# Constant intensity acoustic propagation in the presence of non-uniform
properties and impedance discontinuities: Hermitian and non-Hermitian
solutions

Andrew N. Norris  Mechanical and Aerospace Engineering, Rutgers University,
Piscataway, NJ 08854<EMAIL_ADDRESS>
###### Abstract

Propagation of sound through a non-uniform medium without scattering is
possible, in principle, if the density and acoustic compressibility assume
complex values, requiring passive and active mechanisms, also known as
Hermitian and non-Hermitian solutions, respectively. Two types of constant
intensity wave conditions are identified: in the first the propagating
acoustic pressure has constant amplitude, while in the second the energy flux
remains constant. The fundamental problem of transmission across an impedance
discontinuity without reflection or energy loss is solved using a combination
of monopole and dipole resonators in parallel. The solution depends on an
arbitrary phase angle which can be chosen to give a unique acoustic
metamaterial with both resonators undamped and passive, requiring purely
Hermitian acoustic elements. For other phase angles one of the two elements
must be active and the other passive, resulting in a gain/loss non-Hermitian
system. These results prove that uni-directional and reciprocal transmission
through a slab separating two half spaces is possible using passive Hermitian
acoustic elements without the need to resort to active gain/loss energetic
mechanisms.

##  I Introduction

Sound in non-uniform materials is dominated by multiple scattering with the
result that energy cannot propagate coherently in a given direction without
loss. As a simple example, an acoustic impedance discontinuity in a one
dimensional (1D) acoustic system causes reflection and resultant loss of
energy in the transmitted wave. Even a single impedance discontinuity can
drastically reduce the transmitted energy, as for instance, at an air-water
interface. This effect is reciprocal in that the same fraction of energy is
lost regardless of the incidence direction. Scattering and loss of forward
propagating energy are inevitable in a classical or passive, disordered wave
medium, now often referred to as a Hermitian system.

Allowing the acoustic medium to become non-Hermitian, in particular through
the introduction of active materials which require external energy input,
makes it possible to avoid scattering and achieve so-called constant intensity
waves. This has been demonstrated theoretically and in simulations [1, 2, 3,
4]. The application of these ideas to acoustics was experimentally
demonstrated in a 1D waveguide containing scatterers [5]. By adding discrete
non-Hermitian elements comprising loudspeakers the backscatter was eliminated
and perfect transmission obtained. Similar introduction of non-Hermitian
elements in a multilayered 1D acoustic medium [6] has been shown
computationally to achieve constant intensity wave propagation with no
backscatter.

Non-Hermitian acoustics has been of interest at least since the concept of a
system with balanced gain and loss, also known as a parity-time (PT) symmetric
system, was proposed [7] in 2014. Applications include a metamaterial device
formed by a pair of electro-acoustic resonators realizing an acoustic coherent
perfect absorber [8], the emergence of exceptional points (EPs) in PT
acoustics [9], and coherent acoustic propagation in turbulent fluid flow [10].
A recent review [11] provides an overview of these and other implications and
applications of non-Hermitian acoustics. While certainly of interest, non-
Hermitian acoustic systems require complicated energy intense active elements,
and are to be avoided if purely passive solutions can achieve the same
results.

Our objective is to examine the theory underlying constant intensity acoustic
propagation with no backscatter in disordered systems. Two types of constant
intensity acoustics in one dimension are identified: (i) propagation with
constant magnitude of the acoustic pressure, and (ii) propagation with
constant acoustic energy flux. Two similar but different equations relating
the material proparties are obtained depending on which constant intensity
definition is considered. We then focus on constant intensity propagation
across a material discontinuity. We show that sub-wavelength actuators, either
passive or active, can ensure perfect transmission with no reflection or
energy loss at material discontinuities that otherwise generate reflection and
reduce the transmitted energy. The point actuators are modeled as acoustic
lumped elements, specifically monopole and dipole resonators, with an example
given using Helmholtz and membrane resonators.

The outcome of the analysis is a proposal for a new type of metamaterial that
provides total transmission with no loss in energy across an impedance
discontinuity. The metamaterial, or meta-atom since it is a sub-wavelength
lumped element, is a passive undamped combination of monopole and dipole
resonators in parallel. This demonstrates that constant intensity acoustic
propagation in a discretely disordered system does not require non-Hermitian
elements but may be achieved with the use of standard lossless passive lumped
elements acting in parallel.

The outline of the paper is as follows. Basic equations of constant intensity
acoustics are derived in Section II, where the two definitions of constant
intensity are distinguished. The acoustic elements necessary for constant
intensity propagation across discontinuities are introduced in Section III.
The main result of the paper is in Section IV where it is shown that a
combination of passive undamped monopole and dipole resonators achieve perfect
transmission across an impedance discontinuity with no scattering or energy
loss. Discussion and conclusions are provided in Section V.

##  II Constant intensity equations

###  II.1 Basic equations

Time harmonic motion of frequency $\omega$ is considered with the time-
dependent factor $\text{e}^{-\text{i}\omega t}$ omitted. The acoustic momentum
and constitutive equations are

$\displaystyle-\text{i}\omega\rho{\bf v}$ $\displaystyle=-\nabla p,$ (1a)
$\displaystyle-\text{i}\omega Cp$ $\displaystyle=-\nabla\cdot{\bf v},$ (1b)

where $p$, ${\bf v}$, $\rho$, and $C$ are the acoustic pressure, particle
velocity, density, and compressibility (inverse of bulk modulus),
respectively, all possibly functions of position ${\bf x}$. Eliminating ${\bf
v}$ yields an equation for the pressure only

$\nabla\cdot\big{(}\rho^{-1}\nabla p\big{)}+\omega^{2}Cp=0.$ (2)

The density and compressibility may be complex valued, corresponding to a
passive system if $\text{Im}\,\rho\geq 0$ and $\text{Im}\,C\geq 0$. The system
is active, or non-Hermitian, if $\text{Im}\,\rho<0$ and/or $\text{Im}\,C<0$,
requiring an external source of energy to maintain the steady state. For the
remainder of the paper we specialize to uni-dimensional acoustics ${\bf x}\to
x$, so that Eq. (2) is

$\big{(}\rho^{-1}p^{\prime}\big{)}^{\prime}+\omega^{2}Cp=0$ (3)

where $p^{\prime}=\text{d}p/\text{d}x$.

We consider $\rho$ and $C$ to be functions of position, which, as we will see
in Section IV, is critical in dealing with impedance discontinuities. We also
note that the fundamental paper in the field of constant intensity acoustics
[5] assumes constant $C$ with variable density $\rho(x)$, but uses the
incorrect equation $p^{\prime\prime}+\omega^{2}\rho C_{0}p=0$ (Eq. (4) of
Rivet2018 [5]).

In order to obtain constant intensity uni-directional wave motion we start
with the ansatz

$p(x)=p_{0}\,\text{e}^{\text{i}\omega\int_{0}^{x}s(x^{\prime})\text{d}x^{\prime}}$
(4)

for constant $p_{0}$, where $s(x)$ has dimensions of slowness. Equation (3)
becomes

$s^{\prime}-\frac{\rho^{\prime}}{\rho}s+\text{i}\omega\big{(}s^{2}-\rho
C\big{)}=0.$ (5)

If both the density and compressibility are constant, Eq. (5) has the solution

$s(x)=\sqrt{\rho C}=\text{constant},$ (6)

as expected for uni-directional non-dispersive wave propagation. Otherwise $s$
is a function of $x$, and $s\neq\sqrt{\rho C}$.

According to Eqs. (1a) and (4) the particle velocity is

$v=\frac{s}{\rho}p.$ (7)

The ratio $v/p$ is therefore the local acoustic admittance $y=s/\rho$, or
equivalently, we can identify $p/v$ as the local impedance $z=\rho/s$
$(=1/y)$. The acoustic impedance is $z=\rho c$ where $c=1/s$ is the local
phase speed. Equation (5) can be rewritten as an equation for the impedance

$z^{\prime}+\text{i}\omega\big{(}Cz^{2}-\rho\big{)}=0$ (8)

which shows that in regions where $\rho(x)/C(x)=$ constant that the impedance
$z$ (equivalently the admittance $y$) is constant and given by
$z=\sqrt{\rho/C}$.

The Riccati equations (5) and (8) provide relations for the possibly complex-
valued density and/or compressibility in terms of the variation in phase
derivative $s$ and impedance $z$, respectively. We require that $s$ remains
real, so setting $\rho=\rho_{1}+\text{i}\rho_{2}$ and $C=C_{1}+\text{i}C_{2}$
the real and imaginary parts of Eq. (5) imply

$\displaystyle
s^{\prime}+\omega(\rho_{1}C_{2}+\rho_{2}C_{1})-s(\log|\rho|)^{\prime}$
$\displaystyle=0,$ (9a) $\displaystyle
s^{2}-\rho_{1}C_{1}+\rho_{2}C_{2}+s\frac{\rho_{1}\rho_{2}}{\omega|\rho|^{2}}(\log\rho_{1}/\rho_{2})^{\prime}$
$\displaystyle=0.$ (9b)

###  II.2 Example: Total energy absorption

Assuming $\rho$ to be constant and real, Eq. (9) gives

$C=\frac{1}{\rho}\big{(}s^{2}-\text{i}\omega^{-1}s^{\prime}\big{)}.$ (10)

This complex-valued compressibility is passive or active depending as
$s^{\prime}\leq 0$ or $s^{\prime}>0$, respectively. The physical consequences
can be seen through the energy flux averaged over a period,

$F(x)=s(x)\frac{|p_{0}|^{2}}{2\rho}.$ (11)

Energy flux is constant if $s$ is also constant. A decreasing $s$ as a
function of $x$ leads to a diminishing energy flux, consistent with the
passive damping caused by $\text{Im}\,C>0$. Conversely, $F$ increases with $x$
if $s^{\prime}>0$, corresponding to an active non-Hermitian acoustic
compressibility.

