# The Elastic Spiral Phase Pipe 111This is the submitted version of G.J.
Chaplain, and J.M. De Ponti, The Elastic Spiral Phase Pipe, Journal of Sound
and Vibration, 523, 116718 (2022). The final publication is available at
https://doi.org/10.1016/j.jsv.2021.116718

G. J. Chaplain∗<EMAIL_ADDRESS>Department of Mathematics, Imperial
College London, 180 Queen’s Gate, South Kensington, London SW7 2AZ
Electromagnetic and Acoustic Materials Group, Department of Physics and
Astronomy, University of Exeter, Exeter EX4 4QL, United Kingdom  J.M. De Ponti
Department of Civil and Environmental Engineering, Politecnico di Milano,
Piazza Leonardo da Vinci, 32, 20133 Milano, Italy

###### Abstract

We design a device for the passive mode conversion of guided, axisymmetric,
ultrasonic waves in hollow elastic pipes into arbitrary non-axisymmetric
flexural waves that have a constant angular profile along the pipe axis. To
achieve this we create an elastic analogue to optical spiral phase plates -
the elastic spiral phase pipe. Three possible configurations of the elastic
spiral phase pipe are presented which allow the efficient generation of non-
axisymmetric flexural waves from an axisymmetric, longitudinal forcing. The
theory leverages the dispersive nature of the guided elastic waves that are
supported in pipes through a defined relative refractive index. As such we
include a spectral collocation method used to aid the design of the elastic
spiral phase pipe that is corroborated with numerical simulations and then
experimentally verified.

###### keywords:

Flexural waves , Guided ultrasonic waves , Spiral Phase Plate , Elastic Pipe ,
Mode conversion

##  1  Introduction

The generation, manipulation and inspection of (ultrasonic) guided waves along
structures has played an important role in the development of non-destructive
testing (NDT) techniques. These techniques enable elastic material properties
to be determined and have uses in the inspection of welding defects, cracks
and adhesive joints, all crucially without causing damage to the sample. Given
the ubiquity of large pipe networks across many sectors of industry, from
petrochemical to power generation, these techniques are particularly pertinent
for assessing the structural integrity of pipe structures [1]. To utilise NDT
techniques effectively it is therefore essential to have efficient means of
exciting the guided ultrasonic waves, which requires analysis of the types of
waves supported by a given structure.

There is a long and rich history of the analysis of propagating guided waves
in hollow cylinders. The first general solution for harmonic waves propagating
in an infinitely long hollow cylinder was derived by Gazis [2, 3], who showed
there are three families of modes: longitudinal, torsional and flexural.
Following the conventional labelling system of Silk and Bainton [4], these are
classed as $L(m,n)$, $T(m,n)$ and $F(m,n)$ respectively, where $m$ denotes the
circumferential order and $n$ is the group order, or labelling index, with
each mode possessing unique dispersive properties. The mode shapes for which
$m=0$ are axisymmetric i.e. their angular profile is constant. Therefore only
the $T(0,n)$ are of pure torsion, and the $L(0,n)$ are essentially axial
extensional modes, with varying radial profiles [5]. The flexural modes
$F(m>0,n)$ are non-axisymmetric modes whose mode shapes vary sinusoidally in
the circumferential direction. Alternative naming conventions only use
$L(m,n)$ and $T(m,n)$ where it is understood that only $m=0$ corresponds to
axisymmetric modes with $T(m>0,n)$ being flexural modes [6, 7].

The axisymmetric guided wave modes $L(0,1)$, $L(0,2)$ and $T(0,1)$ can be
easily excited over typical pipe sizes and frequencies [8, 9]. Most commonly
used in pipe inspection are the $L(0,2)$ and $T(0,1)$ modes, due to their
ability to be efficiently excited and their dispersive properties [10, 11, 12,
13]; the $L(0,2)$ mode has the highest phase velocity and is only weakly
dispersive, whilst the $T(0,1)$ is non-dispersive across the whole frequency
spectrum. These modes are sensitive to circumferential defects and had have
much success in the field of non-destructive testing and evaluation [14, 15,
16, 17]. Higher order torsional modes, e.g. $T(0,2)$, have also have uses in
NDT but require more complex electromagnetic acoustic transducers (EMATs) for
excitation [18].

However, the $L(0,2)$ modes are insensitive to axial cracks, and, despite the
particle displacement for the $T(0,1)$ mode being perpendicular to the plane
of propagation, this mode requires deep cracks for detectable signals [19,
20]. There has therefore been several efforts to investigate the potential of
guided non-axisymmetric flexural modes for pipe inspection [21, 22]. These
modes too have their own complications due to their dispersive nature and mode
shapes that change along the direction of propagation; the circumferential
distribution of particle displacements (the angular profile) for non-
axisymmetric guided waves is complex, and changes with propagation distance,
frequency, and mode [23]. This is due to the unique dispersive properties of
the groups of modes present; the differences in phase velocities result in a
superposition leading to a varying angular profile [24]. For such dispersive
wave packets there is a limit on the long range resolution in NDT applications
[25]. The non-trivial mode shapes of the flexural modes require complex
arrangements of transducers (e.g comb arrays and non-axisymmetric partial
loading [26, 24, 27, 28, 22]) for their excitation, or by exploiting mode
conversion from non-axisymmetric defects [16] or bends [29, 6]. In order to
tune the angular profile Li and Rose [30] developed a circumferential phased
array for implementing a circumferential scan with focused guided wave beams,
requiring careful calibration of the transducer elements.

Figure 1: The Elastic Spiral Phase Pipe: (a) FEM frequency domain simulations
showing normalised total solid displacement, $|\boldsymbol{u}|$, and the
normalised radial, angular and axial displacements $u_{r}$, $u_{\theta}$,
$u_{z}$. A harmonic axisymmetric source is used. The position of the eSPP is
shown in the dashed rectangle. Absorbing boundary conditions are used on the
right end of the pipe. (b) Deformed cross sectional solid displacement field
after the eSPP conversion (at position of circular arrow in (a)). (c) Artists
impression of passive conversion of $L(0,2)$ to $F(3,2)$ in the steel pipe.
Throughout this article perceptually uniform colour maps are used [31].

