# Sound from rotors near a trailing edge

Michael Carley
Department of Mechanical Engineering
University of Bath
Bath BA2 7AY
United Kingdom
<EMAIL_ADDRESS>
###### Abstract

A method of noise reduction proposed for the next generation of aircraft is to
shield noise from the propulsion system, by positioning the noise source over
a wing or another surface. In this paper, an approximate analysis is developed
for the acoustic field far from a circular source placed near the edge of a
semi-infinite plate, a model problem for shielding of noise by a wing and for
scattering by a trailing edge. The approximation is developed for a source of
small radius and is found to be accurate when compared to full numerical
evaluation of the field.

##  1 Introduction

The effect of airframe configurations on the propagation of noise from
propulsion systems is a question which has been studied over a number of
years. In particular, studies have been conducted of the possibility of
positioning propulsion systems in such a way that the airframe acts to
“shield” the noise source and reduce noise levels reaching the ground. One
example is the “Silent Aircraft” Initiative, in which a study was conducted of
the scattering of noise from sources placed above the blended wing body
configuration [1]. Another study [2] examined the effect of propeller
positioning on an otherwise conventional airframe, making use of the tailplane
for acoustic shielding. As electric propulsion improves, and novel
configurations using airframe shielding for noise reduction become more
feasible, there is a greater need for reliable methods which can be used in
design and to provide insight into the propagation of sound around airframes.

In this paper, we consider a rotating source located near the trailing edge of
a semi-infinite plate in a uniform flow, a model problem for shielding of
noise from over-wing rotors. In a study of shielding of noise from a counter-
rotation open rotor [3], two theories for shielding attenuation were compared
to experimental data, barrier theory and half-plane diffraction. It was found
that neither gave accurate results, over-predicting the attenuation by 5 d B
when a point source was used to approximate the rotor.

We present here an analysis of far-field noise from a rotating source which is
comparable in its methods to established theory for noise from rotating
sources, and which compares well to full numerical evaluation. It should be
useful to designers and researchers as a starting point for parametric studies
of rotor positioning in new configurations.

##  2 Analysis

The problem of scattering by a semi-infinite plane is an old one [4], which
has been extended to include the effect of mean flow in the acoustic case [5].
We begin by presenting existing results for the problem in a form which can be
incorporated into standard methods for the calculation of the acoustic far
field of rotating sources.

###  2.1 Green’s function

Figure 1: Notation for semi-infinite plate scattering

The coordinate system used to analyze the scattering of sound by a semi-
infinite plate is shown in Figure 1. A uniform flow of Mach number $M_{0}$ is
imposed normal to the trailing edge. A Cartesian coordinate system $(x,y,z)$
is centred on the edge of the plate, with $x$ parallel to the flow, $y$ normal
to the plate and $z$ aligned with the edge. A cylindrical coordinate system
$(r,\theta,z)$ is defined, with $r=\sqrt{x^{2}+y^{2}}$ and
$\theta=\tan^{-1}y/x$. The Green’s function for the field at a point $(x,y,z)$
due to a source of time dependence $\exp[-\mathrm{i}\omega t]$ at
$(x_{0},y_{0},z_{0})$ is given by [5]

$\displaystyle G(\mathbf{x},\mathbf{x}_{0})$
$\displaystyle=\frac{\mathrm{e}^{-\mathrm{i}KM_{0}(X-X_{0})}}{\beta}\left[g_{1}(\mathbf{x},\mathbf{x}_{0})+g_{2}(\mathbf{x},\mathbf{x}_{0})\right],$
(1) $\displaystyle g_{i}(\mathbf{x},\mathbf{x}_{0})$
$\displaystyle=\frac{\mathrm{i}K}{8\pi}\int_{-\infty}^{S_{i}}\frac{H_{1}^{(1)}(K\overline{r}_{i}\sqrt{1+u^{2}})}{\sqrt{1+u^{2}}}\,\mathrm{d}u,$
(2)

