∎ 11institutetext: lnstitut d’Astronomie et de Geophysique G.Lemaitre,
Catholic University of Louvain, Louvain-la-Neuve, Belgium
_Present address:_ Blue Abyss, Newquay, Cornwall, United Kingdom,
<EMAIL_ADDRESS>11institutetext: ORCID 0000-0003-4884-3827

# Annular structures in perturbed low mass disc-shaped gaseous nebulae II :
general and polytropic models

Vladimir Pletser

(Received: date / Accepted: date)

###### Abstract

This is the second of two papers where we study additional analytical
solutions of a bidimensional low mass gaseous disc rotating around a central
mass and submitted to small radial perturbations. In a first Paper,
hydrodynamics equations were solved for the equilibrium and perturbed
configurations and a wave-like equation for the gas perturbed specific mass
was deduced and solved analytically for several cases of exponents of the
power law distributions of the unperturbed specific mass and sound speed. In
this paper, two other general cases of exponents, including a polytropic case,
are solved analytically for small frequencies of the perturbations. Similar
conclusions to the ones of Paper I are found, namely that the maxima of the
gas perturbed specific mass are exponentially spaced and that their distance
ratio is a constant, function of disc characteristics and of the perturbations
frequency. Gaseous annular structures would eventually be formed in the disc
by inward and outward gas flows from zones of minima toward zones of maxima of
perturbed specific mass.

Keywords: Interdisciplinary astronomy, Astrophysical fluid dynamics,
Hydrodynamics, Protoplanetary nebulae

##  1 Introduction

In a previous paper (Paper1 (1) hereafter referred to as Paper 1), we
presented analy1ical solutions of perturbations propagating in a
differentially rotating, axisymmetric, thin gaseous nebular disc, undergoing
poly1ropic transformations of index $\gamma$. Viscous and magnetohydrodynamic
forces were neglected in the disc. The specific mass and sound speed in the
disc at equilibrium had power law distributions in the radial distance $r$,
respectively $\rho\sim r^{d}$ and $c\sim r^{\frac{s}{2}}$. Searching for
solutions yielding annular structures to appear in the disc when submitted to
small radial periodic perturbations, hydrodynamics equations were solved
analy1ically for the equilibrium and perturbed configurations, for two
particular cases ($d=0$ and $d<2(2\gamma-1)$; $s=2$) and for a general case
($d=(s-2)$; $s<2$) for small frequencies. In each case, the maxima of
perturbed specific mass were found to be exponentially spaced and their
distance ratio $\beta$ was found to be a constant, depending on
characteristics of the disc (and on the perturbations frequency for the first
two cases). Inward and outward flows of gas appeared with negative and
positive radial velocities between minima and maxima of gas perturbed specific
mass, leading the nebular gas eventually to accumulate in the zones of maxima
of perturbed specific mass. We present here analy1ical solutions for two other
general models for small frequencies of the periodic perturbations. In section
2, the disc model notations and the equations deduced in Paper 1 are recalled.
The general case for $d=(s-2)/2$ is solved in section 3. We present in section
4 a method to solve a second general model for $d=s/(\gamma-1)$, called the
"polytropic model", and a complete solution is found for the particular values
$\gamma=3/2$, $d=-3$ and $s=-1$. We do not know of any previous similar
general analy1ical resolutions. The conclusions are drawn in section 5 and are
similar to the ones of Paper 1. Both papers are reworked excerpts of
Pletser1990 (2).

##  2 Model notations

In Paper 1 Paper1 (1), we considered a gaseous disc of mass $M_{d}$ and of
specific mass $\rho_{0}=\rho_{c}R^{d}$ with a sound speed
$c_{0}=c_{c}R^{s/2}$, rotating at a circular velocity $v_{0}=v_{0}(R)$ around
a central mass $M^{*}$ ($M^{*}>>M_{d}$)· $R=(r/r_{c})$ is a dimensionless
radial distance and the indexes $0$ and $c$ denote equilibrium characteristics
and reference characteristics at the inner edge of the disc. Allowing for
small radial periodic perturbations of circular frequency $\omega$ to appear,
a wave-like equation was deduced for the spatial term $D$ of the gas perturbed
specific mass, with the prime sign $"\,^{\prime}"=\partial\,/\partial r$,

$D^{\prime\prime}+\left(2s+1-\frac{d+s}{\gamma}\right)\frac{D^{\prime}}{R}+\left(B^{2}R^{d+2-s}+\omega^{2}A^{2}R^{2-s}+s\left(s-\frac{d+s}{\gamma}\right)\right)\frac{D}{R^{2}}=0$
(1)

where

$A^{2}=\frac{r_{c}^{2}}{c_{c}^{2}}\,\,\,;\,\,B^{2}=\frac{4\pi
G\rho_{c}r_{c}^{2}}{c_{c}^{2}}$

are constants. The spatial terms of the associated perturbed radial velocity
$U$ and specific mass flux radial momentum $\Phi$ were found in function of
$D$

$\displaystyle U\left(R\right)$
$\displaystyle=-\kappa\frac{r_{c}}{\rho_{c}}R^{-\left(d+1\right)}\int
D\left(R\right)R\,dR$ (2) $\displaystyle\Phi\left(R\right)$
$\displaystyle=r_{c}\rho_{c}R^{d+1}U\left(R\right)=-\kappa r_{c}^{2}\int
D\left(R\right)R\,dR$ (3)

Two boundary conditions were defined by, first, the gas perturbed specific
mass matching at the disc inner edge, for $R=1$, a certain value independent
from disc physical characteristics (see Paper 1) and, second, decreasing
perturbed specific mass for increasing $R$, vanishing far away from the
central mass.