We are motivated by the prospect of addressing the difficulties of exciting
non-axisymmetric flexural modes in pipe structures and seek to generate them
by passive mode conversion. Conversion of compressional to torsional motion
has been studied in both dynamic [32] and static [33] regimes. Here we focus
specifically on the conversion of compressional waves to flexural waves. The
proposed devices then have the advantages of (i) removing the need for complex
transducer arrangements, replacing these with a single, axial source (which
can be point-like); (ii) exciting (theoretically) a single, desired flexural
mode, rather than a group of modes; and (iii) ensuring only one ‘handedness’
(or chirality) of the desired flexural modes is present (conventional
excitation methods simultaneously excite both clockwise and anti-clockwise
modes). We do so by taking inspiration from phased array effects, notably from
the field of optics, and here employ a spatial inhomogeneity that can be
incorporated into the pipe structure in one of three configurations. To
demonstrate this capability, in Figure 1 we introduce the elastic spiral phase
pipe (eSPP) - an elastic analogue to the optical spiral phase plate (oSPP)
[34].

Figure 1(a) shows motivational results of a frequency domain finite element
method (FEM) simulation displaying capabilities of mode converting a purely
$L(0,2)$ mode almost completely into the (arbitrarily chosen) $F(3,2)$ mode.
We show the normalised total solid displacement, $|\boldsymbol{u}|$, and the
normalised radial, angular and axial displacements $u_{r}$, $u_{\theta}$,
$u_{z}$ respectively in a steel pipe of length $2.0~{}$\mathrm{m}$$,
$5~{}$\mathrm{cm}$$ inner diameter and thickness $4.5~{}$\mathrm{mm}$$. Here
the $L(0,2)$ mode is excited with an axial forcing at $59.7~{}$\mathrm{kHz}$$
from the left. At the centre of the pipe is an eSPP (position marked by dashed
rectangle in $u_{\theta}$ plot and highlighted by the bronzed region in the
artists impression in Figure 1(c)). This component is a hollowed circular
helicoid (same diameter and thickness of the steel pipe) with a specifically
designed azimuthally varying length profile that ensures efficient coupling to
the $F(3,2)$ mode in the steel pipe at this design frequency. The azimuthal
profile depends on the ratio of the speeds of the incident wave in the eSPP
and the speed of desired mode to be excited after the device (in this
illustrative example a tungsten join configuration is used - see Section 2).
The passive phase change endowed to the incident $L(0,2)$ as it traverses the
eSPP results in mode conversion to a flexural $F(3,2)$ mode on the other side
of the eSPP device. This is highlighted by the rotational displacement field
$u_{\theta}$ in Figure 1(a); before the device there is no circumferential
displacement, whilst afterwords there is significant rotational motion. Figure
1(b) shows a cross section after the device (at the position of the circular
arrow in (a)), which shows the characteristic field distribution of the
$F(3,2)$ mode. Figure 1(c) shows an artists impression of the mode conversion
effect: to the left of the eSPP an incident $L(0,2)$ mode is present (which is
easily excited by an axial forcing), represented by the circular phase fronts
shown in gold. After the device, helical phase fronts are shown consistent
with the excitation of the single, left-handed flexural mode $F(3,2)$. We
remark that it is only the phase front of the guided wave which is helical in
nature, not the hollow cylinder itself - we are not considering helical
waveguides such as those analysed by e.g. Treyssède [35].

.

Figure 2: Constant Angular Profile: Comparisons of angular profiles
(normalised $|\boldsymbol{u}|$) of the $F(3,2)$ mode calculated by a spectral
collocation method (blue) against those extracted from the FEM simulation in
Figure 1(a) (orange), at various distances along the pipe after the eSPP
device. The field is taken on the outer surface of the pipe. Shown too is a
zoom of the solid displacement component from Figure 1(a).

We highlight the uni-modal nature of the conversion in Figure 2, plotting the
angular profiles of the normalised solid displacement around the outer pipe
radius and at various distances along the length of the pipe, after the eSPP
device. Additionally shown here are the mode shapes of a perfect $F(3,2)$
mode, reconstructed from the eigensolution of the spectral collocation method
outlined in A. We emphasise here that the non-axisymmetric mode shapes
produced after the device are the result of an axisymmetric excitation.
Without the phase profile introduced by the eSPP these patterns are not
producible with axisymmetric forcings due to the orthogonality of the mode
shapes [24]. It can be seen here that the angular profile is constant along
the length of the pipe, up to a modulation by the propagation factor
$\mathrm{\exp}[\mathrm{i}(k_{z}z-\omega t)]$, i.e. the mode shape simply
rotates around the pipe, where $k_{z}$ is the wavenumber along the axial
direction of the pipe, $\omega$ the frequency of excitation and $t$ being
time. Therefore unlike the complicated mode shapes produced by non-
axisymmetric partial loading (e.g. Li and Rose [24]), here a predictable
angular profile is present due to the single flexural $F(3,2)$ mode that has
been excited due to the tailored mode conversion.

Throughout this paper we draw on the literature from the electromagnetic
community to facilitate the design of the eSPP. Recent translations of optical
phenomena to elastic devices have focused on concepts centred around
metamaterials, gradient index (GRIN) lenses and topological insulators, and
the discrete (often subwavelength) components that comprise them [36, 37, 38,
39, 40]. Some of these concepts have also recently been applied to slowing and
focussing waves in pipe structures [41, 42]. Unlike these discrete, resonant
devices we aim to leverage and emulate classical, continuous optical
components - namely the optical spiral phase plate. In Section 2 we outline
the conventional oSPP, including its history, uses and how its corresponding
physics can be translated across to the elastic spiral phase pipe. Important
differences between the analogous devices, that arise due from the nuances of
elasticity, shall be presented and prove key to its design. We then introduce
three possible configurations of the proposed eSPP devices, highlighting
potential advantages and disadvantages of each. In Section 3 we choose the
simplest of these configurations for experimental verification and proof of
concept, detailing the design process and presenting the experimental
confirmation of the passive mode conversion produced by the eSPP. We then
finally draw conclusions and outline future perspectives and applications for
the eSPP.

##  2 Optical Spiral Phase Plates and Elastic Spiral Phase Pipes

The first optical spiral phase plates, or spiral-phase-delay plates, were
realised in the early 1990s [43], enabling light fields with an azimuthally
varying phase profile to be created. These laser profiles enabled the probing
of optical vortices [44], have uses in particle acceleration experiments [45],
and crucially enabled investigations into angular momentum and spin-orbit
coupling of photons [46, 34]. Spiral phase plates can be used to generate so
called Laguerre-Gaussian (LG) modes, helical-wavefront beams, or “doughnut”
shaped beams of light that satisfy the paraxial wave equation which, after
Allen et al’s realisation that these carry optical angular momentum [47], and
the subsequent experimentation [48], re-ignited interest in the field of
optical tweezers [49, 50, 51]. The helical phase profiles generated by the
humble spiral phase plate have inspired many analogues across a variety of
wave systems [52, 53, 54].