where $H_{n}^{(1)}(\cdot)$ is the Hankel function of the first kind, and

$\displaystyle\beta$ $\displaystyle=\sqrt{1-M_{0}^{2}},\quad
X=\frac{x}{\beta},\quad K=\frac{\omega}{c}\frac{1}{\beta},$
$\displaystyle\overline{r}_{1,2}$ $\displaystyle=\left[(X-X_{0})^{2}+(y\mp
y_{0})^{2}+(z-z_{0})^{2}\right]^{1/2},$
$\displaystyle=\left[\overline{r}^{2}+\overline{r}_{0}^{2}-2\overline{r}\overline{r}_{0}\cos\left(\overline{\theta}\mp\overline{\theta}_{0}\right)+(z-z_{0})^{2}\right]^{1/2},$
$\displaystyle S_{1}$
$\displaystyle=\frac{2\sqrt{\overline{r}\overline{r}_{0}}}{\overline{r}_{1}}\cos\frac{\overline{\theta}-\overline{\theta}_{0}}{2},\quad
S_{2}=-\frac{2\sqrt{\overline{r}\overline{r}_{0}}}{\overline{r}_{2}}\cos\frac{\overline{\theta}+\overline{\theta}_{0}}{2},$
$\displaystyle\overline{r}$
$\displaystyle=\sqrt{X^{2}+y^{2}},\quad\overline{\theta}=\tan^{-1}\frac{y}{X}.$

For convenience, we note that the form of (1) is that of calculating the sound
from a source and its image in the plane $y=0$. If we take $\theta_{1}$ as the
coordinate of the source and define $\theta_{2}=\theta_{1}-2\pi$ as the
coordinate of the image, the acoustic field can be calculated as the sum of
the fields $g_{i}$ from the source and its image, using

$\displaystyle\overline{r}_{i}$
$\displaystyle=\left[\overline{r}^{2}+\overline{r}_{0}^{2}-2\overline{r}\overline{r}_{0}\cos\left(\overline{\theta}-\overline{\theta}_{i}\right)+(z-z_{0})^{2}\right]^{1/2},$
(3) $\displaystyle S_{i}$
$\displaystyle=\frac{2\sqrt{\overline{r}\overline{r}_{0}}}{\overline{r}_{i}}\cos\frac{\overline{\theta}-\overline{\theta}_{i}}{2}.$
(4)

We rewrite (2) in a form which facilitates physical interpretation when
applied to the rotor noise problem, and can be used to efficiently approximate
the field in different regions. Using the result

$\displaystyle\int_{0}^{\infty}\frac{H_{1}^{(1)}(K\overline{r}_{i}\sqrt{1+u^{2}})}{\sqrt{1+u^{2}}}\,\mathrm{d}u$
$\displaystyle=-\mathrm{i}\frac{\mathrm{e}^{\mathrm{i}K\overline{r}_{i}}}{K\overline{r}_{i}},$

we can write $g_{i}$ in a number of alternative forms,

$\displaystyle g_{i}(\mathbf{x},\mathbf{x}_{0})$
$\displaystyle=\frac{\mathrm{e}^{\mathrm{i}K\overline{r}_{i}}}{4\pi\overline{r}_{i}}-\frac{\mathrm{i}K}{8\pi}\mathcal{L}^{\prime}(K\overline{r}_{i},S_{i}),$
(5)
$\displaystyle=\frac{\mathrm{e}^{\mathrm{i}K\overline{r}_{i}}}{8\pi\overline{r}_{i}}+\frac{\mathrm{i}K}{8\pi}\mathcal{L}(K\overline{r}_{i},S_{i}),$
(6)

where for convenience, we define the functions

$\displaystyle\mathcal{L}(\alpha,s)$
$\displaystyle=\int_{0}^{s}\frac{H_{1}^{(1)}(\alpha\sqrt{1+u^{2}})}{\sqrt{1+u^{2}}}\,\mathrm{d}u,$
(7) $\displaystyle\mathcal{L}^{\prime}(\alpha,s)$
$\displaystyle=\int_{s}^{\infty}\frac{H_{1}^{(1)}(\alpha\sqrt{1+u^{2}})}{\sqrt{1+u^{2}}}\,\mathrm{d}u.$
(8)

The evaluation of the Green’s function for scattering by the plate requires
the evaluation of the integrals of (7) and (8). A solution for most of the far
field will be found by approximating (8), while an exact expansion for (7)
will be developed and used to find a solution for the remaining part of the
field.