##  3 Solutions for d = (s - 2)/2

Searching for solutions yielding annular structures to appear in the disc, we
consider a fourth case where the exponents $d$ and $s$ are linked by the
relation $d=(s-2)/2$. The equation (1) reads

$\displaystyle
D^{\prime\prime}+\left(4d+5-\frac{3d+2}{\gamma}\right)\frac{D^{\prime}}{R}$
$\displaystyle+\left(\omega^{2}A^{2}R^{-2d}+B^{2}R^{-d}+2\left(d+1\right)\left(2\left(d+1\right)-\left(\frac{3d+2}{\gamma}\right)\right)\right)\frac{D}{R^{2}}=0$
(4)

Substituting the variable $R$ for

$z=j\left(\frac{2}{d}\omega AR^{-d}\right)$

with $j=\sqrt{-1}$, yields a confluent hypergeometric equation

$\displaystyle z^{2}\frac{\partial^{2}D}{\partial
z^{2}}-\frac{1}{d}\left(3d+4-\left(\frac{3d+2}{\gamma}\right)\right)z\frac{\partial
D}{\partial z}$ $\displaystyle-\left(\frac{z^{2}}{4}+j\frac{B^{2}}{2d\omega
A}z-2\left(\frac{d+1}{d^{2}}\right)\left(2\left(d+1\right)-\left(\frac{3d+2}{\gamma}\right)\right)\right)D=0$
(5)

With $d\neq-2\,/\left(3+i\gamma\right)$ for all integers $i$, (5) has a
complex solution Whittaker-Watson1927 (3, 4, 5)

$D_{C}=R^{-\frac{1}{2}\left(3d+4-\left(\frac{3d+2}{\gamma}\right)\right)}\exp\left(\frac{-z}{2}\right)\left(K_{1}z^{\frac{1}{2}\left(1+\left(\frac{3d+2}{d\gamma}\right)\right)}\phi_{1}+K_{2}z^{\frac{1}{2}\left(1-\left(\frac{3d+2}{d\gamma}\right)\right)}\phi_{2}\right)$
(6)

with $K_{1}$ and $K_{2}$ constants and where
$\phi_{1}=\phi\left(a_{1},b_{1},z\right)$ and
$\phi_{2}=\phi\left(a_{2},b_{2},z\right)$ are Kummer confluent hypergeometric
function of arguments

$\displaystyle a_{1}$
$\displaystyle=\frac{1}{2}\left(1+\frac{3d+2}{d\gamma}\right)+jy\,\,\,;\,\,b_{1}=1+\frac{3d+2}{d\gamma}$
$\displaystyle a_{2}$
$\displaystyle=\frac{1}{2}\left(1-\frac{3d+2}{d\gamma}\right)+jy\,\,\,;\,\,b_{1}=1-\frac{3d+2}{d\gamma}$

where

$y=\frac{B^{2}}{d\omega A\left(\frac{3d+2}{\gamma}-\left(3d+4\right)\right)}$

For $b_{1}$ and $b_{2}$ non null and different from negative integers, the
Kummer function $\phi$ expands for both sets of arguments as Slater1965 (6)

$\phi\left(a,b,z\right)=\Gamma\left(\nu\right)\left(\frac{z}{4}\right)^{-\nu}\exp\left(\frac{z}{2}\right)\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}\left(2\nu\right)_{k}\left(b-2a\right)_{k}}{k!\left(b\right)_{k}}\,I_{\nu+k}\left(\frac{z}{2}\right)$
(7)

where $\Gamma$ is the Legendre Gamma function, $(x)_{k}$ are Pochhammer
polynomials,

$(x)_{k}=\prod_{q=0}^{k-1}(x+q)\,\,\,;\,\,(x)_{0}=1$

and $I_{\nu+k}$ is the complex valued hyperbolic Bessel function of order
$\left(\nu+k\right)$, with $\nu$ either of

$\displaystyle\nu_{1}$
$\displaystyle=\left(b_{1}-a_{1}-\frac{1}{2}\right)=\frac{3d+2}{2d\gamma}-jy$
$\displaystyle\nu_{2}$
$\displaystyle=\left(b_{2}-a_{2}-\frac{1}{2}\right)=-\frac{3d+2}{2d\gamma}+jy$

Developing the complex coefficients $H_{k}$ of $I_{\nu+k}$ in (7) as in
Appendix A (Appendix A), $D_{C}$ (6) becomes