Consider the following example for $s(x)$ with associated passive
compressibility with damping

$s=-\frac{x}{a}s_{0},\quad\Rightarrow\quad
C=C_{0}\Big{(}\frac{x^{2}}{a^{2}}+\frac{\text{i}}{\omega s_{0}a}\Big{)}$ (12)

where $C_{0}=s_{0}^{2}/\rho$. According to Eq. (11) the energy flux on either
side of $x=0$ is directed towards $x=0$ with $F(0)=0$. The compressibility of
Eq. (12), which is a symmetric function of $x$, causes waves from either
direction to be totally absorbed, with no backscatter. The material acts as an
acoustic black hole.

###  II.3 Constant amplitude or constant energy?

The above example shows that even though the pressure amplitude remains
constant the associated wave energy can decrease or increase. This suggests a
more reasonable definition of "constant intensity" that the energy flux is
constant even as the impedance changes. With that in mind the following ansatz
is more appropriate than Eq. (4):

$p(x)=p_{0}\,\sqrt{\frac{z}{z_{0}}}\,\text{e}^{\text{i}\omega\int_{0}^{x}s(x^{\prime})\text{d}x^{\prime}}$
(13)

for constant $p_{0}$, $z_{0}$ where from now on the impedance is defined
$z(x)\equiv\sqrt{\rho(x)/C(x)}$. This solution provides the same energy flux
in regions of different but constant $z$. A Riccati equation analogous to Eq.
(8) follows from Eq. (13)

$q^{\prime}+\text{i}\omega\big{(}\rho q^{2}-C\big{)}=0$ (14)

where $q=s/\rho+z^{\prime}/(2\text{i}\omega\rho z)$. We do not pursue this
equation further. We will however provide an exact solution to Eq. (13) in the
next Section for the important case of a point discontinuity in acoustic
properties.

Our focus on a point discontinuity in impedance as compared with a continuous
variation is because the latter requires a continuous variation in possibly
active material properties. Any discontinuity in impedance leads to scattering
(reflection) and loss of transmitted energy. We therefore consider what is
necessary to maintain the forward propagating wave at constant intensity
according to Eq. (13) in the presence of point changes in impedance. In order
to address this we will take a step back to the fundamental equations (1)
since Eq. (14) is based on smoothly varying impedance and is not suitable for
point discontinuities. First, we introduce the elements to achieve one-way
constant intensity propagation across the discontinuities.

##  III Monopole and dipole resonators

Monopole and dipole resonators have been the workhorses of acoustic
metamaterials: Helmholtz resonators (monopoles) provide the basis for negative
effective compressibility [12, 13, 14, 15, 16], while dipole membrane
resonators yield negative effective inertia [17, 14, 18, 16]. Together they
lead to the possibility of so-called doubly negative properties in acoustic
metamaterials; in fact the earliest notions in metamaterials are generally
agreed to revolve around concepts of negative properties, see for instance the
historical review of Carlin1964 [19]. Monopole and dipole resonators in
combination can provide total energy absorption [20], using a design that is
closely related to the present one proposed in the next Section [21]. Despite
the recent successful use of sub-wavelength acoustic lumped elements it should
be borne in mind that they have a century long history in the development of
analog acoustic control, particularly in acoustic wave filters [22, 23, 24].

Lumped parameter acoustic elements consisting of a Helmholtz resonator and a
sprung mass or membrane are shown in Figure 1. In preparation for dealing with
constant intensity across an interface we first consider each element situated
between acoustic fluids with properties $\rho_{-}$, $C_{-}$ on the left and
$\rho_{+}$, $C_{+}$ on the right. The membrane is placed so that it separates
the fluids. As suggested in the Appendix, the Helmholtz resonator fluid can be
taken as one of the two with the neck inserted in the respective fluid close
to the interface. Alternatively, the monopole element could be provided by a
Hybrid Membrane (HM) resonator [21] situated the same way as the HR in Figure
1, or by a pair of Decorated Membrane (DM) resonators [21] placed parallel to
the membrane resonator separating the fluids as in Fig. 1(a) of [21].

Figure 1: Reflection and transmission from a monopole (top) and a dipole
membrane resonator, both considered as sub-wavelength elements at the
interface between different acoustic fluids on the left and right. The
monopole shown is a Helmholtz resonator, but other types are possible, such as
the Hybrid Membrane (HM) resonator or a pair of Decorated Membrane (DM)
resonators [21].

With $p$ as acoustic pressure, $U=Sv$ as volumetric velocity, the incident,
reflected and transmitted quantities are related by the $U_{i}=p_{i}/Z_{-}$,
$U_{r}=-p_{r}/Z_{-}$, $U_{t}=p_{t}/Z_{+}$, where $Z=z/S$, with
$z=\sqrt{\rho/C}$ the acoustic impedance and $S$ the waveguide cross-sectional
area, Figure 1. The Helmholtz and membrane resonators are defined by
impedances $Z_{H}$ and $Z_{M}$ such that $U_{H}=p_{H}/Z_{H}$ and
$U_{M}=p_{M}/Z_{M}$, respectively. The systems are passive if $\Re\,Z_{H}\geq
0$, $\Re\,Z_{M}\geq 0$, they are otherwise active. Models for both types of
acoustic elements are discussed in the Appendix.

The conditions for the attached Helmholtz resonator transmission-reflection
problem are

$\displaystyle p_{i}+p_{r}$ $\displaystyle=p_{t}=p_{H},$ (15) $\displaystyle
U_{i}+U_{r}$ $\displaystyle=U_{t}+U_{H},$

while those for the in-line mass-spring system are

$\displaystyle p_{i}+p_{r}$ $\displaystyle=p_{t}+p_{M},$ (16) $\displaystyle
U_{i}+U_{r}$ $\displaystyle=U_{t}=U_{M}.$

Solving for the reflection and transmission coefficients, defined by
$p_{r}=Rp_{i}$, $p_{t}=Tp_{i}$, gives

$\displaystyle R_{H}$
$\displaystyle=\frac{\frac{1}{Z_{-}}-\frac{1}{Z_{+}}-\frac{1}{Z_{H}}}{\frac{1}{Z_{-}}+\frac{1}{Z_{+}}+\frac{1}{Z_{H}}},\
\ T_{H}=1+R_{H},$ (17a) $\displaystyle R_{M}$
$\displaystyle=\frac{Z_{M}+Z_{+}-Z_{-}}{Z_{M}+Z_{+}+Z_{-}},\ \
T_{M}=\frac{Z_{+}}{Z_{-}}\big{(}1-R_{M}\big{)}.$ (17b)

If $Z_{+}>Z_{-}$ a transmission coefficient of unity can be achieved with the
Helmholtz resonator by choosing
$\frac{1}{Z_{H}}=\frac{1}{Z_{-}}-\frac{1}{Z_{+}}$, yielding $R_{H}=0$ and
$T_{H}=1$. This passive but dissipative element achieves one way propagation
but at the expense of energy loss since a transmission coefficient of
magnitude $\sqrt{Z_{+}/Z_{-}}$ is necessary to maintain the same energy flux
in $x>0$ as for the incident wave in $x<0$ (see next section for details).

Similarly, one way propagation across the discontinuity with $Z_{+}<Z_{-}$ is
achieved with the passive membrane of impedance $Z_{M}=Z_{-}-Z_{+}$, which
gives $R_{M}=0$ and $T_{M}=\frac{Z_{+}}{Z_{-}}$. Again, this is less than the
required transmission amplitude $\sqrt{Z_{+}/Z_{-}}$ for maintaining the wave
energy, with the energy loss associated with the damping effect of the
membrane.

##  IV Propagation across impedance discontinuities

Here we show that constant intensity propagation through an impedance
discontinuity can be achieved with two passive undamped elements at the point
of discontinuity, or by a combination of one active element with a damped
passive element.

Figure 2: The combined monopole+dipole elements in parallel.

####  IV.0.1 Monopole and dipole strengths

We consider an acoustic waveguide with an impedance discontinuity at $x=0$,
i.e. $Z(\pm 0)=Z_{\pm}$. This could be due to different fluids in a waveguide
of constant cross-section, or a change in cross-sectional area with the same
fluid on either side. In the neighborhood of $x=0$ we therefore have
$Z(x)=Z_{-}+\Delta Z\,H(x)$ where $\Delta Z=Z_{+}-Z_{-}$ and $H$ is the
Heaviside step function. We consider the ansatz

$\displaystyle\rho(x)$ $\displaystyle=\rho_{0}(x)+\text{i}\alpha\,\delta(x),$
(18a) $\displaystyle C(x)$ $\displaystyle=C_{0}(x)+\text{i}\beta\,\delta(x),$
(18b)

where $\rho_{0}(x)=\rho_{-}+\Delta\rho\,H(x)$, $C_{0}(x)=C_{-}+\Delta
C\,H(x)$, $\delta=H^{\prime}$ is the delta function, and $\alpha$ and $\beta$
are constants. The concentrated density and compressibility of Eq. (18)
correspond to a combination of dipole and monopole resonators in parallel, as
exemplified in Fig. 2. A similar parallel combination of monopole and dipole
resonators was used [21] to achieve a quite different effect: total
absorption. Apart from the different application, we are here considering the
pair of elements positioned right at the discontinuity of impedance, whereas
Yang et al. [21] had no discontinuity.

