Acoustics

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2.12. Cylindrical coordinates

In problems where there is axial symmetry, cylindrical coordinates are often useful, as shown in Fig. 2.16. We shall use these for planar circular radiators. In the xy plane of the rectangular coordinate system, the x and y ordinates are replaced by polar ordinates w and ϕ where the radial ordinate w is given by

w = \sqrt{x^{2} + y^{2}}

$w = \sqrt{x^{2} + y^{2}}$

(2.155)

and the azimuthal ordinate ϕ is given by

ϕ = arctan (y / x) .

$ϕ = arctan (y / x) .$

(2.156)

Conversely

x = w \cos ϕ,

$x = w \cos ϕ,$

(2.157)

y = w \sin ϕ .

$y = w \sin ϕ .$

(2.158)

The rectangular z ordinate simply becomes the axial cylindrical ordinate. The three-dimensional wave equation in cylindrical coordinates is

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

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

(2.159)

where the Laplace operator is given by

\nabla^{2} = \frac{\partial^{2}}{\partial w^{2}} + \frac{1}{w} \frac{\partial}{\partial w} + \frac{1}{w^{2}} \frac{\partial^{2}}{\partial ϕ^{2}} + \frac{\partial^{2}}{\partial z^{2}},

$\nabla^{2} = \frac{\partial^{2}}{\partial w^{2}} + \frac{1}{w} \frac{\partial}{\partial w} + \frac{1}{w^{2}} \frac{\partial^{2}}{\partial ϕ^{2}} + \frac{\partial^{2}}{\partial z^{2}},$

(2.160)

which is often written in the following short form:

\nabla^{2} = \frac{1}{w} \frac{\partial}{\partial w} (w \frac{\partial}{\partial w}) + \frac{1}{w^{2}} \frac{\partial^{2}}{\partial ϕ^{2}} + \frac{\partial^{2}}{\partial z^{2}} .

$\nabla^{2} = \frac{1}{w} \frac{\partial}{\partial w} (w \frac{\partial}{\partial w}) + \frac{1}{w^{2}} \frac{\partial^{2}}{\partial ϕ^{2}} + \frac{\partial^{2}}{\partial z^{2}} .$

(2.161)

Let the solution to Eq. (2.159) be of the form

\tilde{p} (w, ϕ, z) = \sum_{n = 0}^{\infty} {\tilde{p}}_{n} W_{n} (w) Φ_{n} (ϕ) Z (z) .

$\tilde{p} (w, ϕ, z) = \sum_{n = 0}^{\infty} {\tilde{p}}_{n} W_{n} (w) Φ_{n} (ϕ) Z (z) .$

(2.162)

Substituting this in Eq. (2.159), multiplying through by w ², and dividing through by W _n(w)Φ_n(ϕ)Z(z) yields

\frac{w^{2}}{W_{n}} \frac{\partial^{2} W_{n}}{\partial w^{2}} + \frac{w}{W_{n}} \frac{\partial W_{n}}{\partial w} + k_{w}^{2} w^{2} = - \frac{1}{Φ_{n}} \frac{\partial^{2} Φ_{n}}{\partial ϕ^{2}} - \frac{w^{2}}{Z} \frac{\partial^{2} Z}{\partial z^{2}} - k_{z}^{2} w^{2},

$\frac{w^{2}}{W_{n}} \frac{\partial^{2} W_{n}}{\partial w^{2}} + \frac{w}{W_{n}} \frac{\partial W_{n}}{\partial w} + k_{w}^{2} w^{2} = - \frac{1}{Φ_{n}} \frac{\partial^{2} Φ_{n}}{\partial ϕ^{2}} - \frac{w^{2}}{Z} \frac{\partial^{2} Z}{\partial z^{2}} - k_{z}^{2} w^{2},$

(2.163)

where

k^{2} = k_{w}^{2} + k_{z}^{2} .

$k^{2} = k_{w}^{2} + k_{z}^{2} .$

(2.164)

If both sides of Eq. (2.163) are equated to a constant of separation n ², then Eq. (2.163) can then be separated into three equations for each ordinate as follows.

The radial equation in w

(\frac{\partial^{2}}{\partial w^{2}} + \frac{1}{w} \frac{\partial}{\partial w} + k_{w}^{2} - \frac{n^{2}}{w^{2}}) W_{n} (w) = 0.

$(\frac{\partial^{2}}{\partial w^{2}} + \frac{1}{w} \frac{\partial}{\partial w} + k_{w}^{2} - \frac{n^{2}}{w^{2}}) W_{n} (w) = 0.$

(2.165)

The solution to this equation is of the form

W_{n} (w) = W_{n +} H_{n}^{(2)} (k_{w} w) + W_{n -} H_{n}^{(1)} (k_{w} w),

$W_{n} (w) = W_{n +} H_{n}^{(2)} (k_{w} w) + W_{n -} H_{n}^{(1)} (k_{w} w),$

(2.166)

where H _n ⁽¹⁾(x) and H _n ⁽²⁾(x) are Hankel functions defined by

H_{n}^{(1)} (x) = J_{n} (x) + j Y_{n} (x),

$H_{n}^{(1)} (x) = J_{n} (x) + j Y_{n} (x),$

(2.167)

H_{n}^{(2)} (x) = J_{n} (x) - j Y_{n} (x),

$H_{n}^{(2)} (x) = J_{n} (x) - j Y_{n} (x),$

(2.168)

where J _n(x) and Y _n(x) are Bessel functions of the first and second kind respectively, as plotted in Figs. 2.17 and 2.18. The “2” in parentheses denotes an outgoing cylindrical wave and the “1” denotes an incoming one.

Figure 2.17 Bessel functions of the first kind.

Figure 2.18 Bessel functions of the second kind.

The azimuthal equation in ϕ

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

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

(2.169)

The solution to this equation is of the form

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

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

(2.170)

It can be seen that the integer n denotes the nth harmonic of the azimuthal modes of vibration where ϕ = 2π represents a full rotation about the z axis. The values of A _n and B _n depend on where the nodes and antinodes lie on the circumference. For example, setting B _n = 0 would place the nodes at ϕ = 0, π, and 2π.

The axial equation in z

(\frac{\partial^{2}}{\partial z^{2}} + k_{z}^{2}) Z (z) = 0.

$(\frac{\partial^{2}}{\partial z^{2}} + k_{z}^{2}) Z (z) = 0.$

(2.171)

The solution to this plane wave equation is of the form

Z (z) = Z_{+} e^{- j k_{z} z} + Z_{-} e^{j k_{z} z},

$Z (z) = Z_{+} e^{- j k_{z} z} + Z_{-} e^{j k_{z} z},$

(2.172)

where the + sign denotes a forward traveling wave and the − sign a reverse one. From Eq. (2.164) we observe that

k_{z} = {\begin{matrix} \sqrt{k^{2} - k_{w}^{2}}, & k \geq k_{w} · \\ - j \sqrt{k_{w}^{2} - k^{2}}, & k < k_{w} \end{matrix},

$k_{z} = {\begin{matrix} \sqrt{k^{2} - k_{w}^{2}}, & k \geq k_{w} · \\ - j \sqrt{k_{w}^{2} - k^{2}}, & k < k_{w} \end{matrix},$

(2.173)

Hence for k < k _w the forward traveling term becomes an evanescent decaying one. Evanescent waves typically occur close to sound sources in the form of nonpropagating standing waves.

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Table of Contents for Acoustics

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2.12. Cylindrical coordinates

The radial equation in w

The azimuthal equation in ϕ

The axial equation in z

Table of Contents for
Acoustics