$\displaystyle D_{C}$
$\displaystyle=CR^{-2\left(d+1\right)+\left(\frac{3d+2}{2\gamma}\right)}\frac{z}{4}jy\left(K_{1}C_{1}\sum_{k=0}^{\infty}\left(H_{1k}I_{\nu_{1}+k}\left(\frac{z}{2}\right)\right)\right.$
$\displaystyle\left.+K_{2}C_{2}\sum_{k=0}^{\infty}\left(H_{2k}I_{\nu_{2}+k}\left(\frac{z}{2}\right)\right)\right)$
(8)

where

$C=\sqrt{\frac{\omega
A}{d}}\exp\left(j\frac{\pi}{4}\right)\,\,\,;\,\,C_{1}=2^{1+\frac{3d+2}{d\gamma}}\Gamma\left(\nu_{1}\right)\,\,\,;\,\,C_{2}=2^{1-\frac{3d+2}{d\gamma}}\Gamma\left(\nu_{2}\right)$

are complex constants. Simple analytical expressions of zeros and extrema of
the real part of $D_{C}$ (8) were not found. However, for small arguments $z$,
$\left(\left|z\right|/2\right)<<1$, i.e. for small frequencies

$\omega<<\left|d\right|\frac{c_{c}}{r_{c}}R^{-\left|d\right|}$ (9)

where vertical bars denote the absolute value, the terms other than the first
in the convergent series of (7) can be neglected. By the multiplication
theorem Olver1965 (7), the hyperbolic Bessel function reduces then to

$I_{\nu}\left(\frac{z}{2}\right)\approx\left(\frac{z}{4}\right)^{-\nu}\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{\left(-1\right)^{m}}{m!n!\Gamma\left(\nu-m+n+1\right)}=\left(\frac{z}{4}\right)^{-\nu}\frac{G\left(\nu\right)}{\nu\Gamma\left(\nu\right)}$
(10)

where terms of second order were neglected in front of unity and where
$G(\nu)$ is a complex function developed in Appendix B (Appendix B). The
relation (7) reads now

$\phi\left(a,b,z\right)=\frac{G\left(\nu\right)}{\nu}\left(\frac{z}{4}\right)^{-2\nu}\exp\left(\frac{z}{2}\right)$
(11)

The complex solution (8), written for the variable $R$, becomes

$D_{C}=\left(K_{1}\varLambda
R^{-2\left(d+1\right)+\frac{3d+2}{\gamma}}+K_{2}MR^{-2\left(d+1\right)}\right)\exp\left(2jy\ln\left(\frac{2}{\left|d\right|}\omega
AR^{-d}\right)\right)$ (12)

where $\varLambda=\left|\varLambda\right|\exp\left(j\lambda\right)$ and
$M=\left|M\right|\exp\left(j\mu\right)$ are complex constants depending on
$y$, $\nu$ and $G(\nu)$ (see Appendix B). The real part of (12) reads as the
sum of two terms

$\displaystyle D_{R}$
$\displaystyle=K_{1}\left|\varLambda\right|R^{-2\left(d+1\right)+\frac{3d+2}{\gamma}}\cos\left(2y\ln\left(\frac{2}{\left|d\right|}\omega
AR^{-d}\right)+\lambda\right)$
$\displaystyle+K_{2}\left|M\right|R^{-2\left(d+1\right)}\cos\left(2y\ln\left(\frac{2}{\left|d\right|}\omega
AR^{-d}\right)+\mu\right)$ (13)

Due to the second boundary condition (decrease of $D$ for increasing $R$),
either the first or the second or both terms of (13) should be considered for
the general solution, depending on the respective values of $d$ and $\gamma$
as indicated in Table 1.

Table 1: Terms of solution $D$ (13) for respective values of $d$ and $\gamma$ | $\gamma<\frac{3}{2}$ | $\gamma=\frac{3}{2}$ | $\gamma>\frac{3}{2}$
---|---|---|---
| $d<\frac{2\left(\gamma-1\right)}{3-2\gamma}$ | $d\geq\frac{2\left(\gamma-1\right)}{3-2\gamma}$ | all values of $d$ | $d\leq\frac{2\left(\gamma-1\right)}{3-2\gamma}$ | $d>\frac{2\left(\gamma-1\right)}{3-2\gamma}$
$d>-1$ | Both parts | $K_{1}=0$ | Both parts | Not Applicable | Both parts
$d\leq-1$ | $K_{2}=0$ | Not Applicable | $K_{2}=0$ | No decrease | $K_{2}=0$

Without loss of generality in the resolution, we consider from now on only the
case $s<0$ and $d<-1$, yielding $K_{2}=0$ in (13), the other constant $K_{1}$
being fully determined by the first boundary condition.