Equations (1a) and (1b) now become

$\displaystyle\text{i}\omega\big{(}\rho_{0}(x)+\text{i}\alpha\,\delta(x)\big{)}v$
$\displaystyle=p^{\prime},$ (19a)
$\displaystyle\text{i}\omega\big{(}C_{0}(x)+\text{i}\beta\,\delta(x)\big{)}p$
$\displaystyle=v^{\prime}.$ (19b)

Both $p$ and $v$ are smooth functions of $x$ near the interface but they can
be discontinuous at $x=0$. To be specific, we assume they have distinct values
on either side of the interface: $p(0\pm)=p_{\pm}$ and $v(0\pm)=v_{\pm}$.
Equations (19) imply jump conditions connecting the values of $p$ and $v$
across the interface. The functions multiplying the Dirac delta functions in
Eqs. (19) are themselves discontinuous precisely at $x=0$, and therefore some
care needs to be taken in deriving the jump conditions in the usual manner by
integration. We use the identity
$\int\delta(x)v(x)\text{d}x=\frac{1}{2}(v_{+}+v_{-})$, which can be seen by
noting that the delta function is symmetric in $x$ and can be written in
various forms as a limit. For instance, taking $\delta(x)=\lim_{\epsilon\to
0}\frac{1}{\epsilon}\big{(}H(x+\frac{\epsilon}{2})-H(x-\frac{\epsilon}{2})\big{)}$
and integrating from $x=-\frac{\epsilon}{2}$ to $x=\frac{\epsilon}{2}$ gives
the identity. Note that this identity is not universally applicable but
depends upon the context, with some embarrassing results when used where it
should not be [25]. In summary, Eqs. (19) give

$\displaystyle\Delta p$ $\displaystyle=-\frac{\omega\alpha}{2}(v_{+}+v_{-}),$
(20a) $\displaystyle\Delta v$
$\displaystyle=-\frac{\omega\beta}{2}(p_{+}+p_{-}),$ (20b)

where $\Delta p=p_{+}-p_{-}$ and $\Delta v=v_{+}-v_{-}$ provide the required
monopole and dipole strengths.

The connection between the point scatterers of Eq. (18) and the membrane and
Helmholtz resonators can be found by first considering reflection and
transmission from the monopole scatterer, i.e. Eq. (18) with $\alpha=0$. In
this case $p$ is continuous at $x=0$, and a simple analysis gives

$Z_{H}=\frac{1}{\omega\beta S}.$ (21)

Conversely, reflection and transmission from the dipole scatterer, i.e. Eq.
(18) with $\beta=0$ has $v$ continuous at $x=0$ with

$Z_{M}=\frac{\omega\alpha}{S}.$ (22)

For wave incidence from $x<0$, we have

$p(x)=\begin{cases}e^{\text{i}k_{-}x}+Re^{-\text{i}k_{-}x},&x<0,\\\
Te^{\text{i}k_{+}x},&x>0,\end{cases}$ (23)

where $k_{\pm}=\omega/c_{\pm}$. From Eqs. (20) through (23) we find

$\displaystyle T$
$\displaystyle=\frac{2Z_{+}(Z_{H}-\frac{1}{4}Z_{M})}{Z_{M}Z_{H}+Z_{+}Z_{-}+(Z_{H}+\frac{1}{4}Z_{M})(Z_{+}+Z_{-})},$
(24a) $\displaystyle R$
$\displaystyle=\frac{Z_{M}Z_{H}-Z_{+}Z_{-}+(Z_{H}+\frac{1}{4}Z_{M})(Z_{+}-Z_{-})}{Z_{M}Z_{H}+Z_{+}Z_{-}+(Z_{H}+\frac{1}{4}Z_{M})(Z_{+}+Z_{-})}.$
(24b)

Setting $R=0$ requires that the impedances are related by

$Z_{M}=\frac{Z_{+}Z_{-}-Z_{H}\Delta Z}{Z_{H}+\frac{1}{4}\Delta Z}$ (25)

and the resulting transmission coefficient is

$T=\frac{Z_{+}}{Z_{-}}\Big{(}\frac{2Z_{-}-Z_{M}}{2Z_{+}+Z_{M}}\Big{)}.$ (26)

Our objective is to find a metamaterial that achieves a designated transmitted
wave amplitude. With that in mind we rewrite the previous equations to express
$Z_{M}$ and $Z_{H}$ in terms of $T$ and the impedances on either side of the
interface:

$\displaystyle Z_{M}$ $\displaystyle=\frac{2Z_{+}Z_{-}(1-T)}{Z_{+}+Z_{-}T},$
(27a) $\displaystyle Z_{H}$
$\displaystyle=\frac{Z_{+}Z_{-}(1+T)}{2(Z_{+}-Z_{-}T)}.$ (27b)

####  IV.0.2 Explicit expressions for the monopole and dipole impedances

The incident energy flux, averaged over a cycle, is
$\frac{1}{2}S\,\Re\,(pv^{*})$ for $x<0$, which is
$\frac{1}{2}\Re\,Z_{-}^{-1}$. The corresponding transmitted energy flux is
$\frac{1}{2}|T|^{2}\Re\,Z_{+}^{-1}$. Assuming real-valued impedances
$Z_{\pm}$, this means that in order to maintain the energy flux, or wave
energy, we need

$T=e^{\text{i}\phi}\,\sqrt{\frac{Z_{+}}{Z_{-}}}$ (28)

for arbitrary phase angle $\phi$. Equations (27) become

$\displaystyle Z_{M}$ $\displaystyle=\frac{-2\sqrt{Z_{+}Z_{-}}(\Delta
Z\cos\phi+\text{i}(Z_{+}+Z_{-})\sin\phi)}{Z_{+}+Z_{-}+2\sqrt{Z_{+}Z_{-}}\cos\phi},$
(29a) $\displaystyle\frac{1}{Z_{H}}$ $\displaystyle=\frac{\Delta
Z\cos\phi-\text{i}(Z_{+}+Z_{-})\sin\phi}{\frac{1}{2}\sqrt{Z_{+}Z_{-}}(Z_{+}+Z_{-}+2\sqrt{Z_{+}Z_{-}}\cos\phi)},$
(29b)

or after some simplification,

$\displaystyle Z_{M}$
$\displaystyle=-\text{i}2Q\,\sqrt{Z_{+}Z_{-}}\,\text{e}^{-\text{i}\theta},$
(30a) $\displaystyle{Z_{H}}$
$\displaystyle=(-\text{i}2Q)^{-1}\,\sqrt{Z_{+}Z_{-}}\,\text{e}^{-\text{i}\theta},$
(30b)

where

$Q=\sqrt{\frac{Z_{+}+Z_{-}-2\sqrt{Z_{+}Z_{-}}\cos\phi}{Z_{+}+Z_{-}+2\sqrt{Z_{+}Z_{-}}\cos\phi}},\
\ \tan\theta=\frac{Z_{+}-Z_{-}}{Z_{+}+Z_{-}}\cot\phi.$ (31)

The explicit expressions of Eq. (30), or the relations

$Z_{M}={-}{4Q^{2}}Z_{H},\ \ Z_{M}Z_{H}^{*}=Z_{M}^{*}Z_{H}=-Z_{+}Z_{-},$ (32)

indicate that if one of the resonators is passive and lossy, e.g.
Re$\,Z_{M}>0$, then the other is active with gain, Re$\,Z_{H}<0$. Since the
transmitted acoustic pressure has the same energy as that of the incident wave
and because there is no reflection, the gain and loss of the two elements must
cancel each other in order to conserve wave energy.

Both elements are passive if $\cos\phi=0$, in which case, taking $\sin\phi=\mp
1$, we have a unique pair of undamped passive solutions

$Z_{M}=\pm 2\text{i}\sqrt{Z_{+}Z_{-}},\quad Z_{H}=-\frac{Z_{M}}{4}.$ (33)

The elements are lossless by virtue of the fact that both have zero real part.
Furthermore, if one is mass-like $(\text{Im}\,Z<0)$ the other acts as a
stiffness $(\text{Im}\,Z>0)$.

####  IV.0.3 Example: transmission through an air-water interface

As an example we consider a water-air interface, a common acoustic boundary
but with a rather extreme impedance mismatch of approximately 3,644 to 1. We
assume that the interface is equipped with membrane and Helmholtz resonators
in parallel with impedances given by Eqs. (36) and (40)1. In order to satisfy
the inequality $Z_{M}^{*}Z_{H}<0$, from Eq. (32), the target frequency for
full transmission, $\omega=\omega_{0}$ must lie between the smaller and larger
of $\\{\omega_{M},\omega_{H}\\}$. To be specific, we assume
$\omega_{M}<\omega_{H}$ and therefore $\omega_{M}<\omega_{0}<\omega_{H}$.
Furthermore, for the sake of simplicity we assume that $S_{+}=S_{-}=S$, and so
$Z_{\pm}=z_{\pm}/S$. Equation (33) with $Z_{M}=-2\text{i}\sqrt{Z_{+}Z_{-}}$
and Eqs. (36) and (40)1 then imply

$\displaystyle\frac{\omega
m}{2S}\Big{(}1-\frac{\omega_{M}^{2}}{\omega^{2}}\Big{)}$
$\displaystyle=\frac{2S}{\omega
C_{b}V}\Big{(}1-\frac{\omega^{2}}{\omega^{2}_{H}}\Big{)}$
$\displaystyle=\sqrt{z_{+}z_{-}}\quad\text{for}\quad\omega=\omega_{0}.$ (34)

The material impedances $z_{\pm}$ are defined by the water-air interface, with
$\rho$ and $c$ for water and air as $1,000$ kg/m3, $1,500$ m/s and $1.2$
kg/m3, $343$ m/s, respectively. We then select the following parameters: the
HR bulb and neck fluids and hence the values of $C_{b}$ and $\rho_{n}$, the
waveguide cross-sectional area $S$, the membrane and Helmholtz resonance
frequencies $f_{M}$, $f_{H}$ and the frequency for total transmission
$f_{0}=\omega_{0}/(2\pi)$. Equation (IV.0.3) then implies values for the
membrane mass $m$ and Helmholtz resonator volume $V$, and Eq. (40)2 then
yields the ratio $A/l$ for the HR neck (see the Appendix).

For the example considered, the HR bulb and neck fluids are both air, the
full-transmission frequency is $f_{0}=170$ Hz, the resonators frequencies are
$f_{M}=100$ Hz, $f_{H}=200$ Hz, and the waveguide cross-section is circular of
radius $0.1$ m. We then find from Eq. (IV.0.3) that $m=2.23$ kg and $V=92.7$
cm3, and from Eq. (40)2 that $A/l=1.2\,\text{x}10^{-3}$ m. The latter
corresponds, for instance, to a cylindrical HR neck of radius $0.75$ cm and
length $l=14.2$ cm. These parameters ensure perfect transmission at the chosen
operating frequency, in this case $f_{0}=170$ Hz.