Despite the prevalence of the oSPP, and the designs it has inspired, there has
been, until now, no elastic equivalent. This is perhaps surprising, especially
in pipe structures which already posses the topological features necessary to
support helical phased wave fronts. Below we outline the original oSPP and
then translate this to form the eSPP; nuances in the elastic system due to the
dispersive nature of the elastic waves in pipes require additional thought in
its design.

Figure 3: Optical spiral phase plates: Classical designs capable of producing
LG beams with example topological charges $\mathcal{Q}=1,2$ respectively. The
step height $h_{s}$ is shown, such that the total extent of the device is
$h=h_{s}+h_{0}$.

Figure 3(a) shows the classical optical spiral phase plate. It consists of a
plate whose thickness varies proportionately to the azimuthal angle $\theta$
defined about the central middle point of the plate and is thus parameterised
as a circular helicoid. The helical surface resembles one turn of a staircase
[34], the height of which we denote $h_{s}$. Figure 3(b) shows another
configuration, which includes a region of uniform thickness, $h_{0}$, before
the spiral region. The total height at any given point on the spiral is then
given as $h=h_{s}+h_{0}$. Spiral phase plates are used to produce optical
vortex beams, characterised by a phase singularity at the beam centre with
locally vanishing intensity carrying a topological charge defined as [55]

$\mathcal{Q}=\frac{1}{2\pi}\oint d\chi,$ (1)

where $\chi$ is the phase of the field. This quantity locally characterises
the complex field amplitude, $u$ through

$u(r,\theta,z)=u^{\prime}(r,z)\mathrm{\exp(\mathrm{i}\mathcal{Q}\theta)},$ (2)

where $u^{\prime}$ is a complex valued function with $r$ and $z$ the radial
and axial directions, with propagation along $z$.

When inserted, for example in the waist of a Gaussian beam, the oSPP endows
the incident field with an azimuthally dependent phase retardation. In optics
this phase profile is given by the difference in optical path length
experienced by the beam as it traverses the plate, i.e.

$\varphi(\theta,\lambda)=\frac{2\pi}{\lambda}\zeta,$ (3)

where $\zeta$ is the optical path difference, i.e. the different lengths
‘seen’ by the wave due to the change in phase speed by refraction. This yields
the phase profile

$\varphi(\theta,\lambda)=\frac{2\pi}{\lambda}\left[\frac{(\Delta
n)h_{s}\theta}{2\pi}+nh_{0}\right],$ (4)

where $\Delta n=(n-n_{0})$, with $n$ the refractive index of the plate and
$n_{0}$, $\lambda$ being the refractive index of, and wavelength in, the
surrounding medium respectively. Given a uniform phase change is imposed
throughout the initial height $h_{0}$. the vortex charge $\mathcal{Q}$ is then
given as [55]

$\mathcal{Q}=\frac{(n-n_{0})h_{s}}{\lambda}.$ (5)

The sign of $\mathcal{Q}$ determining the handedness of the helical phase-
front. We adopt the convention of $\mathcal{Q}>0$ being left-handed and
$\mathcal{Q}<0$ being right-handed. Alternatively the step height can be
partitioned equally across $\mathcal{Q}$ number of turns [56] as we show in
Figure 3(b). This simple equation then allows the step height to be determined
to generate a particular topological charge for a given wavelength, displaying
the chromatic nature of the oSPP. In optics, generation of low $\mathcal{Q}$
values requires either the step height to be commensurate to the order of the
wavelength or for the surroundings to be almost index matched such that
$\Delta n\approx 0$ [55, 34].

We now turn our attention to the elastic analogue of the oSPP, and outline the
similar design for the elastic spiral phase pipe. Figure 4 outlines an example
eSPP structure, which in essence is an elastic, hollowed spiral phase plate.
Here we show a pipe of inner radius $r_{a}$ and $r_{b}$ with three spirals.
Due to the coupled nature of the elastic wave system (A) endowing an incident
wave with, say a helical phase profile, does not merely change the amplitude
profile of the wave, but couples the motion into, potentially, all three
coordinate directions (i.e. excites $u_{r}$, $u_{\theta}$ and $u_{z}$).
Therefore in the elastic system it is not appropriate to just consider the
phase change of one component, but necessary to consider mode conversion from
one wave type to another. We shall demonstrate that the eSPP is capable of
mode conversion from a given incident compressional wave to a desired flexural
wave.

Figure 4: An Elastic Spiral Phase Pipe: Schematics of a three-step elastic
spiral phase pipe of inner radius $r_{a}$ and outer radius $r_{b}$.

Fortunately, in the elastic systems we shall consider, having step heights of
comparable size to the incident wavelength is straightforward. However, the
analogy to (5) is more nuanced in elasticity due to the lack of the notion of
a refractive index; in optical systems the refractive index is defined as the
ratio of the phase speeds of light in vacuo compared to that in a given
material, i.e. $n=c/v$. For most materials in electromagnetism, the
dispersionless nature of the supported waves and the universal speed limit of
$c$ then uniquely defines this refractive index. In elastic materials however,
even those that are isotropic and homogeneous, there exists distinct wave
types (e.g. compression and shear) with unique phase speeds and that couple at
interfaces [57]. This is clearly seen by the dispersion curves for the first
four circumferential orders ($m=0,\ldots,3$) in Figure 5, where we show two
conventional representations used in physics and engineering.

Figure 5: Dispersion curves of a steel pipe: (a) Conventional representation
in physics, frequency vs wavenumber (b) Conventional representation in
engineering, phase velocity vs frequency. Evaluated by the spectral
collocation method (SCM) shown in A, for inner diameter $5~{}$\mathrm{cm}$$
and thickness $4.5~{}$\mathrm{mm}$$.

We account for the lack of definiteness in the definition of refractive index
in elastic systems by defining a relative refractive index, relating the phase
speeds of a given incident mode, $c_{i}$ (with wavelength
$\lambda_{i}=2\pi/k_{i}$), to the desired flexural mode with a helical-phase
front and phase speed $c_{f}$. We therefore define this relative refractive
index $\tilde{n}=c_{f}/c_{i}$. Continuing with this we arrive at the following
formula for the step profile of the eSPP

$h_{s}=\frac{2\pi\mathcal{Q}}{k_{i}(\tilde{n}-1)}.$ (6)

The simple derivation from the optical analogue highlights another nuance with
the elastic system. Rearranging (5) we see that
$h_{s}=\lambda|\mathcal{Q}(n-n_{0})|$, and as such is unchanged to matter
which direction the incident wave traverses the spiral phase plate. However
for the eSPP, due to the form of $\tilde{n}$, reciprocity of the mode
conversion effects in the eSPP is not guaranteed - it is dependent on the form
of the incident wave upon the spiral region.

