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2.13. Spherical coordinates

So far, we have only considered the one-dimensional spherical wave equation and its solution. In many problems where there are spherical surfaces but no axial or rotational symmetry, it is necessary to use spherical coordinates as shown in Fig. 2.19. The x, y, and z ordinates are replaced by spherical ordinates r, ϕ, and θ, where the radial ordinate r is given by

r = \sqrt{x^{2} + y^{2} + z^{2}},

$r = \sqrt{x^{2} + y^{2} + z^{2}},$

(2.174)

the inclination angle θ is given by

θ = arc \cot (z / \sqrt{x^{2} + y^{2}}),

$θ = arc \cot (z / \sqrt{x^{2} + y^{2}}),$

(2.175)

and the azimuth angle ϕ is given by

ϕ = arctan (y / x) .

$ϕ = arctan (y / x) .$

(2.176)

Conversely

x = r \sin θ \cos ϕ,

$x = r \sin θ \cos ϕ,$

(2.177)

y = r \sin θ \sin ϕ,

$y = r \sin θ \sin ϕ,$

(2.178)

z = r \cos θ .

$z = r \cos θ .$

(2.179)

The three-dimensional wave equation in spherical coordinates is

(\nabla^{2} + k^{2}) \tilde{p} (r, θ, ϕ) = 0,

$(\nabla^{2} + k^{2}) \tilde{p} (r, θ, ϕ) = 0,$

(2.180)

where the Laplace operator is given by

\nabla^{2} = \frac{\partial^{2}}{\partial r^{2}} + \frac{2}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{1}{r^{2} \tan θ} \frac{\partial}{\partial θ} + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ϕ^{2}},

$\nabla^{2} = \frac{\partial^{2}}{\partial r^{2}} + \frac{2}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{1}{r^{2} \tan θ} \frac{\partial}{\partial θ} + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ϕ^{2}},$

(2.181)

which is often written in the following short form:

\nabla^{2} = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial}{\partial r}) + \frac{1}{r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial}{\partial θ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ϕ^{2}} .

$\nabla^{2} = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial}{\partial r}) + \frac{1}{r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial}{\partial θ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ϕ^{2}} .$

(2.182)

Let the solution to Eq. (2.180) be of the form

\tilde{p} (r, θ, ϕ) = \sum_{n = 0}^{\infty} \sum_{m = 0}^{n} {\tilde{p}}_{m n} R_{n} (r) Z_{m n} (θ) Φ_{m} (ϕ) .

$\tilde{p} (r, θ, ϕ) = \sum_{n = 0}^{\infty} \sum_{m = 0}^{n} {\tilde{p}}_{m n} R_{n} (r) Z_{m n} (θ) Φ_{m} (ϕ) .$

(2.183)

Substituting this in Eq. (2.180), multiplying through by r ², and dividing through by R _n(r) Z _mn(θ)Φ_n(ϕ) yields

\frac{r^{2}}{R_{n}} \frac{\partial^{2} R_{n}}{\partial r^{2}} + \frac{2 r}{R_{n}} \frac{\partial R_{n}}{\partial r} + k^{2} r^{2} = - \frac{1}{Z_{m n}} \frac{\partial^{2} Z_{m n}}{\partial θ^{2}} - \frac{1}{Z_{m n} \tan θ} \frac{\partial Z_{m n}}{\partial θ} - \frac{1}{Φ_{m} \sin^{2} θ} \frac{\partial^{2} Φ_{m}}{\partial ϕ^{2}} .

$\frac{r^{2}}{R_{n}} \frac{\partial^{2} R_{n}}{\partial r^{2}} + \frac{2 r}{R_{n}} \frac{\partial R_{n}}{\partial r} + k^{2} r^{2} = - \frac{1}{Z_{m n}} \frac{\partial^{2} Z_{m n}}{\partial θ^{2}} - \frac{1}{Z_{m n} \tan θ} \frac{\partial Z_{m n}}{\partial θ} - \frac{1}{Φ_{m} \sin^{2} θ} \frac{\partial^{2} Φ_{m}}{\partial ϕ^{2}} .$

(2.184)

If both sides of Eq. (2.184) are equated to a constant of separation n(n + 1), then Eq. (2.184) can then be separated into three equations for each ordinate as follows.

The radial equation in r

After equating the left-hand side of Eq. (2.184) to n(n + 1), we have

(\frac{\partial^{2}}{\partial r^{2}} + \frac{2}{r} \frac{\partial}{\partial r} + k^{2} - \frac{n (n + 1)}{r^{2}}) R_{n} (r) = 0 .