Figure 3: The transmitted and reflected energy fluxes, $E_{T}$ and $E_{R}$ of
Eq. (35), as a function of frequency. The dashed curve is the value of $E_{T}$
in the absence of the passive resonators: $E_{T}=1.1\,\text{x}10^{-3}$.

Figure 3 shows the fractional reflected and transmitted energy fluxes as a
function of frequency for these parameters, where

$E_{R}=|R|^{2},\quad E_{T}=\frac{Z_{-}}{Z_{+}}|T|^{2},$ (35)

with $E_{R}+E_{T}=1$. The inset in Fig. 3 indicates perfact transmission at
the selected frequency $f_{0}=170$ Hz. Significant transmission $(E_{T}\geq
0.5)$ occurs in a narrow bandwidth of about 4 Hz. However, $E_{T}$ is greater
than the ambient value $1.1\,\text{x}10^{-3}$ of an air-water interface for
every frequency over the broad bandwidth between the two resonance frequencies
$f_{M}$ and $f_{H}$. Figure 3 shows that the transmitted energy equals the
ambient value at both $f_{M}$ and $f_{H}$, and it lies below the ambient value
for frequencies less than $f_{M}$ and greater than $f_{H}$. The pair of
passive resonators therefore act as a broadband transmission amplification
device that achieves full transmission over a narrow range of frequencies
centered at $f_{0}$.

##  V Discussion and conclusions

The main result of this paper is that zero-scatter one-way propagation in the
presence of an impedance discontinuity can be achieved by purely passive
methods. There is no need to resort to more complicated active systems using
gain/loss mechanisms characteristic of non-Hermitian acoustics.

Two ways of looking at constant intensity acoustics for one dimensional wave
motion have been considered: constant pressure amplitude or constant energy
flux. Of the two, the constraint of constant energy is more sensible. This is
evident in the solution for transmission across a material discontinuity. A
change in impedance causes scattering, but introduction of the combined
monopole+dipole acoustic "meta-atom" of Eq. (29) results in total transmission
with no loss in energy. This solution contains an arbitrary phase $\phi$ which
"rotates" the solution in the space of passive/active metamaterials. When
$\cos\phi=0$ the metamaterial becomes passive and undamped. Otherwise, the two
elements form a gain/loss pair with one active and the other passive and
damped that together ensure that the transmitted wave has the same energy as
the incident one.

Transmission through the monopole+dipole system (29) or (33) is reciprocal in
that it provides transmission with full energy conservation and zero
reflection for incidence from either side of the impedance discontinuity. This
means, for instance, that perfect transmission through a finite slab or layer
of material can be achieved by placing the same monopole+dipole system at each
end.

In conclusion, the paper provides fundamental results relevant to sound wave
amplification and scattering cancellation. The proposed acoustic metamaterial
comprising a pair of tuned resonators, passive/passive or active/passive,
could enable a huge increase in acoustic communication efficiency across
disparate interfaces, as shown in the example for passive transmission
amplification between air and water.

###### Acknowledgements.

This work was supported by NSF EFRI award no. 1641078.

*

##  Appendix A Impedance models

###  A.1 Membrane or dipole impedance

The impedance $Z_{M}$ for a lossless membrane resonator follows from e.g.
Lee2009 [17, 16] as

$Z_{M}=\frac{-i\omega
m}{S^{2}}\Big{(}1-\frac{\omega_{M}^{2}}{\omega^{2}}\Big{)},\ \
\omega^{2}_{M}=\frac{\kappa}{m}$ (36)

This can be envisaged as a membrane separating the fluid on either side, and
subject to a pressure differential across it. The mass $m$ is the total mass,
and the stiffness $\kappa$ arises from the elasticity as it is stretched under
pressure.

A more precise model, due to Ingard1954 [26] and based on Morse1948 [27, p.
201], considers the dynamics of a membrane of wavenumber $k_{M}=\omega/c_{M}$,
radius $a$, yielding

$Z_{M}=\frac{i\omega m}{S^{2}}\frac{J_{0}(k_{M}a)}{J_{2}(k_{M}a)}.$ (37)

The resonant frequency of the membrane, $\omega_{M}$, can be calculated
analytically by considering free flexural vibration of circular plate clamped
all around the circumference. The general solution in polar coordinates is
[28] $W_{n}=\big{(}A_{n}J_{n}(k_{M}r)+C_{n}I_{n}(k_{M}r)\big{)}\cos n\theta$,
where $k_{M}^{4}=\rho h\omega^{2}/D$; $J_{n}$ and $I_{n}$ are the Bessel
functions of the first kind and the modified Bessel functions of the first
kind, respectively. Clamped boundary conditions $W=\frac{\partial W}{\partial
r}=0$ at $r=a$ lead to the eigenvalue problem
$J_{n}(k_{M}a)I_{n+1}(k_{M}a)+I_{n}(k_{M}a)J_{n+1}(k_{M}a)=0$. The natural
frequencies of the circular membrane plate can be calculated from the roots
$k_{M}a$. Here we are only interested in the fundamental mode, i.e. $(0,1)$
mode. Taking the number of nodal diameter as $n=0$ and only solving for the
first mode, we have $k_{M}a=3.1962206\cdots$. Among related models for the
acoustic impedance of internal mass-spring resonators [29, 30] and membrane
oscillators we note the explicit expression of BongardLissekMosig2010 [31]:

$Z_{M}=\frac{-i\omega
m}{S^{2}}\frac{J_{1}(k_{M}a)I_{0}(k_{M}a)+I_{1}(k_{M}a)J_{0}(k_{M}a)}{J_{1}(k_{M}a)I_{2}(k_{M}a)-I_{1}(k_{M}a)J_{2}(k_{M}a)}.$
(38)

###  A.2 Helmholtz resonator or monopole impedance

The Helmholtz resonator (HR) impedance is a classic result [32, 33] usually
considered in the context of a uniform ambient acoustic medium. The HR
comprises a volume of fluid - the bulb- which acts as a mechanical spring, and
a volume of fluid - the neck - acting as a mass. The present application
envisages the Helmholtz resonator located at an impedance discontinuity, Fig.
2. The placement of the neck can therefore be in either of the fluids, but is
perhaps simpler if the fluid in the neck is the same as that into which the
neck enters. In the example considered in §IV the fluid is air. The classical
HR is normally considered to contain a single acoustic fluid but here we
consider the possibility of different fluids in the bulb and neck. In order to
avoid mixing of the two acoustic fluids it would be necessary to introduce an
impermeable membrane with ignorable acoustic influence. That means a material
both light and compliant enough that it does not cause pressure or velocity
jumps, while separating the acoustic fluids. The reason we consider the
possibility of two fluids is to increase the potential range of the HR
parameters, which for the purposes considered here can be very demanding. At
the same time, the fluids should be such that the potential and kinetic
energies over one cycle are predominantly located in the bulb and neck fluids,
respectively.

With these considerations in mind, generalizing §10 of KinslerFrey [34] to the
two distinct fluids in the bulb and neck, we model the undamped HR as

$Z_{H}=\frac{1}{-\text{i}\omega C_{b}V}-\text{i}\omega\rho_{n}\frac{l}{A}$
(39)

where $V$ is the volume of the bulb acoustic fluid with compressibility
$C_{b}$, and $\rho_{n}$, $A$, $l$ are the neck fluid density, cross-sectional
area and length, respectively. The impedance is therefore

$Z_{H}=\frac{1}{-\text{i}\omega
C_{b}V}\Big{(}1-\frac{\omega^{2}}{\omega^{2}_{H}}\Big{)},\ \
\omega^{2}_{H}=\frac{A}{\rho_{n}C_{b}Vl}$ (40)

where the resonance frequency $\omega_{H}$ depends upon the bulb
compressibility $C_{b}$ and the neck fluid density $\rho_{n}$, along with the
geometric parameters $V$, $A$ and $l$.

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# A Stochastic Analysis of the Linguistic Provenance of English Place Names

Michael Dalvean

School of Computer Science and Software Engineering
University of Western Australia
<EMAIL_ADDRESS>

###### Abstract

In English place name analysis, meanings are often derived from the
resemblance of roots in place names to topographical features, proper names
and/or habitation terms in one of the languages that have had an influence on
English place names. The problem here is that it is sometimes difficult to
determine the base language to use to interpret the roots. The purpose of this
paper is to stochastically determine the resemblance between 18799 English
place names and 84687 place names from Ireland, Scotland, Wales, Denmark,
Norway, Sweden, France, Germany, the Netherlands and Ancient Rome. Each
English place name is ranked according to the extent to which it resembles
place names from the other countries, and this provides a basis for
determining the likely language to use to interpret the place name. A number
of observations can be made using the ranking provided. In particular, it is
found that ‘Harlington’ is the most archetypically English place name in the
English sample, and ‘Anna’ is the least. Furthermore, it is found that the
place names in the non-English datasets are most similar to Norwegian place
names and least similar to Welsh place names.

Word count: 9011

## A. Introduction

English place names are often analysed through the lenses of Germanic, Celtic
and Latin languages, and the starting point for analysis is what the place
name may mean in a given language. Highly technical linguistic, etymological
and philological methods are used to determine the meaning of place names by
technical experts with a knowledge of the history and traditions underlying
place name conventions. The problem is that different but equally plausible
explanations for the same place name can be derived using these methods. The
differences often come from different starting points in terms of the
underlying language used for the analysis. In many cases, there seems to be no
objective way to determine which language to use to analyse place names.

The purpose of this paper is to use objective methods to derive the language
origins of English place names. The approach is to use computational
linguistics to determine the influence of ten European languages on English
place names. In this way we can identify the probable language to use to begin
the etymological exercise or, alternatively, eliminate languages on the basis
of probable non-influence.

Without written records we may never know the actual original meaning of
words. However, we can determine the linguistic origin of place names with a
high degree of probability. By using the place names of other countries, we
can determine the extent to which the morphology of a given English place name
resembles the morphology of place names from other countries. In this way, we
can determine the likely linguistic origin of individual place names.