Given that the ultrasonic waves we consider are guided along the length of the
pipe we show, in Figure 6, three variations on the design of the eSPP. Figure
6(a) shows a notch configuration whereby a portion of the pipe is milled out,
leaving an impression of the eSPP (bronzed region in Figure 6(a)). This then
delineates the pipe into two regions with different thicknesses. We define
these as $h_{1}$ and $h_{2}$ such that $h_{1}<h_{2}$. The profile of the eSPP
is chosen such that the $L(0,2)$ mode in the eSPP (thickness $h_{1}$) mode
converts to the $F(3,2)$ mode (or whichever desired mode) in the thicker
region of thickness $h_{2}$. The advantages of this is that the region
comprising the eSPP is the same material of the pipe. Disadvantages are that
the pipe has to be machined, and that the thickness $h_{1}$ affects the
coupling of the incident $L(0,2)$ wave to the desired converted mode.

Figure 6(b) shows the join configuration, used in Figures 1&2, in which an
eSPP of different material is inserted between two regions of a pipe. The
advantages of this is that the entirety of the $L(0,2)$ mode is incident on
the face of the eSPP and so has the greatest conversion efficiency.
Additionally the flexibility of having an eSPP of different material at the
join enables a wider range of dispersive properties to be designed. Again
drawbacks from milling are now present in two pipes and joining them with
either welds or adhesives is required.

The final design is shown in Figure 6(c) and is termed the collar
configuration. This can be thought of as an inverted notch, whereby either the
whole pipe, apart from the eSPP region, is milled (leaving $h_{1}>h_{2}$).
However a more practical design is to have a separate collar which fits or
fastens around the pipe, possibly in several components. Again this has the
advantage that it can be a different material from the pipe, and be removed
after its used; its conversion efficiency is however dependent on the contact
between the pipe and the eSPP.

Figure 6: eSPP configurations: (a) Notch (b) Join and (c) Collar; the bronzed
regions represent the spiral phase pipe.

For simplicity we opt to verify the operation of the eSPP with the notch
configuration. In Section 3 we highlight its design and validate the
conversion from $L(0,2)$ into $F(3,2)$ in an aluminium pipe.

##  3  Experimental Verification

###  3.1 Design

In order to design a suitable eSPP for any given mode conversion the effective
refractive index between the two wave speeds must be evaluated. As such, the
dispersion curves for each region of the pipe (i.e. the eSPP and the rest of
the pipe) are required. We evaluate these for an infinitely long region of
each pipe by employing a spectral collocation method (A). Once obtained, the
dispersion curves are then used to infer the supported modes in each section
of the pipe. We first consider an infinitely long aluminium pipe, of inner
diameter $40~{}$\mathrm{mm}$$ and thickness $h_{2}=10~{}$\mathrm{mm}$$ of
density $\rho=2710~{}$\mathrm{k}\mathrm{g}\mathrm{m}^{-3}$$, Young’s Modulus
$E=70~{}$\mathrm{GPa}$$ and Poisson’s ratio $\nu=0.33$. We denote the
thickness of this region $h_{2}$; this is to be the section of the pipe where
the desired $F(3,2)$ mode to be excited, after conversion by the eSPP. We then
design an eSPP region by considering an aluminium pipe with the same internal
diameter but now with $h_{1}=4~{}$\mathrm{mm}$$. The thickness of the eSPP is
chosen such that $h_{1}<h_{2}/2$ to maximise the coupling to the spiral
region. The infinite dispersion curves for the eSPP region are then
calculated, and at a desired frequency the relevant step height can be
evaluated through Eq. (6).

Figures 7(a,b) show the $m=3$ dispersion curves for the pipe of thickness
$h_{2}$ and the $m=0$ dispersion curves for the eSPP region with thickness
$h_{1}$. We wish to excite the $L(0,2)$ in the eSPP to mode convert this to
the $F(3,2)$ mode in the thicker portion of the pipe, and so operate above
$60~{}$\mathrm{kHz}$$ where the frequency spectra of the two families of modes
overlap. We select, arbitrarily, a design frequency of $62~{}$\mathrm{kHz}$$
for the conversion. The phase speeds for each mode are then extracted from the
dispersion curves in Figure 7(b), and used to design the step height for the
eSPP, determined to be $h_{s}=190~{}$\mathrm{mm}$$. Figure 7(c) shows the
angular phase of the $F(3,2)$ mode, confirming it indeed has topological
charge $\mathcal{Q}=3$, as expected. In Figures 7(d)i-iv and Figures 7(e)i-iv
we show the mode shapes for the longitudinal $L(0,2)$ mode in the eSPP region
and the $F(3,2)$ mode in the thick pipe, showing that they are indeed
orthogonal and so require the inhomogeneity introduced by the eSPP to couple.

Figure 7: Dispersion Design and Mode Shapes: (a) Physicists representation of
dispersion curves (b) Engineers representation of dispersion curves. (c) Phase
profile of $\varphi(\theta)$ in region of thickness $h_{2}$, confirming
$\mathcal{Q}=3$. (di-iv) are the normalised radial, angular, axial and total
solid displacement fields respectively in $h_{1}$. (ei-iv) show the same
fields but in $h_{2}$. These fields are reconstructed from the SCM
eigensolutions (A).

###  3.2 Experimental Verification

To validate our findings, we perform time domain experiments, and compared to
numerical simulations, on an aluminium pipe where the eSPP is CNC milled out
removing the material from the thickness. This ensures a region of $h_{2}$
before and after the eSPP of thickness $h_{1}$. The pipe is
$900~{}$\mathrm{mm}$$ long, with internal and external diameter equal to
$40~{}$\mathrm{mm}$$ and $60~{}$\mathrm{mm}$$ respectively. The eSPP is
obtained by milling over a thickness of $6~{}$\mathrm{mm}$$. To decrease the
length of the step height, $h_{s}$, we partition it over three steps, rather
than a single step. We also include a length of $h_{0}$ to ensure the $L(0,2)$
mode is excited in the eSPP. The total portion of the pipe of length is then
of $h=h_{0}+h_{s}/3=158~{}$\mathrm{mm}$$. The experimental setup is shown in
Figure 8, where the pipe is suspended through elastic cables in order to avoid
disturbances from the boundaries. At the right boundary, an aluminium disk of
radius $60~{}$\mathrm{mm}$$ and thickness $10~{}$\mathrm{mm}$$ is connected
through a set of screws to the pipe, while a piezoelectric disk with diameter
$35~{}$\mathrm{mm}$$ and thickness $12~{}$\mathrm{mm}$$ is glued to provide
excitation.

