Wave Equation with Axial or Central Symmetry

4.2. Wave Equation with Axial or Central Symmetry

4.2.1. Equations of the Form ➚

This is the one-dimensional wave equation with axial symmetry, where ➴

+ is the radial coordinate. In the problems considered in Paragraphs 4.2.1-1 through 4.2.1-3, the solutions bounded

at ➺ = 0 are sought (this is not specially stated below).

4.2.1-1. Domain: 0 ≤ ➷ ≤ ➬ . First boundary value problem. The following conditions are prescribed:

= 0 ( ➷ ) at ➮ =0

(initial condition),

) at ➮ =0

(initial condition),

(boundary condition). Solution:

Here, the ➠

are positive zeros of the Bessel function, 0 ( ➠ ) = 0. The numerical values of the first ten ❰▲Ï ➠

are specified in Paragraph 1.2.1-3.

Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

4.2.1-2. Domain: 0 ≤ ➷ ≤ ➬ . Second boundary value problem. The following conditions are prescribed:

= ( ➷ ) at ➮ =0

(initial condition),

) at ➮ =0

(initial condition),

(boundary condition). Solution:

Here, the × are positive zeros of the first-order Bessel function,

1 ( ) = 0. The numerical values of the first ten roots ❰▲Ï

are specified in Paragraph 1.2.1-4.

References : M. M. Smirnov (1975), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

4.2.1-3. Domain: 0 ≤ ➷ ≤ ➬ . Third boundary value problem. The following conditions are prescribed:

= 0 ( ➷ ) at ➮ =0

(initial condition),

) at ➮ =0

(initial condition),

(boundary condition). The solution ➒

at ➷ = ➬

( ➷ , ➮ ) is determined by the formula in Paragraph 4.2.1-2 where Ø Ö Ö Ö Ø Ö Ù ❐ ❐

Here, the × are positive roots of the transcendental equation ❐ ❐

The numerical values of the first six roots × can be found in Carslaw and Jaeger (1984); see also Abramowitz and Stegun (1964).

4.2.1-4. Domain: ➬ 1 ≤ ➷ ≤ ➬ 2 . First boundary value problem.

The following conditions are prescribed:

0 ( ➷ ) at ➮ =0

(initial condition),

) at ➮ =0

(initial condition),

1 ( ➮ ) at ➷ = ➬ 1 (boundary condition),

= Ñ 2 ( ➮ ) at ➷ = ➬ 2 (boundary condition). Solution:

The numerical values of the first five roots î ï = î ï ( ê ) can be found in Abramowitz and Stegun (1964) and Carslaw and Jaeger (1984).

4.2.1-5. Domain: æ 1 ≤

≤ 2 . Second boundary value problem.

The following conditions are prescribed:

= ò 0 ( ß ) at à =0

(initial condition),

= ò 1 ( ß ) at à =0

(initial condition),

( à ) at ß =

(boundary condition),

= Ñ 2 ( à ) at ß = 2 (boundary condition). Solution:

where ( ð ) and í ( ð ) are the Bessel functions ( é ü é = 0, 1); the î ï are positive roots of the transcen- dental equation

The numerical values of the first five roots î ï = î ï ( ê ) can be found in Abramowitz and Stegun (1964).

4.2.1-6. Domain: æ 1 ≤

≤ 2 . Third boundary value problem.

The following conditions are prescribed:

= ò 0 ( ß ) at à =0

(initial condition),

= ò 1 ( ß ) at à =0

(initial condition),

1 = Ñ 1 ( à ) at ß = 1 ó (boundary condition), Ð

2 = Ñ 2 ( à ) at ß = 2 (boundary condition).

The solution ñ ( ß , à ) is determined by the formula in Paragraph 4.2.1-5 where

( ð ) and í ( ð ) are the Bessel functions ( ý ý ý ü = 0, 1); and the ý ï ý are positive roots of the transcendental ý ý ý

4.2.2. Equation of the Form ➾ ✁ 2 = ✂ 2 ➾

4.2.2-1. Domain: 0 ≤ æ

≤ . Different boundary value problems.

. The solution to the first boundary value problem for a circle of radius is given by the formula from Paragraph 4.2.1-1 with the additional term

which allows for the equation’s nonhomogeneity.

. The solution to the second boundary value problem for a circle of radius is given by the formula from Paragraph 4.2.1-2 with the additional term (1).

. The solution to the third boundary value problem for a circle of radius is the sum of the solution presented in Paragraph 4.2.1-3 and expression (1).