The perturbed radial velocity $U$ and the specific mass flux momentum
$\varPhi$ read, from (2) and (3),

$\displaystyle U$ $\displaystyle=-\kappa
K_{1}C_{3}\frac{r_{c}}{\rho_{c}}R^{-\left(3d+1\right)+\frac{3d+2}{\gamma}}\sin\left(2y\ln\left(\frac{2}{\left|d\right|}\omega
AR^{-d}\right)+\lambda+\zeta\right)$ (14) $\displaystyle\varPhi$
$\displaystyle=-\kappa
K_{1}C_{3}r_{c}^{2}R^{-2d+\frac{3d+2}{\gamma}}\sin\left(2y\ln\left(\frac{2}{\left|d\right|}\omega
AR^{-d}\right)+\lambda+\zeta\right)$ (15)

with

$C_{3}=\frac{\left|\varLambda\right|}{\sqrt{\left(\frac{3d+2}{\gamma}-2d\right)^{2}+\left(-2dy\right)^{2}}}\,\,\,;\,\,\zeta=\arctan\left(\frac{\frac{3d+2}{\gamma}-2d}{-2dy}\right)$

The extrema (minima and maxima) of $D_{R}$ are found from

$D_{R}^{\prime}=-K_{1}C_{4}R^{-2\left(d+3\right)+\frac{3d+2}{\gamma}}\sin\left(2y\ln\left(\frac{2}{\left|d\right|}\omega
AR^{-d}\right)+\lambda+\xi\right)=0$ (16)

with

$C_{4}=\left|\varLambda\right|\sqrt{\left(\frac{3d+2}{\gamma}-2\left(d+1\right)\right)^{2}+\left(2dy\right)^{2}}\,\,\,;\,\,\xi=\arctan\left(\frac{\frac{3d+2}{\gamma}-2\left(d+1\right)}{2dy}\right)$

The zeros of $D_{R}$ (13), $U$ (14), $\varPhi$ (15) and $D_{R}^{\prime}$ (16)
read in this fourth case

$R=\alpha_{4}\left(\beta_{4}^{\backprime}\right)^{n}\,\,\,;\,\,\alpha_{4}=\left(\frac{2}{\left|d\right|}\omega
A\right)^{\frac{1}{d}}\exp\left(\frac{\lambda+\varphi_{4}}{2\left|d\right|y}\right)\,\,\,;\,\,\beta_{4}^{\backprime}=\exp\left(\frac{\pi}{2\left|d\right|y}\right)$
(17)

$n$ being non-negative integers and $\varphi_{4}=\left(\pi/2\right)$
for$D_{R}$, $\varphi_{4}=\zeta$ for $U$ and $\varPhi$, and $\varphi_{4}=\xi$
for $D_{R}^{\prime}$. Provided that $\omega A$ is small enough, within the
condition (9), one has $y>>1$, yielding $\zeta<<1$ and $\xi<<1$. The initial
phase between $D$ and $U$ is
$\left(\pi/2\right)-\zeta\approx\left(\pi/2\right)$, while the initial phase
between $U$ (or $\varPhi$) and $D^{\prime}$ is $(\xi-\zeta)\approx 0$.

The distances ratio of two successive maxima of $D$ is

$\beta_{4}=\left(\beta_{4}^{\backprime}\right)^{2}=\exp\left(\frac{\left(3d+4-\frac{3d+2}{\gamma}\right)\omega
c_{c}}{4\pi G\rho_{c}r_{c}}\right)$ (18)

which is a real constant depending on the perturbations circular frequency
$\omega$ and the disc reference characteristics. The period of the
perturbations must be larger than a minimum value

$P_{m}=\frac{2\pi}{\left|d\right|}\frac{r_{c}}{c_{c}}R_{max}^{d}$ (19)

deduced from the condition (9) applied to the whole range of radial distances
of the disc ($R_{max}$ is the ratio of the outer and inner radii of the disc).

##  4 Solution for the polytropic case

###  4.1 General formulation

In the previous section and in Paper 1, we considered the exponents $d$ and
$s$ taking particular values or linked by non-causal relations. However, a
relation between $d$ and $s$ can be found if one considers that the specific
mass and sound speed are fully governed by polytropic processes in the disc.
Considering the two polytropic relations between the pressure $p$, the
specific mass $\rho$ and the sound speed $c$

$p=\frac{c^{2}\rho}{\gamma}\,\,\,;\,\,p\rho^{-\gamma}=\text{constant}$ (20)

one has successively, with the power law radial distributions
$\rho_{0}=\rho_{c}R^{d}$ and $c_{0}^{2}=c_{c}^{2}R^{s}$,

$\displaystyle
c^{2}\rho^{1-\gamma}=c_{c}^{2}\rho_{c}^{1-\gamma}R^{s+d\left(1-\gamma\right)}=\text{constant}$
(21) $\displaystyle
s+d\left(1-\gamma\right)=0\,\,\,\text{or}\,\,\gamma=1+\frac{s}{d}$ (22)

Replacing $\gamma$ for $s$ and $d$ in the equation (1) yields

$D^{\prime\prime}+\left(2s+1-d\right)\frac{D^{\prime}}{R}+\left(B^{2}R^{d+2-s}+\omega^{2}A^{2}R^{2-s}+s\left(s-d\right)\right)\frac{D}{R^{2}}=0$
(23)

which becomes a differential Schrödinger type equation by posing

$D=\frac{Y}{\sqrt{R^{2s+1-d}}}$

yielding

$Y^{\prime\prime}+\left(B^{2}R^{d-s}+\omega^{2}A^{2}R^{-s}-\left(\frac{d^{2}-1}{4}\right)R^{-2}\right)Y=0$
(24)