The method entails analysing English place names by comparing them to the
place names of ten other major European countries that have had a high level
of linguistic and cultural influence on England and the place names of
England. These countries are France, Germany, Ancient Rome, Norway, Sweden,
Denmark, the Netherlands, Scotland, Ireland and Wales. The idea is to rank
each individual English place name in terms of the extent to which it
resembles the place names of England as opposed to the place names of the
other ten European countries.

By distilling the languages of place names, we get an objective probability
that a particular language is the most likely source of the place names. By
assigning probabilities to several languages in this way, we can partial out
the relative contributions of each language on a given place name. In the case
of English place names, there are several candidate languages that can enter
into the provenance of a place name. A traditional explanation of the name is
often based on a local topographical feature and the resemblance of the
topographical feature in a given language to the place name. The etymology of
‘London’, for example, has been linked to its location on the Thames and words
connoting water. Gigot, for example, links the ‘Burgundo-Gothic’ root ‘lohna’,
meaning ‘puddle’ or ‘pond’ with ‘London’ on the basis that this root connotes
the Thames [1] cited in [2]. The problem with this approach is that elements
of a place name can be linked with numerous topographical words in other
languages. Thus, ‘Londonthorpe’ in Lincolnshire is described as being derived
from the Old Scandinavian lundr [grove] + thorp [outlying settlement] [3].
Thus, we have two suggestions for the origin of ‘London’ from two different
languages using two different topographical features. The selection of the
association between a topographical term in a given language and a
topographical feature associated with the relevant place name is a rational
basis upon which to determine the meaning of a place name. However, if we do
not know the linguistic provenance of a place name, we will not know what
language with which to begin the analysis.

The approach taken in this paper is that the first step in place name analysis
is to determine the linguistic provenance of place names. Linguistic
provenance in this study is determined by looking at the similarity of the
place names of England to the place names of countries where the languages
spoken are likely to have influenced the formation and/or the development of
English place names. For a given place name and for a given comparison
country, we generate a score based on the extent to which the English place
name resembles the place names from the other country. If the score indicates
that the place name is more like other English place names than like the place
names of the other country, then we can conclude that the other language did
not have a significant influence on that place name. However, if the score
indicates that the place name is like the place names of the other country, we
can conclude that the country did have an influence on the place name and we
would direct our etymological investigation accordingly.

It should be stressed that the search for the meaning of place names is not
the purpose of this paper. This paper merely provides a means for determining
a starting point for the search for meaning in that it provides a likely
language basis for a place name.

One of the benefits of determining the likely language origins of English
place names is that it enables us to rank English place names in terms of
their English or ‘Anglo-Saxon’ content, and this enables us to identify the
‘archetypal’ English place name. We will see that ‘Harlington’ is the most
archetypal English place name, while ‘Anna’ is the least. We will also be able
to determine the relative linguistic difference between English place names
and the place names of other countries. In particular, we will see that the
difference between English place names and Scottish place names is the
smallest in the cohort of ten languages, whereas the difference between
English and Ancient Roman place names is the largest. Finally, we will see
that there are 475 English place names that defy classification as English.

Please note that there are some differences in the terminology used in
computational linguistics and formal linguistics. The terminology of
computational linguistics is used in this paper.

## B. Previous Work

The standard analysis of English place names seeks to derive an understanding
of English culture and settlement patters based on the interpretation of the
meaning of place names [4]. The method

> …focuses largely on empirical, at times empiricist, enquiry, working through
> extant historical records in order to chart the history of name spellings,
> and infer name origins from established philological paradigms, drawing
> where possible on comparative material from related disciplines such as
> history and archaeology. [4, p29]

There is a great deal of emphasis on the linguistic provenance of place names
in the analysis because the interpretation of the meaning of the place name
will depend on the language basis of the place name. For example, Coates [2]
which looks at the possible roots of the place name ‘London’ together with
explanations as to how phonological influences over time in Britain may have
affected the morphology of the name. Coates holds that ‘London’ is based on a
river name ‘plowonidon’, which is made up of two roots from Old European, an
early Indo-European language now largely known only from hydronyms.

There is another vein of place name research in which the provenance of the
place name is not at issue, although the meaning is. An example is Hough [5],
which examines the use of Scottish place names as an aid in the interpretation
of Old English legal history. Similarly, O’Grady [6] uses place names and
archaeology to examine the nature of court locations in medieval Scotland.
Cuthbertson [7] uses Scottish place names to examine Scots lexicography. For
example, the topographical evidence from Scottish place names is that ‘doors‘
meant ‘passage‘ in Scots.

In yet another vein of place name research, neither the provenance nor the
meaning are at issue, but the patterns of types of place names are used to
shed light on other cultural and historical phenomena. For example, Fekete [8]
analyses the Scandinavian and Anglo-Saxon elements of English place names and
finds that in the Danelaw area in the north-east of England, there is evidence
of ‘code-mixing’ such that many place names have both Scandinavian and Anglo-
Saxon elements. Fekete’s study is an excellent example of pattern analysis in
that the computational process involves looking for specific patterns such as
the Germanic habitation elements ‘-ton’, ‘-by’, and ‘-thorpe’.

A recent innovation in place name research has been the use of machine
learning. In this approach, instead of looking for specific patterns, the
machine learning algorithm trawls through the data in an attempt to derive
patterns, and the analysis proceeds using the patterns derived thereby. Much
of the previous machine learning work on place names has been associated with
information retrieval and database applications. Marchardo et al [9], for
example, use computational methods to extract geographical names and terms
from web documents. Here, the issue is that many words can be names of
entities other than places, and an algorithm designed to extract place names
must be trained to identify the word as denoting a place name. Similarly,
Berragan et al [10] use computational methods to extract geographical place
names from geographical texts. Hu et al [11] use rules, gazetteers and deep
leaning to extract place names from micro blogs. A study that combines text
analysis with machine learning is Bensalem [12], which details an algorithm
for disambiguating identical place names by determining the likely reference
using contextual data in the text from which it is drawn.

The current study brings together aspects of the standard approach to place
name analysis as well as machine learning aspects. The focus of the current
study is broadly similar to the standard approach to place name analysis in
that it seeks to determine the language origin of place names. In contrast to
the standard approach, however, the current study uses machine learning to
determine the provenance of place names. It is beyond the scope of the current
paper to attempt to analyse the place name using methods such as those used by
Gigot [1], Mills [3] and Coates [2]. The application of etymological,
linguistic and philological expertise is the next step after determining the
language provenance of the place name.

## C. A Stochastic Approach to Place Name Analysis

The current study takes a stochastic approach to place name analysis. It seeks
to determine the language basis of English place names, but not the meaning.

The most significant difference in the approach of the current study and that
of the more traditional approaches is that the current study uses the place
names of other countries as benchmarks for ‘English’ morphological structure.
The structure of the place names of England is compared with the structure of
the place names of ten other European countries. The commonalities in the
English place names are derived from this comparison. For example, the element
‘-ton’ comes from Anglo Saxon ‘-tun’, meaning ‘enclosed space. It is the most
common element in English place names. The algorithm in this study does not
specifically look for the element ‘-ton’, which is the approach of the more
standard rule-based ‘molecular’ approaches to place name analysis. Instead,
the current study compares the ‘atomic’ letter placement of the sample of
English place names with the letter placement of the place names of other
countries. It transpires that, in comparison to the other place names, English
place names have a structure such that where the last letter = ‘n’, the second
last letter = ‘o’, and the third last letter = ‘t’, there is a high
probability that the place name is English. Importantly, each of these
variables is independent of the other. Thus, in the element ‘-tan’, the
presence of the ‘t’ in the third last place and the ‘n’ in the last place has
the same effect on the algorithm’s assessment of the place name as the ‘t’ and
the ‘n’ in a place name ending in ‘ton’. However, the presence of the ‘a’ in
the ‘-tan’ element has a different effect in comparison to the ‘o’ in ‘-ton’.
The different weightings for the presence/absence of particular letters in
particular positions in the place names are derived by comparison with the
place names of England to the place names of other countries. As we will see
below, nine of the ten most ‘archetypal’ Anglo-Saxon place names have the
element ‘-ton’, and the fact that this element was identified as an important
indicator of Anglo-Saxon origin without any rule-based search indicates the
efficacy of the stochastic ‘atomic’ approach.

A benefit of this approach is that the training algorithm can search for
patterns that are not known in advance. The deterministic rule-based search
for patterns is efficient, but it is not as efficient as a system that can
both derive the same set of rules that the researcher knows as well as detect
rules that the researcher does not know. It is this process inherent in the
machine learning methodology that the current study benefits from.

The focus on individual letter placement as a method of breaking down place
names into the variables used in the analysis is derived from ‘Benford’s Law’
[13]. This law applies to the distribution of single digit numerals in
particular places of multi digit numerals. For a given distribution of multi-
digit numerals, the rate of occurrence of the numeral ‘1’ in the first
position will be 0.301, that of the numeral ‘2’ will be 0.176, with a
decreasing rate up to the digit ‘9’ which will occur in the first place of
multi-digit numerals at a rate of 0.041. The law states that the probability
of a digit ‘d’ being in first place in a sample of multi-digit numerals is

$log_{10}(1+1/d).$

This concept was applied to letters in words in Yan et al [14], in which it
was found that the most common first letter in the words of nine works of
English literature by English and American authors’ novels was ‘t’, which
occurred at an average rate of 16.26%. The second most common first letter was
‘a’, which occurred at a rate of 11.24%. It was found in that study that when
the rates of occurrence of each of the 26 letters in the first letter of words
were ranked from highest to lowest, the decay followed a similar declining
rate to numerals in Benford’s law, with ‘x’ having the lowest rate of
occurrence in first place.

The current study extends the idea of the importance of the position of
letters by creating a set of variables for place names based on letter
placement. The idea here is that it is likely that there will be differences
in the structure of place names of different countries. We will see below, for
example, that there is a significant difference in the place names of Ancient
Rome as opposed to England in that the former are highly statistically
significantly likely to end in ‘a’ or ‘i’ in comparison to the latter.
Furthermore, this study extends the analysis to letters in the second, third
and fourth letters as well as the last, second last, third last and fourth
last letters.