4.2.2-2. Domain: æ 1 ≤

≤ 2 . Different boundary value problems.

1 ✞ . The solution to the first boundary value problem for an annular domain is given by the formula from Paragraph 4.2.1-4 with the additional term

which allows for the equation’s nonhomogeneity.

2 ✞ . The solution to the second boundary value problem for an annular domain is given by the formula from Paragraph 4.2.1-5 with the additional term (2).

3 ✞ . The solution to the third boundary value problem for an annular domain is the sum of the solution presented in Paragraph 4.2.1-6 and expression (2).

4.2.3. Equation of the Form ➾ ✁

This is the equation of one-dimensional vibration of a gas with central symmetry, where ✆

2 + ☛ 2 + ð 2 is the radial coordinate. In the problems considered in Paragraphs 4.2.3-1 through 4.2.3-3, the solutions bounded at ß = 0 are sought; this is not specially stated below.

4.2.3-1. General solution:

where ☞ ( ✌ 1 ) and ✎ ( ✌ 2 ) are arbitrary functions.

4.2.3-2. Reduction to a constant coefficient equation. The substitution ✏ ( ✌ , ✍ )= ✌ ñ ( ✌ , ✍ ) leads to the constant coefficient equation

which is discussed in Subsection 4.1.1.

4.2.3-3. Domain: 0 ≤ ✌ < ✑ . Cauchy problem. Initial conditions are prescribed:

Solution at the center ✌ = 0: ✢

Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

4.2.3-4. Domain: 0 ≤ ✌ ≤ ✬ . First boundary value problem. The following conditions are prescribed: ✢

= ✤ 0 ( ✌ ) at ✍ =0

(initial condition),

= ✤ 1 ( ✌ ) at ✍ =0

(initial condition),

at ✌ = ✬

(boundary condition).

4.2.3-5. Domain: 0 ≤ ✌ ≤ ✬ . Second boundary value problem. The following conditions are prescribed: ✢

= ✤ ( ✌ ) at ✍ =0

(initial condition),

= ✤ ( ✌ ) at ✍ =0

(initial condition),

at ✌ = ✬

(boundary condition).

Here, the ï are positive roots of the transcendental equation tan ✺ − = 0. The numerical values of the first five roots ï ✺ are specified in Paragraph 1.2.3-5.

4.2.3-6. Domain: 0 ≤ ✌ ≤ ✬ . Third boundary value problem. The following conditions are prescribed: ✢

= ✤ 0 ( ✌ ) at ✭ ✍ =0

(initial condition),

= ✤ 1 ( ✌ ) at ✍ =0

(initial condition),

(boundary condition). The solution ( ✌ , ✍ ) is determined by the formula in Paragraph 4.2.3-5 where ✖

Here, the ✺ ï are positive roots of the transcendental equation

cot + ✻ ✬ − 1 = 0.

The numerical values of the first six roots ✺ ï can be found in Carslaw and Jaeger (1984).

4.2.3-7. Domain: ✬ 1 ≤ ✌ ≤ ✬ 2 . First boundary value problem.

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).

Solution:

where

sin ✱

sin ✱

sin ✸ .

4.2.3-8. Domain: 1 ≤ ✁ ≤ 2 . Second boundary value problem.

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:

( ✁ ) = sin[ ✕ ( ✁ − 1 )] + 1 ✕ cos[ ✕ ( ✁ − 1 )].

Here, the ✕ are positive roots of the transcendental equation ( 2 ✕ 1 2 + 1) tan[ ✕ ( 2 − 1 )] − ✕ ( 2 − 1 ) = 0.

4.2.3-9. Domain: 1 ≤ ✁ ≤ 2 . Third boundary value problem.

The following conditions are prescribed:

= ✄ 0 ( ✁ ) at ☎ =0

(initial condition),

= ✄ 1 ( ✁ ) at ☎ =0

(initial condition),

− ✂ ✙ 1 = ✟ 1 ( ☎ ) at ✁ = 1 (boundary condition),

+ ✙ ✂ 2 = ✟ 2 ( ☎ ) at ✁ = 2 (boundary condition). The solution ✂ ( ✁ , ☎ ) is determined by the formula in Paragraph 4.2.3-8 where ✓ ✓ ✓ ✓

( ✁ ✓ )= ✛ 1 sin[ ✕ ( ✁ − 1 )] + 1 ✕ cos[ ✕ ( ✁ − 1 )], ✛ 1 = ✙ 1 1 + 1, ✛ 2 = ✙ 2 2 − 1. Here, the ✕

are positive roots of the transcendental equation ( ✛ 1 ✛

2 − 1 2 ✕ ) sin[ ✕ ( 2 − 1 )] + ✕ ( 1 ✛ 2 + 2 ✛ 1 ) cos[ ✕ ( 2 − 1 )] = 0.