An approximate solution to this equation can be found by the Wentzel-Kramers-
Brillouin (WKB) theory Bender-Orszag1978 (8). Considering the case of small
frequencies such as

$\omega A<<B^{2}\,\,\,\text{or}\,\,\omega<<4\pi G\frac{\rho_{c}r_{c}}{c_{c}}$
(25)

one poses $\epsilon=\omega AB^{-2}$ with $\epsilon<<1$. The equation (24)
reads then

$\epsilon^{2}Y^{\prime\prime}=\left(-\omega^{2}A^{2}\epsilon^{2}R^{-s}-\omega
A\epsilon
R^{d-s}+\epsilon^{2}\left(\frac{d^{2}-1}{4}\right)R^{-2}\right)Y=Q\left(R\right)Y$
(26)

Let us consider the three following functions of $R$

$S_{0}\left(R\right)=\intop^{R}\sqrt{Q(x)}dx;\,S_{1}\left(R\right)=-\frac{\ln\left(Q\left(R\right)\right)}{4};\,S_{2}\left(R\right)=\intop^{R}\left(\frac{QQ^{\prime\prime}-\frac{5}{4}\left(Q^{\prime}\right)^{2}}{8Q^{\frac{5}{2}}}\right)dx$
(27)

where the first two functions are referred to respectively as the eikonal
function and the transport function and where $Q^{\prime}=dQ(x)/dx$.

If $Q(R)\neq 0$ in the range of interest of $R$ (i.e., $1\leq R\leq R_{max}$)
and under the conditions

$S_{1}\left(R\right)<<\frac{S_{0}\left(R\right)}{\epsilon}\,\,\,;\,\,\epsilon
S_{2}\left(R\right)<<S_{1}\left(R\right)\,\,\,;\,\,\epsilon
S_{2}\left(R\right)<<1$ (28)

the leading orders in the WKB physical optics approximation to the exact
solutions in $Y$ and $D$ reads generally

$\displaystyle Y\left(R\right)$
$\displaystyle=K_{3}\exp\left(\frac{S_{0}\left(1,R\right)}{\epsilon}+S_{1}\left(R\right)\right)+K_{4}\exp\left(-\frac{S_{0}\left(1,R\right)}{\epsilon}+S_{1}\left(R\right)\right)$
(29) $\displaystyle D\left(R\right)$
$\displaystyle=R^{-\left(2s+1-d\right)/2}\left(K_{3}\exp\left(\frac{S_{0}\left(1,R\right)}{\epsilon}+S_{1}\left(R\right)\right)\right.$
$\displaystyle\left.+K_{4}\exp\left(-\frac{S_{0}\left(1,R\right)}{\epsilon}+S_{1}\left(R\right)\right)\right)$
(30)

with $K_{3}$ and $K_{4}$ constants determined by the boundary conditions and
where $S_{0}\left(1,R\right)$ is the eikonal function on the interval
$\left[1,R\right]$. Strictly speaking, the above equality sign should be
replaced by an asymptotic equality sign. The eikonal function
$S_{0}\left(1,R\right)$ reads

$\displaystyle S_{0}\left(1,R\right)=\intop_{1}^{R}\sqrt{Q\left(x\right)}dx$
$\displaystyle=j\omega A\epsilon\intop_{1}^{R}\sqrt{x^{-s}+\frac{1}{\omega
A\epsilon}x^{d-s}-\left(\frac{d^{2}-1}{4\omega^{2}A^{2}}\right)x^{-2}}dx$
$\displaystyle=j\omega
A\epsilon\intop_{1}^{R}\frac{\sqrt{T\left(x\right)}}{x}dx$ (31)

with

$T\left(x\right)=x^{2-s}+\frac{1}{\omega
A\epsilon}x^{2+d-s}-\left(\frac{d^{2}-1}{4\omega^{2}A^{2}}\right)$ (32)

The integral (31) has to be evaluated for specific values of $d$ and $s$. This
evaluation involves most of the time elliptic integrals, which makes it
uneasy.

###  4.2 Solution for $\gamma=3/2$, d = -2 and s = -1

In most nebula models, the gas specific mass and sound speed are decreasing
outward from the central body, with the exponents $d$ and $s$ taking negative
values and $s$ usually in the order of or close to $-1$. We consider here the
particular polytropic case with $s=-1$ and $d=-2$, yielding $\gamma=3/2$.
Cases for other values of $d$, $s$ and $\gamma$ can be solved similarly. The
integral (31) reads

$S_{0}\left(1,R\right)=j\left(\frac{\omega
A\epsilon}{3}I_{1}+\frac{2}{3}I_{2}-\frac{3\epsilon}{4\omega A}I_{3}\right)$
(33)

with

$I_{1}=\intop_{1}^{R}\frac{3x^{2}+\frac{1}{\omega
A\epsilon}}{\sqrt{T\left(x\right)}}dx\,\,\,;\,\,I_{2}=\intop_{1}^{R}\frac{dx}{\sqrt{T\left(x\right)}}\,\,\,;\,\,I_{3}=\intop_{1}^{R}\frac{dx}{x\sqrt{T\left(x\right)}}$
(34)

