6.3 Cylindrical waves

A long vibrating cylinder — a wire, a tube, a powerline in the wind — radiates a cylindrically symmetric field. The geometry is different from a sphere in one important way: the field spreads only in the two dimensions perpendicular to the cylinder’s axis, not in all three. This changes both the amplitude falloff and the temporal-frequency structure of the wave.

The radial wave equation in cylindrical coordinates

For a field depending only on cylindrical radius $\rho$ and time, the wave equation reduces to

\frac{1}{c^2} \frac{\partial^2 p}{\partial t^2} \;=\; \frac{1}{\rho} \frac{\partial}{\partial \rho}\!\left(\rho \frac{\partial p}{\partial \rho}\right).

Look for harmonic solutions $p(\rho, t) = \tilde p(\rho)\, e^{i\omega t}$ . Substituting yields Bessel’s equation:

\tilde p'' + \frac{1}{\rho} \tilde p' + k^2 \tilde p \;=\; 0,

with $k = \omega/c$ . Its solutions are the Bessel and Hankel functions. The outgoing-wave solution is the Hankel function of the second kind, $H_0^{(2)}(k\rho)$ (with our $e^{i\omega t}$ convention). In the far field ( $k \rho \gg 1$ ),

H_0^{(2)}(k\rho) \;\approx\; \sqrt{\frac{2}{\pi k \rho}}\, e^{-i(k\rho - \pi/4)},

p(\rho, t) \;\approx\; \frac{P_0}{\sqrt{\rho}}\, \cos\!\big(\omega t - k\rho + \pi/4\big).

The amplitude decays as $1/\sqrt{\rho}$ — slower than the $1/r$ of a spherical wave. The intensity correspondingly decays as $1/\rho$ :

\langle I \rangle \;\propto\; \frac{1}{\rho},

a $-3$ dB per doubling of distance, half as fast as the spherical case.

Why $1/\sqrt{\rho}$ instead of $1/r$

Conservation of energy through a coaxial cylindrical surface of radius $\rho$ and length $L$ . Surface area = $2\pi \rho L$ . Total power radiated per unit length of cylinder: $\langle I \rangle \cdot 2\pi \rho \cdot L$ for some $\langle I \rangle$ at radius $\rho$ . For this to be independent of $\rho$ , $\langle I \rangle \propto 1/\rho$ , and so $P_0 \propto 1/\sqrt{\rho}$ .

The same logic for spheres: surface area $4\pi r^2$ , so $\langle I \rangle \propto 1/r^2$ , and $P_0 \propto 1/r$ .

The pattern: in $d$ spatial dimensions, intensity falls as $1/r^{d-1}$ . Sound in 2-D ( $d = 2$ , i.e. cylindrical) falls as $1/r$ . Sound in 3-D ( $d = 3$ , spherical) falls as $1/r^2$ . Sound in 1-D ( $d = 1$ , plane wave) doesn’t fall at all.

Phase: the $\pi/4$ shift and the Hankel function

The far-field cylindrical wave has an extra phase of $\pi/4$ ahead of the plane-wave argument. This is a famous feature of Bessel-function asymptotics: the cylindrical Green’s function picks up a $-\pi/4$ phase at infinity relative to the spherical case. It has measurable consequences in 2-D acoustic imaging — the resolved positions of sources are slightly different from what you’d predict by naive ray tracing.

When does this matter?

Cylindrical waves are uncommon as a primary radiation pattern but appear all the time as approximations:

Long structures. A vibrating power line, a long pipe carrying flow noise, an extended distributed source. Within the cylinder’s “near field” (distances comparable to the source length) the wave is roughly cylindrical. Far away, the source looks more like a point and the wave looks roughly spherical.
Ducts and waveguides. Inside a tube of constant cross-section, the lowest mode is essentially a 1-D plane wave (no transverse falloff), but higher modes propagate as 2-D cylindrical waves transverse to the tube axis (chapter 7).
2-D problems. Surface waves on water; sound in a thin shallow layer between two reflecting surfaces. The effective dimensionality is two, and the falloff is $1/\sqrt{r}$ .

Looking ahead

We’ve covered isotropic radiation. But many real sources are not isotropic: a violin string moving transversely, a tuning fork prong, a flat speaker cone. These have directional radiation patterns. The simplest non-isotropic source is the dipole — two opposing monopoles. We meet it next.