Integration by parts gives an asymptotic approximation for
$\mathcal{L}^{\prime}(\cdot)$,

$\displaystyle\int_{s}^{\infty}\frac{H_{1}^{(1)}(\alpha\sqrt{1+u^{2}})}{\sqrt{1+u^{2}}}\,\mathrm{d}u$
$\displaystyle\sim\frac{H_{0}^{(1)}(\alpha\sqrt{1+s^{2}})}{\alpha
s},\quad\alpha\to\infty,$ (9)

so that for large $K\overline{r}_{i}$,

$\displaystyle g_{i}(\mathbf{x},\mathbf{x}_{0})$
$\displaystyle\approx-\mathrm{i}\frac{H_{0}^{(1)}(Kv)}{8\pi|\overline{r}_{i}S_{i}|}+\frac{\mathrm{e}^{\mathrm{i}K\overline{r}_{i}}}{4\pi\overline{r}_{i}}H(S_{i}),$
(10) $\displaystyle v^{2}$
$\displaystyle=(\overline{r}+\overline{r}_{0})^{2}+(z-z_{0})^{2},$

and $H(\cdot)$ is the Heaviside step function.

Integration by parts yields an approximation valid throughout the far field
except as $|S_{i}|\to 0$ and allows us to interpret the Green’s function in
terms of the field generated by the source and by its image at
$(x_{0},-y_{0},z_{0})$. For all values of $S_{i}$, the first term in (10)
contributes to the radiated field, corresponding to scattering from the edge.
For points where $S_{i}>0$, the free-field Green’s function
$\exp[\mathrm{i}K\overline{r}_{i}]/4\pi\overline{r}_{i}$ is “switched on” and
contributes to the acoustic field. The acoustic domain can be decomposed into
regions according to which terms of (10) contribute to the acoustic potential.
Upon examination of the definition of $S_{i}$, it is clear that the boundaries
of these regions are the lines through the trailing edge and the position of
the source and of its image. Figure 2 shows these regions for a source placed
above the plate, upstream of the trailing edge. Above the plate and for some
distance behind it, both $S_{1}$ and $S_{2}$ are positive and the acoustic
field can be viewed as that generated by the source and its image in the
plate, with a correction term applied. Below the plate, where both $S_{1}$ and
$S_{2}$ are negative, both free-field Green’s functions are switched off and
this region is a “shadow zone” where only the Hankel function of (10)
contributes for each source. Finally, behind the trailing edge, in the wedge
where $S_{1}>0$ and $S_{2}<0$, the main source contributes its free-field
term, but the image source does not. A physical interpretation of the effect
of $S_{i}\to 0$ is that the observer is approaching the line dividing the
regions where a source does or does not contribute its free-field term, and
$g_{i}$ is better represented by (6).

Figure 2: Contribution of acoustic source and image in different regions

In order to find an expansion which can be used for small $S_{i}$, we apply
the product theorem for Bessel functions [6, 8.530.2] to (6),

$\displaystyle\frac{H_{1}^{(1)}(\alpha\sqrt{1+u^{2}})}{\sqrt{1+u^{2}}}$
$\displaystyle=2\sum_{q=0}^{\infty}(-1)^{q}(2q+1)H_{2q+1}^{(1)}(\alpha)\frac{J_{2q+1}(\alpha
u)}{\alpha u}.$ (11)

Upon integration of the Bessel functions of the first kind,

$\displaystyle\mathcal{L}(\alpha,s)$
$\displaystyle=\frac{2}{\alpha}\sum_{q=0}^{\infty}(-1)^{q}H_{2q+1}^{(1)}(\alpha)\left[J_{2q+1}(\alpha
s)+2\sum_{k=1}^{\infty}J_{2q+2k+1}(\alpha s)\right],$ (12)

which is exact for $|s|<1$, though slow to converge as $|s|\to 1$. This result
can be used to evaluate the acoustic field in the region $s\to 0$.