$(\frac{\partial^{2}}{\partial r^{2}} + \frac{2}{r} \frac{\partial}{\partial r} + k^{2} - \frac{n (n + 1)}{r^{2}}) R_{n} (r) = 0 .$

(2.185)

The solution to this equation is of the form

R_{n} (r) = R_{n +} h_{n}^{(2)} (k r) + R_{n -} h_{n}^{(1)} (k r),

$R_{n} (r) = R_{n +} h_{n}^{(2)} (k r) + R_{n -} h_{n}^{(1)} (k r),$

(2.186)

where h _n ⁽¹⁾ (x) and h _n ⁽²⁾ (x) are spherical Hankel functions, which are also known as Hankel functions of fractional order, as defined by

h_{n}^{(1)} (x) = j_{n} (x) + j y_{n} (x),

$h_{n}^{(1)} (x) = j_{n} (x) + j y_{n} (x),$

(2.187)

h_{n}^{(2)} (x) = j_{n} (x) - j y_{n} (x),

$h_{n}^{(2)} (x) = j_{n} (x) - j y_{n} (x),$

(2.188)

where j _n(x) and y _n(x) are spherical Bessel functions of the first and second kind respectively, as plotted in Figs. 2.20 and 2.21. The “2” in parentheses denotes an outgoing spherical wave and the “1” denotes an incoming one. These spherical Bessel functions are related to the cylindrical Bessel functions of integer order

J_{n + \frac{1}{2}} (x)

$J_{n + \frac{1}{2}} (x)$

and

Y_{n + \frac{1}{2}} (x)

$Y_{n + \frac{1}{2}} (x)$

j_{n} (x) = \sqrt{\frac{π}{2 x}} J_{n + \frac{1}{2}} (x),

$j_{n} (x) = \sqrt{\frac{π}{2 x}} J_{n + \frac{1}{2}} (x),$

(2.189)

y_{n} (x) = \sqrt{\frac{π}{2 x}} Y_{n + \frac{1}{2}} (x) .

$y_{n} (x) = \sqrt{\frac{π}{2 x}} Y_{n + \frac{1}{2}} (x) .$

(2.190)

Figure 2.20 Spherical Bessel functions of the first kind.

Figure 2.21 Spherical Bessel functions of the second kind.

The inclination equation in θ

After equating the right-hand side of Eq. (2.184) to n(n + 1), we have

\frac{1}{Z_{m n}} \frac{\partial^{2} Z_{m n}}{\partial θ^{2}} + \frac{1}{Z_{m n} \tan θ} \frac{\partial Z_{m n}}{\partial θ} + n + 1 = - \frac{1}{Φ_{m} \sin^{2} θ} \frac{\partial^{2} Φ_{m}}{\partial ϕ^{2}} .

$\frac{1}{Z_{m n}} \frac{\partial^{2} Z_{m n}}{\partial θ^{2}} + \frac{1}{Z_{m n} \tan θ} \frac{\partial Z_{m n}}{\partial θ} + n + 1 = - \frac{1}{Φ_{m} \sin^{2} θ} \frac{\partial^{2} Φ_{m}}{\partial ϕ^{2}} .$

(2.191)

Equating the left-hand side of Eq. (2.191) to another constant of separation m ²/sin² θ yields

(\frac{\partial^{2}}{\partial θ^{2}} + \frac{1}{tan θ} \frac{\partial}{\partial θ} + n (n + 1) - \frac{m^{2}}{{sin}^{2} θ}) Z_{m n} (θ) = 0 .

$(\frac{\partial^{2}}{\partial θ^{2}} + \frac{1}{tan θ} \frac{\partial}{\partial θ} + n (n + 1) - \frac{m^{2}}{{sin}^{2} θ}) Z_{m n} (θ) = 0 .$

(2.192)

After substituting z = cos θ, the inclination equation becomes

[(1 - z^{2}) \frac{\partial^{2}}{\partial z^{2}} - 2 z \frac{\partial}{\partial z} + n (n + 1) - \frac{m^{2}}{1 - z^{2}}] Z_{m n} (z) = 0.

$[(1 - z^{2}) \frac{\partial^{2}}{\partial z^{2}} - 2 z \frac{\partial}{\partial z} + n (n + 1) - \frac{m^{2}}{1 - z^{2}}] Z_{m n} (z) = 0.$

(2.193)

The solution to this equation is of the form

Z_{m n} (z) = Θ_{m n} P_{n}^{m} (z)

$Z_{m n} (z) = Θ_{m n} P_{n}^{m} (z)$

(2.194)

Z_{m n} (θ) = Θ_{m n} P_{n}^{m} (\cos θ),

$Z_{m n} (θ) = Θ_{m n} P_{n}^{m} (\cos θ),$

(2.195)

where P _n ^m(cos θ) is the associated Legendre function. In the case of axisymmetry, where m = 0, it reduces to the Legendre function (or Legendre polynomial) denoted by P _n(cos θ), as plotted in Fig. 2.22.

The azimuth equation in ϕ

Equating the right-hand side of Eq. (2.191) to the constant of separation m ²/sin² θ yields

(\frac{\partial^{2}}{\partial ϕ^{2}} + m^{2}) Φ_{m} (ϕ) = 0.