Figure 8: Experimental Setup: (a) Measurements are performed using a 3D laser
vibrometer (i) scanning the velocity field on one side of the prototype (ii)
made of a pipe with a notched eSPP configuration (iii). (b) The pipe is
suspended through elastic cables on a frame in order to avoid edge
disturbances (iv) while on the right side a piezolectric disk is glued to the
cap in order to provide excitation of longitudinal waves (v). (c) Shows
schematic detailing geometric parameters. (d) Time domain source of piezo-disk
for experimental results in Figure 9 (blue) and sweep source for the
experimental Fourier spectra in Figure 10 (orange).

The wavefield is measured on one side of the pipe through a Polytec 3D Scanner
Laser Doppler Vibrometer (SLDV), which is able to separate the out-of-plane
velocity field in both space and time. A narrow-band spectrum excitation of
central frequency $f_{c}=60~{}$\mathrm{kHz}$$ and fractional bandwidth
$B=\Delta f/f_{c}=0.14$ is synchronously started with the acquisition which,
in turn, is averaged in time to decrease the noise. Figure 9 compares the
numerical and experimental wavefield along the pipe at different time
instants. The agreement between the numerical and the experimental data
confirms the capability of the eSPP to generate an helical phase profile, mode
converting incoming longitudinal $L(0,2)$ waves into flexural $F(3,2)$ waves.

Figure 9: Time Domain Results: Snapshots in time of the normalised velocity
field $v_{z}$ for (a)(i-iii) Numerical simulations at $t=0.812$, $0.82$ and
$0.828~{}$\mathrm{ms}$$ respectively and (b)(i-iii) Experimental results at
$1.248$, $1.256$ and $1.264~{}$\mathrm{ms}$$ respectively. Mode conversion of
longitudinal $L(0,2)$ waves into flexural $F(3,2)$ waves after the eSPP
(highlighted in dashed rectangle) is clearly seen. We note the handedness of
the helical phase-fronts differ compared to that in e.g. Figure 1 as we
measure the field right to left in the experiments. The input signal for both
cases is as in Figure 8(d)(i). Due to the curvature of the pipe we can only
extract the displacement fields over the portion shown in the experimental
results, compared to the whole pipe from the simulations (note the different
axes). Figure 10: Normalised Fourier Spectra and conversion efficiency: (a)
FFT of FEM time domain simulation performed with COMSOL MultiPhysics® [58].
The peak at the design frequency of $62~{}$\mathrm{kHz}$$ is clear on the
$F(3,2)$ branch. This is highlighted by the predicted conversion efficiency,
$\eta$ (right panel). (b) FFT of Experimental data. In both cases the
frequency sweep source (Figure 8(dii)) is used and the FFT performed along a
line after the eSPP on the surface of the pipe. Clear excitation of the
$F(3,2)$ mode is seen in both cases.

To assess the degree of conversion we inspect and compare the Fourier spectra
obtained by the spatio-temporal Fast Fourier Transform (FFT) along a line
after the eSPP, shown in Figure 10, for the time domain FEM simulation (a) and
the experimental results (b). Given the finite length of the pipe we are limit
in the resolution in $k_{z}$, and so we additionally show, in B, the results
of the FFT for a frequency domain simulation over a much longer pipe that is
$10~{}$\mathrm{m}$$ in length. Overlaid on these spectra are the dispersion
curves for the infinitely long aluminium pipe of thickness $h_{2}$. Clear
agreement is seen, and that the eSPP is most efficient near the design
frequency. We elucidate this further by showing, in the side panel of Figure
10(a), a measure of the conversion efficiency, $\eta$. We evaluate this from a
frequency domain FEM simulation, and define $\eta$ as the ratio of the
integrals of the absolute values of the compressional and rotational complex
mechanical energy flux, over the same volume before and after the eSPP, such
that

$\eta=\frac{\int|\boldsymbol{F}_{z}|^{2}dV}{\int|\boldsymbol{F}_{\theta}|^{2}dV^{\prime}},$
(7)

where $\boldsymbol{F}=\sigma\cdotp\boldsymbol{u}^{*}$, with $\sigma$ the
stress tensor and $\boldsymbol{u}^{*}$ the complex conjugate of the solid
displacement field. We denote the volume element before and after the eSPP as
$dV$ and $dV^{\prime}$ respectively, ensuring that the total integral volume
is the same in each case. This definition of the conversion efficiency then
gives a measure of the energy present in the compressional mode before the
eSPP and the flexural mode afterwards in the rest of the pipe. As seen in
Figure 10(a) we achieve a theoretical conversion efficiency of $>90\%$ at the
design frequency. We note that this calculation neglects any damping effects.

##  4 Conclusions

We have presented an elastic analogue to the optical spiral phase plate - the
elastic spiral phase pipe - and shown that it can be used to efficiently mode
convert axisymmetric longitudinal guided modes in pipes into arbitrary non-
axisymmetric flexural modes. We achieved this by defining the relative
refractive index, which relates the phase velocities of the two considered
modes to the azimuthally varying step profile of the device. Moreover we
presented three possible configurations of this device, each with their own
advantages and disadvantages. This was enabled by obtaining the dispersion
relations of the two sections of the pipe through a spectral collocation
method. We demonstrated, through numerical simulation corroborated by
experimentation that the $L(0,2)$ mode can be efficiently converted to the
$F(3,2)$ mode at the design frequency, chosen here to be
$62~{}$\mathrm{kHz}$$.

Advantages of this new device include that, compared to more complex non-
axisymmetric forcings, an axisymmetric forcing can be used to generate a
flexural wave with constant angular profile. As such we envisage exciting
applications for this device in the fields of non-destructive testing,
ultrasonic motor design and beyond.

## Acknowledgments

The authors are grateful to Prof. Richard V. Craster, Prof. Michael J.S. Lowe,
and Evan Hill Esq. for helpful conversations. G.J.C gratefully acknowledges
financial support from the EPSRC in the form of a Doctoral Prize Fellowship,
and from the Royal Commission for the Exhibition of 1851 in the form of a
Research Fellowship. J.M.D.P acknowledges the financial support from the H2020
FET-proactive project MetaVEH under grant agreement No. 952039.