Thus, the first set of variables consists of 26 binary variables where the
value = 1 if the fist letter is a…z, and = 0 if the letter is not present. The
second, third and fourth sets of variables are similar but are based on the
second, third and fourth letter of the place name respectively. Thus, there
are four sets of binary variables based on the letters in the first four
positions in place names = 4 * 26 = 104. A similar set of variables is created
for the four last letters of place names.

Another six binary variables were created indicating the presence/absence
anywhere in the place name of individual vowels (A,E,I,O U and Y). Six rate
variables were created by dividing each of these vowel binaries by the length
of the name. One variable was created by taking the total number of vowels in
each name and dividing by the length of the name. Twenty variables were
created by creating binaries for the presence/absence anywhere in the name of
each of the 20 consonants, and another 20 variables were created by dividing
each of these by the length of the name. Name length and entropy (bits per
letter) were included in the variables.

In this way, each place name is broken into 263 variables, most of which are
based on individual letters and letter placement.

Before we continue, it is worthwhile addressing an issue that may be
considered as problematic in the methodology: place names have changed since
their original inception, and therefore using the modern versions of place
names will lead to invalid results. This objection can be addressed by
considering the stochastic nature of the analysis. The modern name ‘Birkby’
has undergone several changes since it was first recorded in 1086 as ‘Bretebi’
[15]. The first letter has remained, and the ‘bi’ has transformed into ‘by’.
The ‘r’ has moved one place forwards and the vowel ‘e’ has transformed into
‘i’. The ‘t’ has become ‘k’. The ‘e’ before the ‘by’ element has gone, thereby
shortening the name by one letter. The ratio of vowels (A,E,I,O,U and Y) to
consonants has fallen from 3:4 to 2:4. Let us first consider the ‘by’ element
in the modern name. In the sample of 18799 English place names used in this
analysis, there are no place names ending in ‘bi’, and there are 391 that end
with ‘by’. We know that the ‘by’ element is a Scandinavian settlement term.
There is clearly a pattern in language change in that the ‘bi’ has gone from
English names and has been replaced with ‘by’. It might be suggested that the
‘bi’ could have transformed in some cases to ‘be’. However, inspection of the
data shows that 183 place names end in ‘be’, 182 of which are due to the
element ‘-combe’, which has a different etymology. In short, there has been a
systematic change in the form of ‘bi’ to by. However, we cannot be as definite
about some of the other changes that have occurred. Consider the change of ‘t’
to ‘k’ in the middle of the original name. Has this shift been systematic?
Without further analysis, we will not know. However, let us assume that the
change has not been systematic and that in half of the English place names
with ‘t’ in the middle of the name the ‘t’ has changed to ‘k’ while the
remaining half have stayed the same. The algorithm will detect a pattern for
English place names such that a ‘t’ or a ‘k’ in the 4th position of the name
is associated with English place names. Thus, whether the change is systematic
or partial, the pattern is likely to be detected because there are binary
variables indicating the presence/absence of individual letters in specific
positions in the place name. Similar arguments could be used to explain why
the other changes in ‘Bretebi’ are not likely to result in the inability of
the algorithm to identify ‘Birkby’ as ‘English’. The vowel and consonant
variables, as well as the length and entropy variables are included so that
even if the changes in language and orthographic conventions are not
systematic, there is the possibility of detecting the relevant signals in the
data. We will address this issue empirically below when we examine the
algorithm’s ability to distinguish between place names of Old Norse origin and
Anglo-Saxon Origin. We will see that despite the fact that modern spellings
are used to train the algorithm, the algorithm is able to distinguish between
the two classes.

## D. Data

### D.1. Data Sources

Where possible, the place names used in this study were accessed from the
official repositories of place name data in the respective countries. However,
in some cases this was not possible, so alternative sources were used. Sources
for the initial place names were as follow:

-England, Scotland, Wales: Index of Place Names in Great Britain (September 2019) [16]

-France: Liste des communes au 1er janvier 2022 [17]

-Germany: Gemeinden in Deutschland nach Fläche, Bevölkerung und Postleitzahl am 30.09.2021 [18]

-Denmark Autoriserede stednavne i Danmark [19]

-Sweden, Norway, Ireland: Country Files (GNS) [20]

-Ancient Rome: Pleiades Ancient Names Database [21]

The US GNS data was used for Sweden, Ireland and Norway because, although
individual place name data is available online for these countries, it was not
possible to download the data in toto. In the case of the Danish data, only
entries labeled as ‘By’ = Town, and ‘Bydel’ = District were included because
many of the names were for features of the landscape such as tunnels and ports
rather than place names.

### D.2. Data Processing and Cleaning

Only single names were used in this study. Therefore all names with more than
one word, such as ‘West London’, or with hyphens (‘Aix-en-Provence’) were
removed. The reasoning here is that the simple structure is more likely to
carry a fundamental signal than a compound structure.

All letters were transformed into the most close approximation to the 26 Latin
characters of the English alphabet. The reason for this is that there are
certain characters that would provide a signal that could not be applied to
all names. The umlaut, for example, occurs predominantly in the Germanic
samples and therefore would provide a biased signal. Letters with diacritics
were transformed to their closest Latin characters using Python’s unidecode
function. Thus, characters such as ‘ø’ and ‘æ’ were changed to ‘o’ and ‘ae’
respectively. The German ‘ß’ was changed to ‘ss’ and the French ‘œ’ was
changed to ‘oe’.

A search for repetitions of names within each country was made and repeated
names were removed. Each country file therefore represents a list of unique
names for the country.

A second removal process involved removing names that were repeated across
countries. The reason for this is that if the same name occurs in two
countries, there is no ground truth as to its most likely origin.

Sample sizes are reported in Table 1.

Table 1: Place Name Sample Sizes Country | Sample Size
---|---
England | 18799
Denmark | 5493
Norway | 5056
Sweden | 20794
Ireland | 9928
Scotland | 4203
Wales | 2814
Ancient Rome | 1688
Germany | 8699
France | 18175
Netherlands | 7837

## E. Modeling

In initial experiments it was found that the most accurate classifiers were
one-to-one binary classifiers for each of the England-Other pairs. Thus, ten
binary classifiers were created using the England sample as the target.

Ten datasets were created using the English names as the ‘cases’ (binary = 1),
and each of the other name samples as ‘controls’ (binary = 0). This led to
significantly imbalanced datasets. The case of the English-Roman data set was
the most extreme, with more than ten times the number of English names than
Ancient Roman names. The solution for this problem was to use a variation on
Synthetic Minority Oversampling Technique (SMOTE). The procedure was
implemented using the Python imbalanced-learn library. The basic SMOTE
methodology is to synthesise cases in the minority sample (‘oversampling’)
using linear interpolation to increase the minority class. The SMOTE-ENN
(Synthetic Minority Oversampling Technique-Edited Nearest Neighbor)
methodology extends this by removing from both classes any case that is
misclassified by the majority class of its three nearest neighbours [22]. This
method has been shown to be efficient in modeling imbalanced datasets. It
should be stressed that the SMOTE-ENN procedure are only applied to the
training procedure: the test results were generated using the held out folds
at each of the ten stages of the 10-fold procedure. Scores for each name are
those generated where the name was in the held out fold. The test for the
efficacy of the SMOTE-ENN procedure is the classification accuracy of the held
out samples. As we will see below, the classification accuracy for all
classifiers is high.

The induction method chosen was random forest classifier. This algorithm has
been found to work well with the SMOTE-ENN procedure [23]. The implementation
in the current study uses the random forest from Python’s scikit-learn library
with n_estimators=100, max_depth=None, min_samples_split=5. Accuracy metrics
(cut point = .5) for each classifier are reported in the results section
below.

## F. Results

### F.1. Accuracy

Table 2 shows the accuracy metrics of the classifiers. The average accuracy of
the classifiers is 92%. The lowest score is for Scotland (83%) and the highest
for Ancient Rome (98%). This level of accuracy indicates that the classifiers
are all able to efficiently distinguish between the English and non-English
place names using the variables derived from the place names.

Table 2: Classifier Accuracies | Accuracy | Sensitivity | Specificity
---|---|---|---
England-Denmark | 0.94 | 0.94 | 0.92
England-Sweden | 0.94 | 0.92 | 0.96
England-Norway | 0.94 | 0.94 | 0.94
England-Ireland | 0.90 | 0.92 | 0.85
England-Scotland | 0.82 | 0.86 | 0.65
England-Wales | 0.92 | 0.96 | 0.66
England-France | 0.91 | 0.88 | 0.93
England-Rome | 0.98 | 0.99 | 0.88
England-Germany | 0.92 | 0.93 | 0.90
England-Netherlands | 0.91 | 0.92 | 0.91
Average: | 0.92 | 0.93 | 0.86

Inspection of the data indicates that the relatively low accuracy for Scotland
is due to the low specificity of 65%. The specificity is low because there is
a large number of place names that are ‘English’ in style and form. This is
also the case for Wales, which explains the low specificity of 67%.

A further test of accuracy is to see the rate correct classification of
English place names by the classifiers. The starting point for this analysis
is to calculate the average score for each place name across the ten
classifiers. For example, the place name ‘Laira’ scores the following on each
classifier: Denmark - 0.56, Norway - 0.13, Sweden \- 0.09, France - 0.05,
Germany - 0.38, Netherlands - 0.15, Rome - 0.05, Ireland - 0.10, Scotland -
0.03, Wales - 0.45. The average of all these scores is 0.20. Using the
standard machine learning cut point of 0.5, we would classify this name as
non-English as it scores below 0.5. Applying the ‘ensemble of classifiers’ to
all 18799 English place names in the same way, the accuracy is 97%. However,
we will see that some of the misclassified names are misclassified because
they have a highly non-English structure. Thus, it is not necessarily that the
classifiers cannot detect the English-Other content of the misclassified
names, but that some of the names themselves are not ‘English’.