###  2.2 Acoustic field of a ring source near a trailing edge

Figure 3: Coordinate system for evaluation of acoustic field of ring source

We now apply the results of Section 2.1 to the estimation of the acoustic
field of a rotating source impinging on the edge of a semi-infinite plate. The
system is modelled as a ring source and its image in the plate, with the field
of each calculated using the “partial” Green’s function $G_{1,2}$ with

$\displaystyle G_{i}(\mathbf{x},\mathbf{x}_{0})$
$\displaystyle=\frac{\mathrm{e}^{-\mathrm{i}KM_{0}(X-X_{0})}}{\beta}g_{i}(\mathbf{x},\mathbf{x}_{0})$
(13)

with the principal source denoted by subscript $1$ and its image by subscript
$2$.

As is standard for rotor noise problems [7, 8, 9, 10, 11], the source is taken
to vary as $\exp[\mathrm{i}n\psi_{0}]$ where $\psi_{0}$ is the angular
variable on a circular source. To respect the image system, the image ring
source has azimuthal variation $\exp[-\mathrm{i}n\psi_{0}]$ and its field can
be calculated using the same analysis, changing $n$ to $-n$.

The ring is centred at a point $(x_{0},h,0)$ with coordinates on the source
given by

$\displaystyle\mathbf{x}_{0}$
$\displaystyle=(x_{0},h+a\cos\psi_{0},a\sin\psi_{0}),$

where $a$ is the radius of the source. We introduce spherical coordinates
centred on each ring source,

$\displaystyle X-X_{0}$ $\displaystyle=R_{i}\cos\phi_{i},$ $\displaystyle y\pm
h$ $\displaystyle=R_{i}\sin\phi_{i}\cos\psi_{i},$ $\displaystyle z$
$\displaystyle=R_{i}\sin\phi_{i}\sin\psi_{i}.$

The acoustic field of each source is then given by

$\displaystyle p_{i}(\mathbf{x})$
$\displaystyle=\frac{\mathrm{e}^{-\mathrm{i}KM_{0}(X-X_{0})}}{\beta}\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}n\psi_{0}}g_{i}(\mathbf{x},\mathbf{x}_{0})\,\mathrm{d}\psi_{0}.$
(14)

We now develop an approximation for $p_{i}$ valid in the far field. The first
part of this calculation, for the effect of the free-field Green’s function is
well-known from previous work and is presented for completeness and to
illustrate the approach taken.

We treat the field as made up of two contributions, corresponding to the
“direct” field generated by the ring source, and that scattered by the
trailing edge. The “ring” term is the familiar integral

$\displaystyle\mathcal{R}_{n}(K,a,R_{i},\phi_{i},\psi_{i})$
$\displaystyle=\int_{0}^{2\pi}\frac{\mathrm{e}^{\mathrm{i}(K\overline{r}_{i}+n\psi_{0})}}{4\pi\overline{r}_{i}}\,\mathrm{d}\psi_{0},$
(15)

for which a number of exact [11] and approximate [7, 9] expressions exist.

To evaluate the part of the field scattered by the trailing edge, we define
the functions,

$\displaystyle\mathcal{E}_{n}(K,a,\overline{r}_{0},\overline{r}_{i},S_{i})$
$\displaystyle=\frac{\mathrm{i}K}{8\pi}\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}n\psi_{0}}\mathcal{L}(K\overline{r}_{i},S_{i})\mathrm{d}\psi_{0};$
(16)
$\displaystyle\mathcal{E}^{\prime}_{n}(K,a,\overline{r}_{0},\overline{r}_{i},S_{i})$
$\displaystyle=\frac{\mathrm{i}K}{8\pi}\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}n\psi_{0}}\mathcal{L}^{\prime}(K\overline{r}_{i},S_{i})\,\mathrm{d}\psi_{0}.$
(17)

In the spherical coordinate system, $\overline{r}_{i}$ is given by

$\displaystyle\overline{r}_{i}^{2}$ $\displaystyle=R_{i}^{2}+a^{2}-2a(y\mp
h)\cos\psi_{0}-2az\sin\psi_{0},$
$\displaystyle=R_{i}^{2}+a^{2}-2R_{i}a\sin\phi_{i}\cos(\psi_{0}-\psi_{i}).$