##  Appendix A  Spectral Collocation Method

Guided propagating waves in a hollow isotropic elastic cylinder are governed
by Navier’s equation

$\mu\nabla^{2}\boldsymbol{u}+(\lambda+\mu)\nabla(\nabla\cdotp\boldsymbol{u})=\rho\ddot{\boldsymbol{u}},$
(8)

where $\boldsymbol{u}$ is the displacement field, $\lambda$ and $\mu$ are
Lamé’s first and second parameters respectively, $\rho$ the material density
and $\ddot{\boldsymbol{u}}$ is the acceleration. As the material is isotropic,
Helmholtz decomposition is used to write the displacement in terms of the
dilatational scalar potential, $\Phi$ and the equivoluminal vector potential
$\boldsymbol{\Psi}$

$\boldsymbol{u}=\nabla\Phi+\nabla\times\boldsymbol{\Psi}.$ (9)

Under this assumption, Navier’s governing wave equation then reduces to two
wave equations for compressional and shear waves:

$\begin{cases}\nabla^{2}\Phi=c_{p}^{-2}\ddot{\Phi},&c_{p}=\sqrt{\frac{\lambda+2\mu}{\rho}}\\\
\nabla^{2}\boldsymbol{\Psi}=c_{s}^{-2}\ddot{\boldsymbol{\Psi}},&c_{s}=\sqrt{\frac{\mu}{\rho}},\end{cases}$
(10)

with $c_{p}$ and $c_{s}$ being the compressional and shear bulk wavespeeds
respectively. Along with these two equations we employ traction free boundary
conditions on the inner and outer radii, $r_{a}$ and $r_{b}$, such that

$\sigma_{rr}=\sigma_{r\theta}=\sigma_{rz}=0\big{\rvert}_{r_{a,b}},$ (11)

along with the infinitely long cylinder gauge invariance through

$\nabla\cdotp\Psi=0.$ (12)

Working in cylindrical coordinates, such that
$\boldsymbol{\Psi}=(\Psi_{r},\Psi_{\theta},\Psi_{z})$, then gives

$\begin{split}\nabla^{2}\Phi&=\left(\frac{\mathrm{\partial}^{2}}{\mathrm{\partial}r^{2}}+\frac{1}{r}\frac{\mathrm{\partial}}{\mathrm{\partial}r}+\frac{1}{r^{2}}\frac{\mathrm{\partial}^{2}}{\mathrm{\partial}\theta^{2}}+\frac{\mathrm{\partial}^{2}}{\mathrm{\partial}z^{2}}\right)\Phi,\\\
\\\
\nabla^{2}\boldsymbol{\Psi}&=\nabla^{2}\left(\Psi_{r}\boldsymbol{\hat{e}}_{r}+\Psi_{\theta}\boldsymbol{\hat{e}}_{\theta}+\Psi_{z}\boldsymbol{\hat{e}}_{z}\right)\\\
&=\left(\nabla^{2}\Psi_{r}-\frac{1}{r^{2}}\Psi_{r}-\frac{2}{r^{2}}\frac{\mathrm{\partial}\Psi_{\theta}}{\mathrm{\partial}\theta}\right)\boldsymbol{\hat{e}}_{r}\\\
&+\left(\nabla^{2}\Psi_{\theta}-\frac{1}{r^{2}}\Psi_{\theta}+\frac{2}{r^{2}}\frac{\mathrm{\partial}\Psi_{r}}{\mathrm{\partial}\theta}\right)\boldsymbol{\hat{e}}_{\theta}\\\
&+\left(\nabla^{2}\Psi_{z}\right)\boldsymbol{\hat{e}}_{z}.\end{split}$ (13)

Following Rose [7] and Gazis [2, 3] we pose the complete solutions for guided
axial waves

$\begin{split}\Phi&=\phi(r)\mathrm{\exp}[\mathrm{i}(m\theta+k_{z}z-\omega
t)],\\\ \Psi_{r}&=\psi_{r}(r)\mathrm{\exp}[\mathrm{i}(m\theta+k_{z}z-\omega
t)],\\\
\Psi_{\theta}&=\psi_{\theta}(r)\mathrm{\exp}[\mathrm{i}(m\theta+k_{z}z-\omega
t)],\\\ \Psi_{z}&=\psi_{z}(r)\mathrm{\exp}[\mathrm{i}(m\theta+k_{z}z-\omega
t)],\\\ \end{split}$ (14)

such that $m\in\mathbb{Z}$. Noninteger values of $m$ correspond to
circumferential waves propagating in an isolated section of an annulus [59].
Other cases for the solutions of circumferential waves include circumferential
shear horizontal waves and circumferential Lamb-type waves [7]. In the case of
these waves any displacement in the $z$-direction must be uniform through the
entire $z$-plane, and it is necessary to distinguish between the circular and
angular wavenumber [60].

At this stage we diverge from the methods suggested by Rose [7], where
substituting (14) into (10) leads to four forms of Bessel’s equation for the
radial components of each potential. Solving for the dispersion relation then
requires incorporating the boundary conditions (11)&(12) that leads to an
exercise in root-finding for the zeros of the determinant of the resulting
system of equations [7, 61]. This method has been used with success for
multilayer and viscoelastic problems [62], yet is tedious and particularly
perilous if non-dimensionalisation is not used which leads to large arguments
in the Bessel functions. Alternative methods also include the Finite Element
Method [63] (FEM) and Semi-Analytical FEM [64, 65] (SAFEM).

Instead we opt to expand the Spectral Collocation Method (SCM) of Adamou and
Craster [59] which solves for the circumferential modes of an infinite
cylinder. This method is especially attractive given its spectral accuracy,
speed and ease of implementation. SCMs have had much success and been applied
to many other scenarios surrounding pipes and pipe structures, and have been
adapted to include viscoelasticity [66], anisotropy [67], and fluid filled
interactions [68]. The success of this method is highlighted by the resulting
commercialisation [69].

The basic idea is to represent (10) in operator form and construct a
generalised eigenvalue problem by replacing the differential operators with
differentiation matrices, thereby directly solving the differential equation
by numerical interpolation by spectral methods. Advances in computational
linear algebra allow this with ease [70], and have advantages over other
matrix methods susceptible to the so called large f-d problem (e.g. the
transfer matric method (TMM) and global matrix formalism [71, 72, 73, 74,
75]).