### F.2. Accuracy Test Using Old Norse and Old English Place Names

In order to confirm the ability of the method to correctly classify place
names, a test was conducted using samples of place names deriving from Old
English and Old Norse/Scandinavian extracted from the Key to English Place-
Names at the Institute for Name-Studies at the University of Nottingham [24].
To generate the Old English sample, a search for names based on the following
languages was made: Old English, Anglian, Kentish, Mercian, Northumbrian
(incl. Old N), West-Saxon. This yielded 12507 names (including duplicates). A
search was made in the Derivation column for the term ‘Old English’, and those
names that did not have the term were excluded, reducing the sample to 12203.
Any name with the text ‘Norse’ or ‘Scand’ in the Derivation column was
removed, giving a sample of 11301. Names of more than one word, hyphenated
names and names for which multiple versions were given, as designated by a
forward slash were removed, and removal of duplicates resulted in final sample
of 6485.

The Old Norse/Scandinavian sample was generated by doing a search for place
names derived from the following languages: Old Norse, Old Danish, Old East
Scandinavian, Old Norwegian, Old West Scandinavian. This generated a sample of
1719 place names. A search for ‘English’, ‘Saxon’, ‘Mercian’, ‘Northumbrian’,
‘Kentish’ and ‘Anglian’ was made in the Derivation column and, if present, the
place name was removed. This reduced the sample to 694. Finally, names of more
than one word, hyphenated names and names for which multiple versions were
given, as designated by a forward slash, were removed, yielding a final sample
of 541. Removal of duplicates resulted in a sample of 494. Average scores on
the Scandinavian classifiers for the Old English and Old Norse samples are
shown in Table 3.

Table 3: OE and ON Place Name Scores on Scandinavian and English Classifiers | Eng-Den | Eng-Swe | Eng-Nor | Ave’ Scand’ | Eng-Other
---|---|---|---|---|---
OE Av (n=6485) | 0.907 | 0.921 | 0.892 | 0.906 | 0.884
ON A(n=494) | 0.799 | 0.879 | 0.752 | 0.810 | 0.851
T-test | <.001 | <.001 | <.001 | <.001 | <.001
Mann-Whitney | <.001 | <.001 | <.001 | <.001 | <.001

Scores for the three Scandinavian classifiers (Eng-Den, Eng-Swe and Eng-Nor)
are averaged to give a Scandinavian average. The Eng-Other score shows the
scores for the ensemble (average) of ten classifiers, including the three
Scandinavian classifiers. T-tests (two tailed) were conducted using Welch’s
test for the Eng-Den, Eng-Swe, Eng-Nor and Ave’ Scand data because the OE and
ON data are from independent samples and the n and variances are unequal.
Results for the Mann-Whitney test are also provided as the data are not
normally distributed. For the Eng-Other OE and ON data, an independent two
tailed t-test was done because the variances are equal. As expected, the
scores for the ON place names are significantly (p<.001) lower than the scores
for the OE place names on all three Scandinavian classifiers as well as the
Eng-Other ensemble on both the T-test and the Mann-Whitney test. This
indicates that the classifiers are efficient in distinguishing between place
names of ON and OE origin. This is an interesting finding given that the
classifiers were trained using modern English and Scandinavian place names.

## G. Discussion

The classification accuracy for the ensemble of classifiers of English place
names of 97% (cutpoint = .5) indicates that the ensemble classifier is
efficient. Furthermore, all ten individual classifiers are efficient, as
indicated by the average accuracy of 92%. These results mean that the actual
scores generated for each place name are likely to be a good indication of the
English-Other content of the place name. The higher this score, the higher the
English content. We can therefore rank each place name by the extent to which
it represents an English (Anglo-Saxon) place name as opposed to a place name
with content from one or more languages from the other ten countries. Table 4
shows the ten top scoring English place names. For comparison, the ten lowest
scoring English place names are also shown.

Table 4: Ten Highest/Lowest Scoring Place Names Place name | England-Other Score
---|---
Harlington | 0.999
Widdington | 0.998
Colworth | 0.998
Beckington | 0.998
Didlington | 0.998
Toddington | 0.998
Lowthorpe | 0.998
Tiddington | 0.998
Bedlington | 0.998
Ridlington | 0.998
Danum | 0.231
Menna | 0.228
Moira | 0.227
Arun | 0.224
Belgravia | 0.223
Lamanva | 0.218
Laira | 0.200
Alma | 0.179
Lana | 0.166
Anna | 0.157

The place name with the highest English content is ‘Harlington’, which scores
0.999 on the English-Other scale. The place name with the lowest English-Other
score is ‘Anna’, which scores 0.157 . Table 5 breaks down the scores for these
place names in relation to all ten classifiers as well as the ensemble
English-Other score.

Table 5: Scores for ‘Harlington’ and ‘Anna’ | Harlington | Anna
---|---|---
Classifier | Score | Score
England-Denmark | 1.000 | 0.436
England-Sweden | 0.994 | 0.001
England-Norway | 1.000 | 0.038
England-France | 1.000 | 0.043
England-Germany | 1.000 | 0.238
England-Netherlands | 1.000 | 0.311
England-Ancient Rome | 1.00 | 0.047
England-Ireland | 1.000 | 0.061
England-Scotland | 0.998 | 0.148
England-Wales | 1.000 | 0.251
Average (England-Other) | 0.999 | 0.157

It should be noted that the scores for all ten top scoring English names are
effectively equal to 1 on the English-Other score. The ranking is based on
five decimal places, but only three are shown. Thus, these are all archetypal
‘English’ names in relation to the ten comparison datasets.

Of the ‘non-English’ names,‘Anna’, scores lowest, having a 15.7% probability
of being English. If we look at the individual classifier scores, we can see
that Anna’s lowest score is on the England-Sweden classifier. The score of
0.001 indicates that ‘Anna’ has no statistical resemblance to the general
sample of English place names but, instead, has a complete statistical
resemblance to general Swedish place names. The highest resemblance of ‘Anna’
to an English name occurs with the England-Denmark classifier. The score of
0.436 indicates that it has a 43.6% probability of being English in comparison
to Danish place names.

The ensemble score is useful in itself to identify the English content in
place names. However, the individual classifiers can be used to individually
distil the underlying contribution of each of the other ten languages entering
into the etymology of English place names. If we rank the place names by score
on the classifier of a given country, we get a ranking of the relative
resemblance of each English place name in relation to the place names of the
given country. For example, if we rank the place names on the basis of the
England-Denmark classifier, the English place name with the highest Danish
resemblance (lowest England-Denmark resemblance) is Sedrup, which scores 0.01
on the England-Denmark classifier. At the other end of the scale, there are
2749 English place names which score 1 on the England-Denmark classifier,
including eight of the ten archetypal English place names in Table 4, the
exceptions being Toddington and Bedlington which score 0.990 and 0.993
respectively.

The foregoing analysis shows that it is possible to distil the ‘English’
content from place names. The implication of this is that most place names are
composites of linguistic components from cultures that have had input into
place names in England. This is not a controversial proposition; the
discussion of place names is characterised by historical accounts of how place
names have been created from various linguistic influences. However, the above
analysis is unique in that it provides a method of measuring the input of
various languages in the creation of place names without specifically
identifying features to search for, such as the elements ‘-by’, ‘-ton’ and
‘-dale’. The benefit of this is that the classifier can find both structures
that are known and those that are not.

In the following section we will see how having a fine-grained place name
classifier can be used to address a number of issues in the place name
research.

### G.1. Resemblance of English to Other Place Names

The scores for place names on each of the ten individual classifiers give us
an insight into which of the ten countries’ place names the English place
names are most similar to. Table 6 shows the average scores for the 18799
English place names for each of the classifiers in the ‘Mean Score’ column
ordered by ascending accuracy.

Table 6: Mean Scores of English Place Names on Ten Classifiers Classifier | Mean Score
---|---
England-Scotland | 0.737
England-Ireland | 0.810
England-France | 0.815
England-Netherlands | 0.837
England-Germany | 0.844
England-Sweden | 0.848
England-Wales | 0.850
England-Denmark | 0.877
England-Norway | 0.889
England-Rome | 0.949

If we consider the Mean Score for the England-Scotland classifier, we can see
that the average score for English place names using this classifier is 0.737.
At the other end of the spectrum, on the last line of the table, the average
score on the England-Rome classifier is 0.949. What this indicates is that the
England-Scotland classifier is less efficient than the England-Rome
classifier. The reason for this is that there is a greater similarity between
Scottish and English place names than there is between Ancient Roman and
English place names. The classifier trained on Scottish and English place
names has a more difficult task distinguishing between English and Scottish
names because they are more similar, whereas the classifier trained on the
English and Roman place names is more efficient because there is a great
difference between the Roman and English place names.

What we can determine from Table 6 above is that there is a declining
similarity between English place names and those of the other ten countries as
we go down the table. The place names of countries used to train classifiers
in the top of the table are more similar to English place names than those
used to train classifiers in the lower segment of the table. In short, the
classifier accuracies are proxies for place name similarity. This gives us a
good insight into the influences on English place names. For example, we can
tell that the similarity of English place names to those of Scotland, Ireland,
France, the Netherlands and Germany is greater than the similarity to any of
the Scandinavian place names. This is an interesting observation given the
significant inflow of Scandinavian settlers in England during the period in
which Anglo-Saxon place names were being applied to English localities.
However, the direction of causation may be difficult to establish. Consider
the case of Scotland: Scots is the Scottish descendant of Old-English, and as
such we might expect that there would be significant similarities between
Scottish and English place names. Therefore, we cannot make the claim that
Scottish influenced English place names. Similarly, the resemblance of English
place names to Irish place names may be due to the adoption of English names
in Irish localities due to the political, military and economic influence of
England on Ireland. In the case of France, the Netherlands and Germany, there
is a good historical basis for claiming that the direction of causation is
from these continental countries to England due to known historical migration
patterns. Importantly, these migration patterns were permanent, in that the
settlers remained in England, unlike the situation with Roman settlers who
returned to Rome in the early fifth century. This may explain the relative
lack of influence of Roman place names on English place names despite four
centuries of occupation. Finally, the low level of similarity between English
and Welsh place names is an interesting observation given the close proximity
of England and Wales. We can speculate that the reason for this is that there
was no systematic Anglo-Saxon settlement in Wales after the departure of the
Romans from Wales in the fifth century, and the subsequent political influence
likely to affect Welsh place names came from the the French-speaking Norman
conquerors of England who turned their attention to Wales in the latter
decades of the eleventh century. Thus, there was little opportunity for Anglo-
Saxon place name conventions to be applied to Wales.