Differentiating with respect to $a$ and neglecting terms less than
$O(1/R_{i})$ yields a far-field approximation,

$\displaystyle\overline{r}_{i}$ $\displaystyle\approx
R_{i}-a\sin\phi_{i}\cos(\psi_{0}-\psi_{i}),$
$\displaystyle\frac{1}{\overline{r}_{i}}$
$\displaystyle\approx\frac{1}{R_{i}},$

which, upon insertion into (14) and use of tables [6, 8.411], yields the
familiar result for the acoustic field of a ring source [7, 8, 9, among many
others],

$\displaystyle\mathcal{R}_{n}(K,a,R_{i},\phi_{i},\psi_{i})$
$\displaystyle\approx(-\mathrm{i})^{n}\mathrm{e}^{\mathrm{i}n\psi_{i}}\frac{\mathrm{e}^{\mathrm{i}KR_{i}}}{2R_{i}}J_{n}(Ka\sin\phi_{i}),$

where $J_{n}(\cdot)$ is the Bessel function of the first kind. The free-field
part of the ring-source field is then given by

$\displaystyle p_{i}^{(f)}(\mathbf{x})$
$\displaystyle\approx\frac{\mathrm{e}^{-\mathrm{i}KM_{0}(X-X_{0})}}{\beta}\mathcal{R}_{n}(K,a,R_{i},\phi_{i},\psi_{i}).$
(18)

We can apply a similar approach to the approximation of the scattered field.
Given

$\displaystyle
v^{2}=\overline{r}_{1}^{2}(1+S_{1}^{2})=\overline{r}_{2}^{2}(1+S_{2}^{2})$
$\displaystyle=\left(\overline{r}+\overline{r}_{0}\right)^{2}+(z-z_{0})^{2},$

to first order in $a$,

$\displaystyle v$ $\displaystyle\approx
v_{0}+\frac{hSa}{v_{0}}\cos(\psi_{0}+\alpha),$ $\displaystyle v_{0}$
$\displaystyle=\left[\left(\overline{r}+\sqrt{X_{0}^{2}+h^{2}}\right)^{2}+z^{2}\right]^{1/2},$
$\displaystyle\alpha$
$\displaystyle=\tan^{-1}\frac{z/h}{1+\overline{r}/\sqrt{X_{0}^{2}+h^{2}}},$
$\displaystyle S^{2}$
$\displaystyle=\left(\frac{\overline{r}}{\sqrt{X_{0}^{2}+h^{2}}}+1\right)^{2}+\left(\frac{z}{h}\right)^{2}.$

Using Neumann’s addition theorem [12, 10.23],

$\displaystyle H_{0}^{(1)}(Kv)$ $\displaystyle\approx
H_{0}^{(1)}\left(Kv_{0}+\frac{KhSa}{v_{0}}\cos(\psi_{0}+\alpha)\right),$
$\displaystyle=\sum_{q=-\infty}^{\infty}H_{q}^{(1)}(Kv_{0})J_{-q}\left(K\frac{hSa}{v_{0}}\cos(\psi_{0}+\alpha)\right).$

With the use of tables [6, 6.681.8, 9], it is readily shown that for $n$ and
$q$ both odd or both even, the integral

$\displaystyle\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}n\psi}J_{q}(\alpha\cos\psi)\,\mathrm{d}\psi$
$\displaystyle=2\pi J_{(q-n)/2}(\alpha/2)J_{(q+n)/2}(\alpha/2),$ (19)

and is identically zero otherwise. This yields

$\displaystyle\int_{0}^{2\pi}H_{0}^{(1)}(Kv)\mathrm{e}^{\mathrm{i}n\psi_{0}}\,\mathrm{d}\psi_{0}$
$\displaystyle\approx
2\pi\mathrm{e}^{-\mathrm{i}n\alpha}\sum_{q=-\infty}^{\infty}H_{2q+\delta}^{(1)}(Kv_{0})J_{-(n^{\prime}+q+\delta)}\left(K\frac{hSa}{2v_{0}}\right)J_{(n^{\prime}-q)}\left(K\frac{hSa}{2v_{0}}\right),$

where $n=2n^{\prime}+\delta$, $\delta=0,1$.