4.2.4. Equation of the Form ✢

4.2.4-1. Reduction to a nonhomogeneous constant coefficient equation. The substitution ✩ ( ✁ , ☎ )= ✁ ✂ ( ✁ , ☎ ) leads to the nonhomogeneous constant coefficient equation

which is discussed in Subsection 4.1.2.

4.2.4-2. Domain: 0 ≤ ✁ < ✫ . Cauchy problem. Initial conditions are prescribed:

Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

4.2.4-3. Domain: 0 ≤ ✁ ≤ . Different boundary value problems.

1 ✴ . The solution to the first boundary value problem for a sphere of radius is given by the formula from Paragraph 4.2.3-4 with the additional term

which allows for the equation’s nonhomogeneity.

2 ✴ . The solution to the second boundary value problem for a sphere of radius is given by the formula from Paragraph 4.2.3-5 with the additional term (1).

3 ✴ . The solution to the third boundary value problem for a sphere of radius is the sum of the solution presented in Paragraph 4.2.3-6 and expression (1).

4.2.4-4. Domain: 1 ≤ ✁ ≤ 2 . Different boundary value problems.

1 ✴ . The solution to the first boundary value problem for a spherical layer is given by the formula from Paragraph 4.2.3-7 with the additional term

which allows for the equation’s nonhomogeneity.

2 ✴ . The solution to the second boundary value problem for a spherical layer is given by the formula from Paragraph 4.2.3-8 with the additional term (2).

3 ✴ . The solution to the third boundary value problem for a spherical layer is the sum of the solution presented in Paragraph 4.2.3-9 and expression (2).

4.2.5. Equation of the Form ✢

For ✧

> 0 and ✪ ≡ 0, this is the Klein–Gordon equation describing one-dimensional wave phenomena with axial symmetry. In the problems considered in Paragraphs 4.2.5-1 through 4.2.5-3, the solutions bounded at ✁ = 0 are sought; this is not specially stated below.

4.2.5-1. Domain: 0 ≤ ✁ ≤ . First boundary value problem. The following conditions are prescribed:

0 ( ✁ ) at ☎ =0

(initial condition),

= ✄ 1 ( ✁ ) at ☎ =0

(initial condition),

(boundary condition). Solution:

where the ✹

are positive zeros of the Bessel function, 0 ( ✺ ) = 0. The numerical values of the first ten ✺ are specified in Paragraph 1.2.1-3.

4.2.5-2. Domain: 0 ≤ ✁ ≤ . Second boundary value problem. The following conditions are prescribed:

= ✄ 0 ( ✁ ) at ☎ =0

(initial condition),

= ✄ 1 ( ✁ ) at ☎ =0

(initial condition),

(boundary condition). Solution:

where the ✹

are positive zeros of the first-order Bessel function, 1 ( ✺ ) = 0. The numerical values of the first ten ✺

are specified in Paragraph 1.2.1-4.

4.2.5-3. Domain: 0 ≤ ✁ ≤ . Third boundary value problem. The following conditions are prescribed:

= ✄ ( ✁ ) at ☎ =0

(initial condition),

= ✄ 1 ( ✁ ) at ☎ =0

(initial condition),

at ✁ =

(boundary condition).

The solution ✂ ( ✁ , ☎ ) is determined by the formula in Paragraph 4.2.5-2 where

Here, the ✺ are positive roots of the transcendental equation

The numerical values of the first six roots ✺ can be found in Abramowitz and Stegun (1964) and Carslaw and Jaeger (1984).

4.2.5-4. Domain: 1 ≤ ✁ ≤ 2 . First boundary value problem.

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). Solution:

) and ❂ 0 ( ❃ ) are the Bessel functions and the ✺ are positive roots of the transcendental equation

where ✹ 0 (

The numerical values of the first five roots ✺ = ✺ ( ❁ ) can be found in Abramowitz and Stegun (1964) and Carslaw and Jaeger (1984).

4.2.5-5. Domain: 1 ≤ ✁ ≤ 2 . Second boundary value problem.

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).