The cubic trinomial $T\left(x\right)$ (32) has a single real root and,
neglecting terms in $\epsilon^{2}$ and of higher order, it becomes

$T\left(x\right)=\left(x-\frac{3\epsilon}{4\omega
A}\right)\left(x^{2}+\frac{3\epsilon}{4\omega A}x+\frac{1}{\omega
A\epsilon}\right)$ (35)

Posing

$C_{5}=\sqrt{\frac{1}{\omega
A\epsilon}+\frac{9}{8}\left(\frac{\epsilon}{\omega A}\right)^{2}}$

one substitutes $\left(x-\frac{3\epsilon}{4\omega A}\right)$ for $C_{5}z$ in
integral $I_{2}$ and for $C_{5}z^{2}$ in integral $I_{3}$ in (34). The
integrals in (34) are evaluated under the two following conditions

$\frac{\omega A}{B}R<<1\,\,\,;\,\,B^{2}R>>\frac{3}{4}$ (36)

in the range $1\leq R\leq R_{max}$, showing also that the real root of the
trinomial $T\left(x\right)$ (35) is outside the range of interest of $R$,
fulfilling the condition $Q(R)\neq 0$. The other conditions (28) of
application of the WKB theory are verified in Appendix C.

Under the above two conditions, $I_{2}$ and $I_{3}$ become Gradshteyn-
Ryzhik1965 (9)

$\displaystyle I_{2}$
$\displaystyle=\frac{1}{\sqrt{C_{2}}}\left(F\left(\boldsymbol{\varphi}\left(R\right),k\right)-F\left(\boldsymbol{\varphi}\left(1\right),k\right)\right)$
(37) $\displaystyle I_{3}$
$\displaystyle=\frac{1}{\sqrt{\left(C_{2}\right)^{3}}}\left(\left(F\left(\boldsymbol{\varphi}\left(R\right),k\right)-2E\left(\boldsymbol{\varphi}\left(R\right),k\right)\right)-\left(F\left(\boldsymbol{\varphi}\left(1\right),k\right)-2E\left(\boldsymbol{\varphi}\left(1\right),k\right)\right)\right)$
$\displaystyle-\frac{2}{C_{2}}\left(\frac{1}{R-\frac{3\epsilon}{4\omega
A}+C_{2}}\sqrt{\frac{R^{2}+\frac{3\epsilon}{4\omega A}R+\frac{1}{\omega
A\epsilon}}{R-\frac{3\epsilon}{4\omega A}}}\right.$
$\displaystyle\left.-\left(\frac{1}{1-\frac{3\epsilon}{4\omega
A}+C_{2}}\sqrt{\frac{1+\frac{3\epsilon}{4\omega A}+\frac{1}{\omega
A\epsilon}}{1-\frac{3\epsilon}{4\omega A}}}\right)\right)$ (38)

where $F$ and $E$ are the incomplete elliptic integrals of the first and
second kinds of argument and modulus

$\displaystyle\boldsymbol{\varphi}\left(R\right)=2\arctan\left(\frac{1}{C_{2}}\sqrt{R-\frac{3\epsilon}{4\omega
A}}\right)\approx 2\arctan\left(\sqrt{\frac{\omega A}{B}R}\right)$ (39)
$\displaystyle k=\sqrt{\frac{1}{2}-\frac{9\epsilon}{16\omega
A\sqrt{C_{2}}}}\approx\frac{1}{\sqrt{2}}$ (40)

where the conditions (36) were used, yielding also $C_{2}=\frac{B}{\omega A}$.
Replacing in (38), (39) and in (34), (33) yields eventually

$S_{0}\left(1,R\right)=j\epsilon\left(s_{0R}-s_{01}\right)$ (41)

with

$\displaystyle s_{0R}$
$\displaystyle=\left(\frac{2}{3}\sqrt{\frac{B^{3}}{\omega
A}}-\frac{3}{4}\sqrt{\frac{\omega
A}{B^{3}}}\right)F\left(\boldsymbol{\varphi}\left(R\right),k\right)+\frac{3}{2}\sqrt{\frac{\omega
A}{B^{3}}}E\left(\boldsymbol{\varphi}\left(R\right),k\right)$
$\displaystyle+\frac{\sqrt{\left(\omega
A\right)^{2}R^{3}+B^{2}R-\frac{3}{4}}\left(\frac{2}{3}+\frac{3}{2}\frac{B^{3}}{\omega
A}\right)}{\left(B^{2}R-\frac{3}{4}\right)\left(B^{2}R-\frac{3}{4}+\frac{B^{3}}{\omega
A}\right)}$ (42)

and a similar relation for $s_{01}$ with $R=1$ .