The part of the ring-source field scattered by the trailing edge,

$\displaystyle\mathcal{E}^{\prime}_{n}(K,a,\overline{r}_{0},\overline{r}_{i},S_{i})$
$\displaystyle=\frac{\mathrm{i}K}{8\pi}\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}n\psi_{0}}\mathcal{L}^{\prime}(K\overline{r}_{i},S_{i})\,\mathrm{d}\psi_{0},$

is then approximated by

$\displaystyle\mathcal{E}^{\prime}_{n}(K,a,\overline{r}_{0},\overline{r}_{i},S_{i})$
$\displaystyle\approx-\frac{\mathrm{i}}{4}\frac{\mathrm{e}^{-\mathrm{i}n\alpha}}{\sqrt{v_{0}^{2}-R_{i}^{2}}}\sum_{q=-\infty}^{\infty}H_{2q+\delta}^{(1)}(Kv_{0})J_{-(n^{\prime}+q+\delta)}\left(K\frac{hSa}{2v_{0}}\right)J_{(n^{\prime}-q)}\left(K\frac{hSa}{2v_{0}}\right).$
(20)

This approximation breaks down between the free-field being switched on or off
where $S_{i}$ is small ($v_{0}\approx R_{i}$) and a different approximation is
required in this region.

If the field point lies in a region which gives rise to small values of
$S_{i}$ in the integral of (14), the field must be evaluated using (12).
Again, we proceed by approximating for the far field and using the properties
of Bessel functions to find a form suitable for evaluation.

Returning to (12),

$\displaystyle\mathcal{L}(\alpha,s)$
$\displaystyle=\frac{2}{\alpha}\sum_{q=0}^{\infty}(-1)^{q}H_{2q+1}^{(1)}(\alpha)\left[J_{2q+1}(\alpha
s)+2\sum_{k=1}^{\infty}J_{2q+2k+1}(\alpha s)\right],$

and noting that $\alpha=K\overline{r}_{i}$, we seek an approximation which can
be inserted into (14) and integrated. In the far field,
$K\overline{r}_{i}\to\infty$, the difficulty lies in capturing the rapid
variation in $K\overline{r}_{i}s_{i}$ as $s_{i}\to 0$. We proceed as follows
to develop a tractable far-field approximation. The Hankel function is
replaced by an approximation valid to first order in $a$, using the standard
relations for derivatives,

$\displaystyle H_{m}(K\overline{r}_{i})$ $\displaystyle\approx
H_{m}(KR_{i})+KaH_{m+1}(KR_{i})\sin\phi_{i}\cos(\psi_{0}-\psi_{i})+O(1/(KR_{i})^{2}).$

The argument of the Bessel function of the first kind is approximated by

$\displaystyle K\overline{r}_{i}s_{i}$ $\displaystyle\approx
2K\sqrt{\overline{r}\overline{r}_{0}}\cos\frac{\overline{\theta}-\overline{\theta}_{0}}{2}+Ka\sqrt{\frac{\overline{r}}{\overline{r}_{0}}}\sin\frac{\overline{\theta}+\overline{\theta}_{0}}{2}\cos\psi_{0},$

and again using the product theorem [6, 8.530.2],

$\displaystyle J_{m}(K\overline{r}_{i}s_{i})$
$\displaystyle\approx\sum_{k=-\infty}^{\infty}J_{m-k}\left(2K\sqrt{\overline{r}\overline{r}_{0}}\cos\frac{\overline{\theta}-\overline{\theta}_{0}}{2}\right)J_{k}\left(Ka\sqrt{\frac{\overline{r}}{\overline{r}_{0}}}\sin\frac{\overline{\theta}+\overline{\theta}_{0}}{2}\cos\psi_{0}\right),$