$(\frac{\partial^{2}}{\partial ϕ^{2}} + m^{2}) Φ_{m} (ϕ) = 0.$

(2.196)

The solution to this equation is of the form

Φ_{m} (ϕ) = A_{m} \cos (m ϕ) + B_{m} \sin (m ϕ) .

$Φ_{m} (ϕ) = A_{m} \cos (m ϕ) + B_{m} \sin (m ϕ) .$

(2.197)

It can be seen that the integer m denotes the mth harmonic of the azimuthal modes of vibration where ϕ = 2π represents a full rotation about the z axis. The values of A _m and B _m depend on where the nodes and antinodes lie on the circumference. For example, setting B _m = 0 would place the nodes at ϕ = 0, π, and 2π. Now the complete solution to Eq. (2.180) may be written as

\begin{matrix} \tilde{p} (r, θ, ϕ) = \sum_{n = 0}^{\infty} \sum_{m = 0}^{n} {\tilde{p}}_{m n} (R_{n +} h_{n}^{(2)} (k r) + R_{n -} h_{n}^{(1)} (k r)) \\ \times P_{n}^{m} (\cos θ) (A_{m} \cos (m ϕ) + B_{m} \sin (m ϕ)) . \end{matrix}

$\begin{matrix} \tilde{p} (r, θ, ϕ) = \sum_{n = 0}^{\infty} \sum_{m = 0}^{n} {\tilde{p}}_{m n} (R_{n +} h_{n}^{(2)} (k r) + R_{n -} h_{n}^{(1)} (k r)) \\ \times P_{n}^{m} (\cos θ) (A_{m} \cos (m ϕ) + B_{m} \sin (m ϕ)) . \end{matrix}$

(2.198)

which in the case of axial symmetry (m = 0) simplifies to

\tilde{p} (r, θ) = \sum_{n = 0}^{\infty} {\tilde{p}}_{n} (R_{n +} h_{n}^{(2)} (k r) + R_{n -} h_{n}^{(1)} (k r)) P_{n} (\cos θ) .

$\tilde{p} (r, θ) = \sum_{n = 0}^{\infty} {\tilde{p}}_{n} (R_{n +} h_{n}^{(2)} (k r) + R_{n -} h_{n}^{(1)} (k r)) P_{n} (\cos θ) .$

(2.199)

Problem 2.1. A piccolo is a side-blown half-flute which is open at both ends (open-open pipe), whereas a pan flute is end blown and blocked at the opposite end (open-closed pipe). Assuming an effective air column length of 294 mm in both, including any end corrections, which instrument has the lowest fundamental resonance frequency? Which instrument does not produce even harmonics and thus has a “hollower” tone? Using Eq. (2.112), calculate the fundamental resonance frequency (n = 0) for the pan flute. Assuming the eigenfrequencies of an open-open pipe to be the same as those of a closed-closed one, use Eq. (2.94) to calculate the fundamental resonance frequency (n = 1) of the piccolo.

Problem 2.2. The wall of an infinitely long cylinder of radius a moves radially with velocity

{\tilde{u}}_{0}

${\tilde{u}}_{0}$

. Hence it may be considered to be “pulsating.” Derive the radial pressure distribution using both the homogeneous and inhomogeneous wave equations and thus verify the Bessel function identity used in Eq. (4.198).

Hint: Because of the infinite length and rotational symmetry of the cylinder, this reduces to a one-dimensional problem in the radial distance from the center w. The axial symmetry ensures that there is zero pressure gradient at the center, which behaves like a rigid termination to the waves transmitted from the wall. Hence, it is analogous to the closed tube of Section 2.7. In the steady state, the solution to the homogenous wave equation (2.23) in cylindrical coordinates is given by Eqs. (2.126) and (2.127). However, we omit the Y ₀ function because of continuity at the center. Find the unknown coefficient by applying the velocity boundary condition at the wall (w = a) using the first line of Eq. (2.48) but replacing the axial ordinate x with the radial ordinate w and noting that ∂/∂wJ ₀(kw) = kJ ₁(kw). To find the solution to the inhomogeneous wave equation, rewrite Eq. (2.115), and Eqs. (2.116) in cylindrical coordinates using the Laplace operator of Eq. (2.23) and replacing the length l with the radius a. Let the solution be in the form

\tilde{p} (w) = \sum_{n = 0}^{\infty} {\tilde{A}}_{n} J_{0} (β_{n} w / a),

$\tilde{p} (w) = \sum_{n = 0}^{\infty} {\tilde{A}}_{n} J_{0} (β_{n} w / a),$

where β _n are the solutions to J ₁(β _n) = 0. Insert this into the inhomogeneous wave equation and solve for

{\tilde{A}}_{n}

${\tilde{A}}_{n}$

using the orthogonal integrals of Eqs. (A2.101a) and (A2.154) of Appendix II. Then equating the solutions to the homogenous and inhomogeneous wave equations gives the identity of Eq. (4.198).

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Table of Contents for Acoustics

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2.13. Spherical coordinates

The radial equation in r

The inclination equation in θ

The azimuth equation in ϕ

Table of Contents for
Acoustics