The incomplete elliptic integrals $F$ and $E$ are evaluated after an ascending
Landen transformation, yielding the new argument and modulus and the
transformed expressions of $F$ and $E$ to be

$\displaystyle\boldsymbol{\varphi}_{t}$
$\displaystyle=\frac{1}{2}\left(\boldsymbol{\varphi}+\arcsin\left(k\sin\boldsymbol{\varphi}\right)\right)\approx\frac{1}{2}\arcsin\left(2\left(1+\frac{1}{\sqrt{2}}\right)\sqrt{\frac{\omega
A}{B}R}\right)$ (43) $\displaystyle k_{t}$
$\displaystyle=\frac{2\sqrt{k}}{1+k}\approx\frac{2^{\frac{5}{4}}}{1+\sqrt{2}}\approx
0.9852$ (44) $\displaystyle
F\left(\boldsymbol{\varphi}\left(R\right),k\right)$
$\displaystyle=\frac{2}{1+k}F\left(\boldsymbol{\varphi_{t}}\left(R\right),k_{t}\right)$
(45) $\displaystyle E\left(\boldsymbol{\varphi}\left(R\right),k\right)$
$\displaystyle=\left(1+k\right)E\left(\boldsymbol{\varphi_{t}}\left(R\right),k_{t}\right)+\left(1-k\right)F\left(\boldsymbol{\varphi_{t}}\left(R\right),k_{t}\right)$
$\displaystyle-\frac{\left(1+k\right)\tan\boldsymbol{\varphi}}{2+\frac{\sec\boldsymbol{\varphi}\left(1+\frac{1}{k^{2}}\right)\sqrt{1-k^{2}}}{\cos\boldsymbol{\varphi}+\sqrt{\frac{1}{k^{2}}-\sin^{2}\boldsymbol{\varphi}}}}$
(46)

As $k_{t}$ is close to unity, one can use the expansions Gradshteyn-Ryzhik1965
(9)

$\displaystyle F\left(\boldsymbol{\varphi_{t}}\left(R\right),k_{t}\right)$
$\displaystyle=\frac{2}{\pi}\boldsymbol{K^{\prime}}\ln\left(\tan\left(\frac{\boldsymbol{\varphi}_{t}}{2}+\frac{\pi}{4}\right)\right)-\sin\boldsymbol{\varphi}_{t}\sec^{2}\boldsymbol{\varphi}_{t}\left(a_{0}-\frac{2}{3}a_{1}\tan^{2}\boldsymbol{\varphi}_{t}+...\right)$
(47) $\displaystyle
E\left(\boldsymbol{\varphi_{t}}\left(R\right),k_{t}\right)$
$\displaystyle=\frac{2}{\pi}\boldsymbol{E^{\prime}}\ln\left(\tan\left(\frac{\boldsymbol{\varphi}_{t}}{2}+\frac{\pi}{4}\right)\right)+\sin\boldsymbol{\varphi}_{t}\sec^{2}\boldsymbol{\varphi}_{t}\left(b_{0}-\frac{2}{3}b_{1}\tan^{2}\boldsymbol{\varphi}_{t}+...\right)$
(48)

where

$\boldsymbol{K^{\prime}}=\boldsymbol{K}\sqrt{1-k_{t}^{2}}\,\,\,;\,\,\boldsymbol{E^{\prime}}=\boldsymbol{E}\sqrt{1-k_{t}^{2}}$

are the complete integrals of the first and second kinds and
$a_{0},a_{1},...,b_{0},b_{1},...$ are decreasing coefficients, functions of
$k_{t}$. As $k_{t}\approx 0.9852$,
$\boldsymbol{K^{\prime}}\approx\boldsymbol{E^{\prime}}\approx\pi/2$ within
$7.5\times 10^{-3}$, $a_{0}\approx 7.5\times 10^{-3}$, $a_{1}\approx 10^{-4}$,
$b_{0}\approx-7.5\times 10^{-3}$, etc … , yielding

$F\left(\boldsymbol{\varphi_{t}}\left(R\right),k_{t}\right)\approx
E\left(\boldsymbol{\varphi_{t}}\left(R\right),k_{t}\right)\approx\ln\left(\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}\right)\sqrt{\frac{\omega
A}{B}R}\right)$ (49)

Replacing in (45), (46) and (42) and using conditions (36) to neglect small
terms, it yields

$s_{0R}=W\ln\left(\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}\right)\sqrt{\frac{\omega
A}{B}R}\right)+\frac{2}{3}\sqrt{\left(\omega
A\right)^{2}R^{3}+B^{2}R-\frac{3}{4}}$ (50)

and similarly for $s_{01}$ for $R=1$, with $W=W(\omega,A,B)$

$W=\frac{8}{3}\left(1-\frac{1}{\sqrt{2}}\right)\sqrt{\frac{B^{3}}{\omega
A}}+3\sqrt{\frac{\omega
A}{2B^{3}}}\approx\frac{8}{3}\left(1-\frac{1}{\sqrt{2}}\right)\sqrt{\frac{B^{3}}{\omega
A}}$ (51)

under the conditions (36).