### G.2. Broader Language Influences

So far we have focused on English place names. However, the scores on the ten
country classifiers enable us to make some observations about the broader
language influences in Europe. We have scores for English place names from ten
classifiers. Each vector of 18799 scores represents the similarity between the
place names of an individual country and the place names of England. If we
look at the correlation between the scores generated by the classifiers of two
countries, we get a measure of how similar the naming conventions are between
the two countries. That is, even though the scores are for English place
names, the differences between scores generated by different classifiers are
due to the interpretation of each classifier of the English place name, and
these differences are due to the classifiers being trained on the place names
of their own country. Correlations between the scores of countries shows how
similar their own place names are to each other.

Table 7 shows the inter-correlations between the scores for all ten
classifiers. The highest score in each column is highlighted in bold. Note
that all correlations are significant at the p<.01 level.

Table 7: Inter-correlations between classifier scores | Den | Nor | Swe | Ire | Scot | Wal | Fra | Ger | Net | Rom
---|---|---|---|---|---|---|---|---|---|---
Den |  | 0.76 | 0.75 | 0.32 | 0.37 | 0.18 | 0.45 | 0.46 | 0.61 | 0.34
Nor | 0.76 |  | 0.84 | 0.39 | 0.43 | 0.33 | 0.5 | 0.53 | 0.69 | 0.47
Swe | 0.75 | 0.84 |  | 0.44 | 0.44 | 0.31 | 0.48 | 0.44 | 0.56 | 0.46
Ire | 0.32 | 0.39 | 0.44 |  | 0.62 | 0.48 | 0.51 | 0.35 | 0.37 | 0.43
Scot | 0.37 | 0.43 | 0.44 | 0.62 |  | 0.42 | 0.46 | 0.46 | 0.42 | 0.45
Wal | 0.18 | 0.33 | 0.31 | 0.48 | 0.42 |  | 0.46 | 0.37 | 0.32 | 0.37
Fra | 0.45 | 0.5 | 0.48 | 0.51 | 0.46 | 0.46 |  | 0.41 | 0.48 | 0.65
Ger | 0.46 | 0.53 | 0.44 | 0.35 | 0.46 | 0.37 | 0.41 |  | 0.75 | 0.36
Net | 0.61 | 0.69 | 0.56 | 0.37 | 0.42 | 0.32 | 0.48 | 0.75 |  | 0.42
Rom | 0.34 | 0.47 | 0.46 | 0.43 | 0.45 | 0.37 | 0.65 | 0.36 | 0.42 |
Mean | 0.47 | 0.55 | 0.52 | 0.43 | 0.45 | 0.36 | 0.49 | 0.46 | 0.51 | 0.44

In the first column, we can see that the Danish scores are most highly
correlated with the Norwegian scores (0.76). The Norwegian and Swedish columns
show that these two sets of scores are the most highly correlated at 0.84.
These correlations are uncontroversial given the linguistic similarities among
the Scandinavian countries. Similarly, the Irish and Scottish scores are most
highly correlated with each other at 0.62, most likely due to their Gaelic
origin. Interestingly, the Welsh scores have a relatively low correlation with
the other two Gaelic languages at 0.48 for Ireland and 0.42 for Scotland.
French scores are most highly correlated with the Roman scores (0.65). Again,
given the Latin origin of French, this is uncontroversial. The German scores
are most highly correlated with the Netherlands scores (0.75), due to both
linguistic and geographical proximity.

The last row of the table gives the average score for each column. This shows
for each classifier the average correlation for that classifier with the
scores of the other 9 countries. The highest average correlation is 0.55 for
the Norwegian classifier. This shows that the scores generated by the
Norwegian classifier are the most highly correlated with the scores of the
other classifiers. In other words, Norwegian place name conventions are more
similar to the naming conventions of the other countries than the place name
conventions of any other individual country. The Welsh naming conventions are
clearly the least similar, with an average correlation with the scores of
other countries of 0.36. Again, these observations are uncontroversial given
the historical perambulatory inclinations of the Norwegians, and the relative
non-perambulatory inclinations of the Welsh. The Norwegians (Vikings) spent
approximately 600 years venturing from their home to other areas of Europe,
and it is likely that, as in England, they left their linguistic mark in the
place names of the areas they visited or settled. This is in contrast to the
Welsh who have tended to be highly centralised in Wales.

One controversial observation is the relative lack of influence of Ancient
Roman naming conventions on the countries represented. The average correlation
of the Ancient Roman scores to the scores of the other classifiers is 0.44.
The Romans conquered the land mass of what is now France, which explains the
high correlation of Roman and French classifier scores. However, the Romans
only conquered the western edge of what is now Germany, thereby leaving the
bulk of ‘Germania’ relatively uninfluenced by their place naming conventions
(r = 0.36). Despite Ancient Roman presence in Wales and southern Scotland,
there was a relatively small influence of Ancient Rome on Welsh and Scottish
place names, with correlations of 0.37 and 0.45 respectively. Thus, their
influence on place names was greatest in France. Despite the lack of physical
conquest, the Romans had a high level of cultural influence on all of Europe.
Therefore, it is interesting to note that their influence on the place names
of Europe was less than seven of the other countries represented. This may be
an indication that in the case of place names, military and political hegemony
are more important than cultural hegemony.

## H. Future Work and Conclusions

The intention of this paper has been to identify the most likely linguistic
influences on English place names. By comparing each English place name with
the place names of ten western European countries, we are able to apply a
score to each English place name indicating whether it is more like other
English place names or more like the place names of other countries. By using
a large sample of place names from both England and Europe, it is possible to
determine with a high degree of probability whether an English place name has
the morphological structure of an English place name or that from one of the
other countries.

The use of the place names of other countries to partial out the influence on
English place names of those other countries is a novel procedure. The number
of place names in each sample is such that the results, albeit stochastic,
have a high degree of validity.

Another novel aspect is the method of breaking down each place name into 263
mainly ‘atomic’ variables. We have seen that this method has been able to find
patterns in the data that more ‘molecular’ or ‘rule-based’ procedures are able
to find, as well as finding others that were not previously known. For
example, the greater the number of vowels divided by the length of a place
name, the more likely it is to be a Roman, as opposed to an English place
name.

The vast majority of English place names (97%) are more like English place
names than those of other countries. However, there are English place names
that are ‘more English’ than others. We have seen that ‘Harlington’ has more
similarity to other English place names than any other English place name. We
have also seen that there are some place names that are unrelated to English
place names.

The relative classifier accuracies were used as a proxy for the difference
between English place names and the place names of other countries. The
classifier accuracy for Ancient Rome was the highest, indicating that Ancient
Roman place names are the most different from English place names, while the
accuracy for the Scottish classifier was the lowest, indicating that the
Scottish place names are the most similar to English place names.

Finally, the analysis enabled us to look at some broad influences on place
name patterns. We saw that the Norwegian names are the most similar to the
names of the other 9 countries, while the Welsh names are the least. We also
saw that Ancient Roman names have a relatively minor resemblance to the place
names of the other countries given the historical influence of Ancient Rome.

There are a number of ways the foregoing analysis could be extended.

One extension would be to see how the geographical distribution of more
‘English’ place names differs from the geographical distribution of less
‘English’ place names. We have scores that can provide a ranking method for
English place names, and a subsequent study could be done indicating the
settlement patterns of the various different language groups. Work has been
done on settlement patterns in the Danelaw region of the UK, showing a
preponderance of Scandinavian names in areas such as Cleveland, Yorkshire,
Lindsay and Leicestershire [25]. It would be interesting to see what the place
name settlement patterns are for the English place names that score low (have
a high ‘other’ content) on each of the ten classifiers. We would most likely
confirm the aforementioned study on settlement patterns in the Danelaw areas,
with the likelihood that place names scoring low on the England-Denmark
classifier most likely being centered around the Danelaw areas. Similarly, we
might expect those place names scoring low in the England-Scotland and
England-Wales classifiers to be clustered around the Scottish and Welsh
borders respectively. However, it would be interesting to see if there is a
particular pattern in relation to the 471 place names that are archetypically
‘non-English’.

Another extension would be to use a finer-grained distribution of languages.
The current study has three countries that are broadly Celtic, and the
influence of these languages has been discussed. However, it would be possible
to incorporate the place names of other Celtic areas such as Breton, Cornwall
and the Isle of Man. In the current study, the place names of Breton were
included in the French sample, and the place names of Cornwall and the Isle of
Man were incorporated into the English sample. A higher level of accuracy and
a subsequent more informative score could be achieved if these language groups
were given their own representation in the ensemble of classifiers and removed
from the French and English samples. Similarly, there are five Germanic
languages included in the current study (six if we include English), and the
possibility exists to use the place names of the Faeroe Islands, and Frisian
place names to get a finer gained analysis.

One issue that could be covered in a subsequent study is the influence of non-
Indo European languages on English place names. We have seen that there are
475 English place names that are not classifiable as English. Inspection of
these suggests that some of them are unrelated to the 11 languages used in
this study. It would be interesting to see how many of these closely resemble
the place names of the Basque country, Finland, Estonia and the Sami.

Data Availability Statement

The data used in this study are available at Figshare (https://figshare.com/),
DOI: https://doi.org/10.6084/m9.figshare.24916158.v1.

Research Funding Statement

This research was supported by an Australian Government RTP Scholarship.

Disclosure Statement

The author reports there are no known competing interests to declare.

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