Assuming time-harmonicity, the governing equations (10) can be written as the
eigen-problems

$\begin{split}&\mathcal{L}\Phi=-\frac{\omega^{2}}{c_{p}^{2}}\Phi,\\\ \\\
&\left[\left(\mathcal{L}-\frac{1}{r^{2}}\right)\Psi_{r}-\frac{2\mathrm{i}m}{r^{2}}\Psi_{\theta}\right]\boldsymbol{\hat{e}}_{r}=-\frac{\omega^{2}}{c_{s}^{2}}\Psi_{r}\boldsymbol{\hat{e}}_{r},\\\
\\\
&\left[\frac{2\mathrm{i}m}{r^{2}}\Psi_{r}+\left(\mathcal{L}-\frac{1}{r^{2}}\right)\Psi_{\theta}\right]\boldsymbol{\hat{e}}_{\theta}=-\frac{\omega^{2}}{c_{s}^{2}}\Psi_{\theta}\boldsymbol{\hat{e}}_{\theta},\\\
\\\ &\mathcal{L}\Psi_{z}=-\frac{\omega^{2}}{c_{s}^{2}}\Psi_{z},\end{split}$
(15)

with

$\mathcal{L}=\left[\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}+\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}-\left(\frac{m^{2}}{r^{2}}+k_{z}^{2}\right)\right].$
(16)

Spectral collocation rests on representing this differential operator as a
differentiation matrix. As outlined in [59], these are introduced as matrices,
$D^{(p)}$ which approximate the $p^{th}$ derivative of a column vector
$\boldsymbol{f}$, whose entries are the values of the function $f(x)$ at the
$N$ interpolation points $x_{i}$, $i=1,\ldots,N$. Therefore the $p^{(th)}$
derivative is given by the matrix multiplication of $\boldsymbol{f}$ with the
$N\times N$ matrix $D^{(p)}$ such that

$\boldsymbol{f^{(p)}}\approx D^{(p)}\boldsymbol{f}.$ (17)

Given the bounded nature of the geometry in the radial direction, it is
appropriate to use Chebyshev differentiation matrices [76, 77]. These are
evaluated using the MATLAB differentiation matrix suite [70] with a suitable
coordinate transform to polar coordinates [59]. Therefore $\mathcal{L}$ can be
represented, with sufficient ease, by the $N\times N$ matrix

$L=D^{(2)}+\text{diag}\left(\frac{1}{r}\right)D^{(1)}-\text{diag}\left(\frac{m^{2}}{r^{2}}+k_{z}^{2}\right).$
(18)

Following this (15) can be written as a matrix eigenvalue equation:

${P}\boldsymbol{u}=-\omega^{2}Q\boldsymbol{u},$ (19)

with
$\boldsymbol{u}=\left(\Phi(r_{i}),\Psi_{r}(r_{i}),\Psi_{\theta}(r_{i}),\Psi_{z}(r_{i})\right)^{T}$
where $i=1,\ldots,N$, and

$Q=\begin{pmatrix}M(c_{p})&0&0&0\\\ 0&M(c_{s})&0&0\\\ 0&0&M(c_{s})&0\\\
0&0&0&M(c_{s})\end{pmatrix},$ (20)

where $M(x)$ is the $N\times N$ matrix

$M(x)=\text{diag}\left(\frac{1}{x^{2}}\right).$ (21)

The matrix $P$ encodes the differential operators such that

$P=\begin{pmatrix}L&0&0&0\\\ 0&L_{1}&-L_{2}&0\\\ 0&L_{2}&L_{1}&0\\\
0&0&0&L\end{pmatrix}.$ (22)

where

$\begin{split}L_{1}&=\left[L-\text{diag}\left(\frac{1}{r^{2}}\right)\right],\\\
\\\
L_{2}&=2\mathrm{i}m\left[\text{diag}\left(\frac{1}{r^{2}}\right)\right]\end{split}$
(23)

The eigenvalue problem is completed by incorporating the traction free
boundary conditions and the gauge condition in these matrices. This involves
constructing the boundary condition matrix $S$ below, and replacing rows of
$P$ and $Q$ to reflect the boundary conditions.

In terms of the elastic potential fields, the traction free conditions read

$\begin{split}{\sigma_{rr}}&=\lambda\nabla^{2}\Phi+2\mu\frac{\mathrm{\partial}u_{r}}{\mathrm{\partial}r}\\\
&=\left[\kappa^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}+(\kappa^{2}-2)\left(\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}-\frac{m^{2}}{r^{2}}-k_{z}^{2}\right)\right]\phi\\\
&-\left(2\mathrm{i}k_{z}\frac{\mathrm{d}}{\mathrm{d}r}\right)\psi_{\theta}+2\mathrm{i}m\left(\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}-\frac{1}{r^{2}}\right)\psi_{z},\end{split}$
(24)
$\begin{split}{\sigma_{r\theta}}&=2\mu\epsilon_{r\theta}=\mu\left[r\frac{\mathrm{\partial}}{\mathrm{\partial}r}\left(\frac{u_{\theta}}{r}\right)+\frac{1}{r}\frac{\mathrm{\partial}u_{r}}{\mathrm{\partial}\theta}\right]\\\
&=2\mathrm{i}m\left(\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}-\frac{1}{r^{2}}\right)\phi+\mathrm{i}k_{z}\left(\frac{\mathrm{d}}{\mathrm{d}r}-\frac{1}{r}\right)\psi_{r}\\\
&+\frac{k_{z}m}{r}\psi_{\theta}+\left(-\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}+\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}-\frac{m^{2}}{r^{2}}\right)\psi_{z}\end{split}$
(25)
$\begin{split}{\sigma_{rz}}&=2\mu\epsilon_{rz}=\mu\left(\frac{\mathrm{\partial}u_{r}}{\mathrm{\partial}z}+\frac{\mathrm{\partial}u_{z}}{\mathrm{\partial}r}\right)\\\
&=\left(2\mathrm{i}k_{z}\frac{\mathrm{d}}{\mathrm{d}r}\right)\phi+\mathrm{i}m\left(\frac{1}{r^{2}}-\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}\right)\psi_{r}\\\
&+\left(\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}+\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}-\frac{1}{r^{2}}+k_{z}^{2}\right)\psi_{\theta}-\frac{mk_{z}}{r}\psi_{z},\end{split}$
(26)

where $\kappa=c_{p}/c_{s}$. The gauge condition then has the form

$\begin{split}\nabla\cdotp\boldsymbol{\Psi}=\left(\frac{1}{r}+\frac{\mathrm{d}}{\mathrm{d}r}\right)\psi_{r}+\frac{\mathrm{i}m}{r}\psi_{\theta}+\mathrm{i}k_{z}\psi_{z}=0.\end{split}$
(27)

The vector we define as
$\boldsymbol{s}=\left(\sigma_{rr}(r_{i})/\mu,\sigma_{r\theta}(r_{i})/\mu,\sigma_{rz}(r_{i})/\mu,\nabla\cdotp\Psi(r_{i})\right)^{T}$
is then related to the displacement vector $\boldsymbol{u}$ through the
$4N\times 4N$ boundary condition matrix $S$ defined by

