**Sound Radiated by a Circular Piston**

**in an Infinite Baffle**

An exact boundary integral solution for a flat piston is described in another section. For the far-field pattern, a simple closed-form result was obtained. I am not aware of a closed form solution or the near-field. One can of course perform a numerical integration to get a result from the boundary integral. Here we develop an alternate exact solution using a plane wave spectrum, which results in a one-dimensional integral (but over an infinite range), that can be numerically evaluated everywhere on the baffle surface.

These boundary integral and plane wave spectrum approches result in analytical expressions, which are ultimately evaluated numerically. We also compare these two results to a purely numerical solution, meaning there is no analytical expression for the answer. It gives me a warm feeling to get (nearly) overlaying curves from three completely different solutions, and this greatly increases confidence in the results.

The numerical solution has the great advantage of being applicable to a more general piston shape, and it is applied to typical loudspeaker cone. Radiation impedance for the flat piston and cone shapes is compared. The numerical method is described in its own section.

The initial problem geometry consists of a flat piston of radius a, flush mounted in an infinite plane. It is assumed that the piston excursion is negligible. (This is a virtually universal assumption, and is quite a good approximation for the questions of interest here. A numerical solution which includes the piston excursion is given in another section). Time dependence exp(jωt) is assumed, and suppressed in the equations.

**Plane Wave Spectrum Solution**

The general solution for the problem at hand can be written as a continuous plane wave spectrum for the axially symmetric case, as derived in the section on wave spectra. For reasons that will soon be obvious, the spectrum amplitude function is written as

As shown in the wave spectra section, D(β ) is the far-field directivity pattern of the sound. (Therefore we could simply use the exact far-field result mentioned above to solve the problem, but that would be cheating). The general solution for the problem at hand is obtained from the potential function equation (W7), using the relations defined by equation (W3):

**Solution of the Boundary Value Problem**

The piston and baffle are at z=0. We assume that the boundary condition at z=0 is

Then the spectrum function must satisfy

Equation (P4) is again in the form of a Hankel transform, and the inversion formula (W8) provides the solution for D(β )

**Sound Pressure in the Far-field**

Using the definition of ɸ_{o} in the wave spectra section, equation
(W19), the sound pressure in the far-field can be written

Apart from minor changes in notation, this is identical to the exact boundary
integral result. A rather remarkable consequence of equation (P6) is that for
a constant value of u_{0}, on the z-axis the pressure is a linear function
of frequency from DC up. Considering the complex behavior that occurs as a function
of frequency, this is extraordinary.

**Sound Pressure in the Near-field **

Substituting the solution for D(β ) into equation (P2), the pressure at an arbitrary point in space is

This integral is tricky to evaluate numerically. To obtain results at the baffle
surface z=0 it is best to divide the integration range into 3 segments. The
first segment from 0 to 4k is evaluated using a change of variables β =sinγ.
From 4k to 10k the form above works best. The final infinite segment can be
evaluated as a Weber-Schefheitlin type integral [NBS
Handbook equations 11.4.33 and 11.4.34] less another numerical integration.
A plot of the pressure at the baffle surface z=0
shows the result of this equation as solid lines, for ka=2π;
[30kb]. The piston velocity profile is
in blue. Pressure is normalized by Z_{o}, so it is expected to approach
the velocity value of 1.0 for large ka. The result of the boundary integral
is shown by the dots. A rather dense set of integration points were used, spaced
.01 wavelengths apart. The pressure was evaluated at z=.05 wavelengths, and
then a phase shift applied to approximate the surface value. Finally, the results
of the numerical solution are shown, as the dashed line. I am quite happy with
the agreement between these three results. The boundary integral is off a bit,
but the agreement between the other two is pretty darn good. In all of the remaining
plots, the exact equation result is shown as a solid line, and the numerical
result as dashed. The agreement so good you have to look carefully in some cases
to see that there really are two curves.

The next example is for ka=π/2 [27kb]. The imaginary part is now much larger, and the pressure is spread out. Note that the surface integral of the pressure pattern is identical to the integral of the velocity pattern (blue shape), even though it looks very different to say the least!

Finally a result for ka=16π [30kb]. It is interesting that the average value of the pressure over the piston face does approach the velocity value, but is quite spiky in detail. (I wonder if this contributes to the cone breakup at high frequencies?)

The results for a cone have been derived using the generalized multipole technique. I think they are downright gorgeous. A contour plot of the near-field pressure [60kb]shows amazing details. The cone loudspeaker is represented by the blue contour on the left. Results on the left-hand side of this contour are not physically meaningful. On the right one sees the main lobe of pressure emanating along the axis, nulls forming off-axis (the dark blue), and then a wide-angle lobe. The null is of course the same null that appears in the far-field pattern [27kb] for this case. The near-field phase pattern [65kb] shows contours near the driver surface taking on the shape of the surface, and then becoming spherical away from the driver. There is a kink in the phase contours in the direction of the null, since there is a 180-degree phase shift in crossing from the main lobe to the sidelobe, in the far-field. Pressure on the surface of the driver is shown in the section describing the numerical technique.

The power produced by the piston is the real part of

Where the asterisk denotes the complex conjugate. Radiation impedance is defined
as 2P divided by the aperture area, u_{o}^{2}, and Z_{o}.
The imaginary part of this integral does not contribute to the radiated power,
but does contribute the reactive part of the radiation impedance. Substituting
equations (P3) and (P7) into (P8), and performing the integral over φ,
results in

The expression in the square brackets is easily evaluated, leaving

A closed form result of this integral is given by Jacobsen. (There may be an easy way to perform this integration but I haven't found it. After much work I could only obtain the result for the real part, and I'll spare you the ugly details).

The function H_{1}(2ka) is the Struve function, defined in the NBS
Handbook, and J_{1}(2ka) is the garden-variety Bessel function.
A plot of equation (P11) [32kb] shows a distinct change
in the vicinity of ka=π,
that is when the diameter is one wavelength. Above this value the impedance
becomes very nearly the value that would occur if the wave at the surface were
a plane wave in free space. Below this value of ka the impedance becomes more
and more reactive. Also shown in this figure are the impedance computed using
the generalized multipole numerical technique, for both a flat piston and a
cone. For the flat piston there is again excellent agreement between equation
(P11) and the numerical result. The results for the cone are fairly close to
the flat piston.

Note: Equation (P11) was originally written with a negative complex part. My thanks to Teodoro Marinucci for pointing out the error.