yielding

$\displaystyle\mathcal{L}(K\overline{r}_{i},s_{i})$
$\displaystyle\approx\frac{2}{KR_{i}}\sum_{q=0}^{\infty}(-1)^{q}\left[H_{2q+1}^{(1)}(KR_{i})+Ka\sin\phi_{i}H_{2q+2}^{(1)}(KR_{i})\cos(\psi_{0}-\psi_{i})\right]$
$\displaystyle\times\biggl{\\{}\sum_{m=-\infty}^{\infty}J_{m}(\gamma_{2}\cos\psi_{0})\left[J_{2q+1-m}\left(\gamma_{1}\right)+2\sum_{k=1}^{\infty}J_{2q+2k+1-m}(\gamma_{1})\right]\biggr{\\}},$
(21) $\displaystyle\gamma_{1}$
$\displaystyle=2K\sqrt{\overline{r}\overline{r}_{0}}\cos\frac{\overline{\theta}-\overline{\theta}_{0}}{2},\quad\gamma_{2}=Ka\sqrt{\frac{\overline{r}}{\overline{r}_{0}}}\sin\frac{\overline{\theta}+\overline{\theta}_{0}}{2},$

where all source coordinates such as $\overline{\theta}$ are evaluated at
$a=0$, the centre of the ring. Using (19), for $n=2n^{\prime}$,

$\displaystyle\mathcal{E}_{n}(K,a,\overline{r}_{0},\overline{r}_{i},S_{i})$
$\displaystyle=\frac{\mathrm{i}K}{8\pi}\int_{0}^{2\pi}\mathcal{L}(K\overline{r}_{i},s_{i})\mathrm{e}^{\mathrm{i}n\psi_{0}}\,\mathrm{d}\psi_{0}\approx$
$\displaystyle\frac{\mathrm{i}}{2R_{i}}\biggl{\\{}$
$\displaystyle\sum_{q=0}^{\infty}(-1)^{q}H_{2q+1}^{(1)}(KR_{i})\sum_{m=-\infty}^{\infty}J_{m-n^{\prime}}\left(\frac{\gamma_{2}}{2}\right)J_{m+n^{\prime}}\left(\frac{\gamma_{2}}{2}\right)M_{1}(\gamma_{1})$
$\displaystyle+\mathrm{e}^{-\mathrm{i}\psi_{i}}\frac{Ka\sin\phi_{i}}{2}$
$\displaystyle\sum_{q=0}^{\infty}(-1)^{q}H_{2q+2}^{(1)}(KR_{i})\sum_{m=-\infty}^{\infty}J_{m-n^{\prime}}\left(\frac{\gamma_{2}}{2}\right)J_{m+n^{\prime}+1}\left(\frac{\gamma_{2}}{2}\right)M_{2}(\gamma_{1})$
$\displaystyle+\mathrm{e}^{\mathrm{i}\psi_{i}}\frac{Ka\sin\phi_{i}}{2}$
$\displaystyle\sum_{q=0}^{\infty}(-1)^{q}H_{2q+2}^{(1)}(KR_{i})\sum_{m=-\infty}^{\infty}J_{m-n^{\prime}}\left(\frac{\gamma_{2}}{2}\right)J_{m+n^{\prime}-1}\left(\frac{\gamma_{2}}{2}\right)M_{3}(\gamma_{1})\biggr{\\}},$
(22)

where the auxiliary quantities are

$\displaystyle
M_{1}(\gamma_{1})=J_{2(q-m)+1}(\gamma_{1})+2\sum_{k=1}^{\infty}J_{2(q+k-m)+1}(\gamma_{1}),$
$\displaystyle
M_{2}(\gamma_{1})=J_{2(q-m)}(\gamma_{1})+2\sum_{k=1}^{\infty}J_{2(q+k-m)}(\gamma_{1}),$
$\displaystyle
M_{3}(\gamma_{1})=J_{2(q-m+1)}(\gamma_{1})+2\sum_{k=1}^{\infty}J_{2(q+k-m+1)}(\gamma_{1}).$

###  2.3 Summary

Figure 4: Regions around rotor labelled by far-field approximation

To summarize, the far-field sound from a rotor near the edge of a semi-
infinite plane can be estimated by summing the contribution from the circular
source of azimuthal order $n$ and from its image in the plane, of order $-n$.
For each source, the acoustic field is estimated by first determining the
appropriate far-field approximation to apply. Figure 4 shows the region around
a rotor divided according to the sign of $S_{i}$. In region I, $S_{i}>0$ for
all source positions on the rotor while in region II, $S_{i}<0$. In region
III, $S_{i}$ changes sign for some value of $\psi_{0}$. The appropriate far-
field approximation to use in each region is then:

