ③ =0

(initial condition),

− ② ❺ 1 ❯ = ❾ 1 ( ❳ , ❩ , ③ ) at ❲ = ❦ ❼ 1 ❽ (boundary condition),

+ ❺ ② 2 = ❾ 2 ( ❳ , ❩ , ③ ) at ❲ = ❦ 2 (boundary condition).

The solution ♥ (

, ❳ , ❩ , ③ ) is determined by the formula in Paragraph 6.3.3-5 where ➀ ♥ ♥ ♥ ♥

(cos ❳ ) ① (cos ◗ ) cos[ ➁ ( ❩

where the ♥

( ❹ ) are the associated Legendre functions (see Paragraph 6.3.3-4), and the ✇ ♣ ♥ are positive roots of the transcendental equation ♥

+1 ✈ 2 ( ❲ ) and ❷ +1 ✈ 2 ( ❲ ) are the Bessel functions, the ①