$S=\begin{pmatrix}S_{11}^{rr}&S_{12}^{rr}&S_{13}^{rr}&S_{14}^{rr}\\\ \\\
S_{21}^{r\theta}&S_{22}^{r\theta}&S_{23}^{r\theta}&S_{24}^{r\theta}\\\ \\\
S_{31}^{rz}&S_{32}^{rz}&S_{33}^{rz}&S_{34}^{rz}\\\ \\\
S_{41}^{\nabla}&S_{42}^{\nabla}&S_{43}^{\nabla}&S_{44}^{\nabla}\end{pmatrix},$
(28)

such that $\boldsymbol{s}=S\boldsymbol{u}$ and the components of $S$ are the
$N\times N$ matrices determined by incorporating the differentiation matrices
$D^{(p)}$ into equations (24)-(27):

$\begin{split}S_{11}^{rr}&=\kappa^{2}D^{(2)}+(\kappa^{2}-2)\left[\text{diag}\left(\frac{1}{r}\right)D^{(1)}-m^{2}\text{diag}\left(\frac{1}{r^{2}}+k_{z}^{2}\right)\right],\\\
S_{12}^{rr}&=Z,\\\ S_{13}^{rr}&=-2\mathrm{i}\text{
diag}\left(k_{z}\right)D^{(1)},\\\
S_{14}^{rr}&=2\mathrm{i}m\left[\text{diag}\left(\frac{1}{r}\right)D^{(1)}-\text{diag}\left(\frac{1}{r^{2}}\right)\right],\\\
S_{21}^{r\theta}&=2\mathrm{i}m\left[\text{diag}\left(\frac{1}{r}\right)D^{(1)}-\text{diag}\left(\frac{1}{r^{2}}\right)\right],\\\
S_{22}^{r\theta}&=\mathrm{i}\text{
diag}\left(k_{z}\right)\left[D^{(1)}-\text{diag}\left(\frac{1}{r}\right)\right],\\\
S_{23}^{r\theta}&=m\text{ diag}{\left(\frac{k_{z}}{r}\right)},\\\
S_{24}^{r\theta}&=-D^{(2)}+\text{diag}{\left(\frac{1}{r}\right)D^{(1)}}-m^{2}\text{diag}\left(\frac{1}{r^{2}}\right),\\\
S_{31}^{rz}&=2\mathrm{i}\text{ diag}\left(k_{z}\right)D^{(1)},\\\
S_{32}^{rz}&=\mathrm{i}m\left[\text{diag}\left(\frac{1}{r^{2}}\right)-\text{diag}\left(\frac{1}{r}\right)D^{(1)}\right],\\\
S_{33}^{rz}&=D^{(2)}+\text{diag}\left(\frac{1}{r}\right)D^{(1)}-\text{diag}\left(\frac{1}{r^{2}}-k_{z}^{2}\right),\\\
S_{34}^{rz}&=-m\text{ diag}{\left(\frac{k_{z}}{r}\right)},\\\
S_{41}^{\nabla}&=Z,\\\
S_{42}^{\nabla}&=\text{diag}\left(\frac{1}{r}\right)+D^{(1)},\\\
S_{43}^{\nabla}&=\mathrm{i}m\text{ diag}\left(\frac{1}{r}\right),\\\
S_{44}^{\nabla}&=\mathrm{i}\text{ diag}\left(k_{z}\right),\\\ \end{split}$
(29)

where $Z$ is an $N\times N$ matrix of zeros.

The boundary conditions are the built in to the eigenvalue problem by
augmenting the rows of $P$ with the rows of $S$ that pertain to the
coordinates for $r_{a}$ and $r_{b}$, namely $r_{i}$ and $r_{N}$, and enforcing
their values by modifying $Q$. The spectral method is then fully described by
the generalised eigenvalue problem

$\hat{P}\boldsymbol{u}=-\omega^{2}\hat{Q}\boldsymbol{u},$ (30)

where $\hat{P}$ is the matrix $P$, but with the $1^{st}$, $N^{th}$,
$N+1^{th}$,$2N^{th}$,$2N+1^{th}$,$3N^{th}$,$3N+1^{th}$ and $4N^{th}$ rows
replaced by the corresponding rows of $S$, and

$\hat{Q}=\begin{pmatrix}\hat{M}&0&0&0\\\ 0&\hat{M}&0&0\\\ 0&0&\hat{M}&0\\\
0&0&0&\hat{M}\end{pmatrix}M,$ (31)

such that $\hat{M}$ is the $N\times N$ matrix

$\hat{M}=\begin{pmatrix}0&&&&\\\ &1&&&\\\ &&\ddots&&\\\ &&&1&\\\
&&&&0\end{pmatrix}.$ (32)

This equation is solved numerically giving the dispersion relation
$\omega(k_{z})$ for a given circumferential order $m$. A pictorial
representation of $\hat{P}$ is shown below.

.

The mode profiles can then be evaluated through each displacement component:

$\begin{split}u_{r}&=\frac{\mathrm{\partial}\Phi}{\mathrm{\partial}r}+\frac{\mathrm{i}m}{r}\Psi_{z}-\mathrm{i}k_{z}\Psi_{\theta}\\\
u_{\theta}&=\frac{\mathrm{i}m}{r}\Phi+\mathrm{i}k_{z}\Psi_{r}-\frac{\mathrm{\partial}\Psi_{z}}{\mathrm{\partial}r}\\\
u_{z}&=\mathrm{i}k_{z}\Phi+\frac{\Psi_{\theta}}{r}+\frac{\mathrm{\partial}\Psi_{\theta}}{\mathrm{\partial}r}-\frac{\mathrm{i}m}{r}\Psi_{r}.\\\
\end{split}$ (33)

We note that using the displacement potentials is most appropriate for this
simple example, but it is more appropriate to reformulate the SCM using the
displacement fields and elasticity tensor coefficients to deal with anisotropy
and more complex geometries, as the gauge and boundary conditions are
difficult to incorporate otherwise.

##  Appendix B Supplementary Results

Here we shown an additional numerical FFT of a frequency domain FEM simulation
for an extended pipe of length $10~{}$\mathrm{m}$$. The eSPP is the same as in
Figure 8. A harmonic forcing is applied on the end cap at each frequency and
the FFT extracted along a line on the outer radius of the extended pipe after
the eSPP. The dispersion curves for an infinitely long pipe are overlaid, just
as in Figure 10.

Figure 11: FEM frequency domain FFT: Simulation of extended pipe to improve
resolution in $k_{z}$. Dispersion curves from SCM are also shown, highlighting
the accuracy of the method for predicting the modes supported by the finite
pipe.

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