) and ❂ ( ❃ ) are the Bessel functions ( ✙ = 0, 1); the ✺ are positive roots of the transcen- dental equation

where ❄ (

The numerical values of the first five roots ✺ = ✺ ( ❁ ) can be found in Abramowitz and Stegun (1964).

4.2.5-6. Domain: 1 ≤ ✁ ≤ 2 . Third boundary value problem.

The following conditions are prescribed:

= ✄ 0 ( ✁ ) at ☎ =0

(initial condition),

= ✞ ✄ 1 ( ✁ ) at ☎ =0

(initial condition),

− ✙ 1 = ✟ 1 ( ☎ ) at ✁ = 1 (boundary condition),

+ ✂ 2 = ✟ 2 ( ☎ ) at ✁ = 2 (boundary condition). The solution ✂ ( ✁ , ☎ ) is determined by the formula in Paragraph 4.2.5-5 where

1 ( ❅ 1 ) ✘ ❂ ❅ 0 ( ✁ ), where the ✗

are positive roots of the transcendental equation ❅ ❅ ✗ ❅ ❅ ❅

4.2.6. Equation of the Form ✢

For ✧

> 0 and ✪ ≡ 0, this is the Klein–Gordon equation describing one-dimensional wave phenomena with central symmetry. In the problems considered in Paragraphs 4.2.6-1 through 4.2.6-3, the solutions bounded at ✁ = 0 are sought; this is not specially stated below.

4.2.6-1. Domain: 0 ≤ ✁ ≤ . First boundary value problem. The following conditions are prescribed:

0 ( ) at

(initial condition),

= ✄ 1 ( ✁ ) at ☎ =0

(initial condition),

(boundary condition). Solution:

4.2.6-2. Domain: 0 ≤ ✁ ≤ . Second boundary value problem. The following conditions are prescribed:

= ✄ 0 ( ✁ ) at ☎ =0

(initial condition),

= ✄ 1 ( ✁ ) at ☎ =0

(initial condition),

(boundary condition). Solution:

2 + ✛ . Here, the ✓

are positive roots of the transcendental equation tan ✺ − ✺ = 0; for the numerical values of the first five roots ✺ , see Paragraph 1.2.3-5.

4.2.6-3. Domain: 0 ≤ ✁ ≤ . Third boundary value problem. The following conditions are prescribed:

= 0 ( ) at

(initial condition),

= ✄ 1 ( ✁ ) at ✆ ☎ ✞ =0

(initial condition),

(boundary condition). The solution ✂ ( ✁ , ☎ ) is determined by the formula in Paragraph 4.2.6-2 where ✓ ✓ ✓ ✓

Here, the ✓

are positive roots of the transcendental equation ✺ cot ✺ + ✙ − 1 = 0. The numerical values of the six five roots ✺

can be found in Carslaw and Jaeger (1984).

4.2.6-4. Domain: 1 ≤ ✁ ≤ 2 . First boundary value problem.

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). Solution:

4.2.6-5. Domain: 1 ≤ ✁ ≤ 2 . Second boundary value problem.

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). Solution:

( ✁ ) = sin[ ✕ ( ✁ − 1 )] + 1 ✕ cos[ ✕ ( ✁ − 1 )], where the ✕ are positive roots of the transcendental equation ( 2 ✕ 1 2 + 1) tan[ ✕ ( 2 − 1 )] − ✕ ( 2 − 1 ) = 0.

4.2.6-6. Domain: 1 ≤ ✁ ≤ 2 . Third boundary value problem.

The following conditions are prescribed:

= ✄ 0 ( ✁ ) at ☎ =0

(initial condition),

= ✞ ✄ 1 ( ✁ ) at ☎ =0

(initial condition),

at

= 1 (boundary condition),

+ ✂ ✙ 2 = ✟ 2 ( ☎ ) at ✁ = 2 (boundary condition). The solution ✂ ( ✁

, ✓ ☎ ) is determined by the formula in Paragraph 4.2.6-5 where

( ✁ )= ✛ 1 sin[ ✕ ( ✁ − 1 )] + 1 ✕ cos[ ✕ ( ✁ − 1 )], ✛ 1 = ✙ 1 1 + 1, ✛ 2 = ✙ 2 2 − 1. Here, the ✕

are positive roots of the transcendental equation ( ✛

1 ✛ 2 − 1 2 ✕ ) sin[ ✕ ( 2 − 1 )] + ✕ ( 1 ✛ 2 + 2 ✛ 1 ) cos[ ✕ ( 2 − 1 )] = 0.