$\displaystyle p_{n}$
$\displaystyle=\frac{\mathrm{e}^{-\mathrm{i}KM_{0}(X-X_{0})}}{\beta}\left(\mathcal{R}_{n}-\mathcal{E}_{n}^{\prime}\right),$
(I) (23a)
$\displaystyle=-\frac{\mathrm{e}^{-\mathrm{i}KM_{0}(X-X_{0})}}{\beta}\mathcal{E}_{n}^{\prime},$
(II) (23b)
$\displaystyle=\frac{\mathrm{e}^{-\mathrm{i}KM_{0}(X-X_{0})}}{\beta}\left(\frac{\mathcal{R}_{n}}{2}+\mathcal{E}_{n}\right),\quad$
$\displaystyle\text{(III)}.$ (23c)

In the far field, the functions $\mathcal{R}(\cdot)$, $\mathcal{E}(\cdot)$,
and $\mathcal{E}^{\prime}(\cdot)$ can be evaluated using the results given
earlier. Finally, with regard to evaluation of the series defining these
functions, we note that the Bessel function of the first kind $J_{n}(x)$
decays exponentially with increasing order for $n>x$, allowing the series to
be truncated with negligible loss of accuracy.

##  3 Results

To illustrate the application of the method, we present sample results for an
arbitrarily selected configuration, with rotor parameters $h=0.9$, $a=0.3$,
$n=6$. The rotor is positioned at $x_{0}=-2a$, i.e. one rotor diameter
upstream of the trailing edge. The rotor tip Mach number $M_{t}$ is set to
0.6, so that the wavenumber $k=nM_{t}/a=12$, and the flow Mach number
$M_{0}=0.15$. The acoustic field is evaluated using the methods of this paper,
and compared to full numerical evaluation, with results calculated above and
below the plate to examine the shielding effect.

Figure 5: Acoustic potential above and below source; left-hand column: real
part; right-hand column: imaginary part; solid line: numerical; dashed line:
approximation.

Figure 5 shows the real and imaginary part of the potential at $-16a\leq
x-x_{0}\leq 16a$, $z=0$, i.e. directly above and below the source centre.
Results are calculated at a constant vertical displacement above and below the
source, $y=h\pm 16a$. The region marked “III” on each plot shows where (23c)
has been applied, in the transition region where the asymptotic approximation
for the edge-scattered field breaks down. The beginning of this breakdown is
apparent at the end of region III, in particular in the lower left-hand plot
of the figure.

In applications, where the rotor may be placed above a wing in order to
benefit from shielding effects, the region of interest is the “shadow zone”
upstream of the trailing edge. The upper and lower fields are computed at
constant distance from the source, so the different scales on the upper and
lower plots make the shielding effect clear. In this case, the maximum
amplitude upstream of the trailing edge is about ten times greater above the
plate than below, and is well predicted by the far field approximation.

We note that while the results presented demonstrate good accuracy, the
analysis is limited by the assumptions made in its development. In particular,
the analysis does assume small source radius $a$ and numerical experimentation
has found that the accuracy of predictions degrades as $a$ is increased. This
may limit the applicability of the method, in particular as $a$ becomes
comparable to $h$, the case of a rotor with a small tip clearance above the
wing.

##  4 Conclusions

An approximate analysis has been developed for the radiation of sound from a
rotating source near the edge of a semi-infinite plate in a uniform flow, a
model problem for shielding and scattering of rotor noise by a wing. Sample
calculations for small source radius show good agreement with full numerical
evaluation, and capture the wing shielding effect reasonably accurately.
Future work will focus on improving the accuracy and range of validity of the
approximation, in particular in the case of small clearance between the rotor
and wing. An open question is that of whether there is a useful exact
expansion for the acoustic field, comparable to the series solutions which
exist for isolated rotors.

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