The complex perturbed specific mass $D_{C}$ (30) and its real part read

$\displaystyle D_{C}$ $\displaystyle=\frac{B}{\sqrt{j\omega
A}\sqrt[4]{\left(\omega
A\right)^{2}R^{3}+B^{2}R-\frac{3}{4}}}\left(K_{3}\exp\left(j\left(s_{0R}-s_{01}\right)\right)\right.$
$\displaystyle\left.+K_{4}\exp\left(-j\left(s_{0R}-s_{01}\right)\right)\right)$
(52) $\displaystyle D_{R}$ $\displaystyle=K_{5}\frac{B}{\sqrt{\omega
A}\sqrt[4]{\left(\omega
A\right)^{2}R^{3}+B^{2}R-\frac{3}{4}}}\cos\left(s_{0R}-s_{01}-\kappa\right)$
(53)

with

$K_{5}=\sqrt{K_{3}^{2}+K_{4}^{2}}\,\,\,;\,\,\kappa=\arctan\left(\frac{K_{3}-K_{4}}{K_{3}+K_{4}}\right)$

The extrema of $D_{R}$ are solutions of

$D_{R}^{\prime}=-K_{5}C_{6}\frac{1}{\sqrt[4]{\left(\omega
A\right)^{2}R^{3}+B^{2}R-\frac{3}{4}}}\sin\left(s_{0R}-s_{01}-\kappa+\tau\right)=0$
(54)

with

$C_{6}=\frac{4}{3}\left(1-\frac{1}{\sqrt{2}}\right)\frac{B^{\frac{5}{2}}}{\omega
A}\,\,\,;\,\,\tau=\arctan\left(\frac{3}{8}\left(1+\frac{1}{\sqrt{2}}\right)\sqrt{\frac{\omega
A}{B^{3}}}\right)$

where the conditions (36) were used to neglect small terms. The zeros and
extrema of $D_{R}$ are given by

$\displaystyle\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}\right)\sqrt{\frac{\omega
A}{B}R}\exp\left(\frac{2}{3}\frac{\sqrt{\left(\omega
A\right)^{2}R^{3}+B^{2}R-\frac{3}{4}}}{W}\right)$
$\displaystyle=\exp\left(\frac{s_{01}+\varphi_{5}+\kappa}{W}\right)\exp\left(\frac{\pi
n}{W}\right)$ (55)

where $n$ are non-negative integers and $\varphi_{5}=\pi/2$ for $D_{R}$ and
$\varphi_{5}=-\tau$ for $D_{R}^{\prime}$. The exponential term in the above
left hand side part reduces to

$\exp\left(\frac{2}{3}\frac{\sqrt{\left(\omega
A\right)^{2}R^{3}+B^{2}R-\frac{3}{4}}}{W}\right)\approx\exp\left(\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}\right)\sqrt{\frac{\omega
A}{B}R}\right)$ (56)

under the conditions (36), and it can be neglected when multiplied by its
argument, small by (36). The zeros and extrema of $D_{R}$ are then given in
good approximation by

$R=\alpha_{5}\left(\beta_{5}^{\backprime}\right)^{n};\,\alpha_{5}=16\left(\frac{3}{2}-\sqrt{2}\right)\frac{B}{\omega
A}\exp\left(\frac{2\left(s_{01}+\varphi_{5}+\kappa\right)}{W}\right);\,\beta_{5}^{\backprime}=\exp\left(\frac{2\pi}{W}\right)$
(57)

The perturbed radial velocity $U$ and the specific mass flux radial momentum
$\varPhi$ are found from (2) and (3), with (53). However, their evaluation
requires the resolution of a new elliptic integral. To avoid this and as there
are no zeros due to the transport function $S_{1}(R)$ (in the 4-th root of the
trinomial term in $R$) in (53), we evaluate $U$ and $\varPhi$ by neglecting
$S_{1}(R)$. This is the geometrical optics approximation, which gives the most
rapidly varying component (controlling factor) of the leading behaviour of the
exact solution. In the geometrical optics (g.o.) approximation, the real part
of the complex perturbed specific mass (53) reduces then to

$D_{R\,g.o.}=K_{5}\frac{B}{\sqrt{\omega
A}}\cos\left(s_{0R}-s_{01}-\kappa\right)$ (58)

Under the conditions (36), the term $s_{0R}$ (50) can be written approximately

$s_{0R}\approx
W\left(\ln\left(\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}\right)\sqrt{\frac{\omega
A}{B}R}\right)+\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}\right)\sqrt{\frac{\omega
A}{B}R}\right)$ (59)

as was already done in (56). The radial velocity $U$ and the specific mass
flux radial momentum $\varPhi$ read then in the geometrical optics
approximation

$\displaystyle U_{g.o.}$ $\displaystyle=-\kappa
K_{5}C_{7}\frac{r_{c}}{\rho_{c}}R^{3}\sin\left(s_{0R}-s_{01}-\kappa+\sigma\right)$
(60) $\displaystyle\varPhi_{g.o.}$ $\displaystyle=-\kappa
K_{5}C_{7}r_{c}^{2}R^{2}\sin\left(s_{0R}-s_{01}-\kappa+\sigma\right)$ (61)

with

$C_{7}=\frac{2B}{\sqrt{\omega
A\left(W^{2}+16\right)}}\,\,\,;\,\,\sigma=\arctan\left(3\left(1+\frac{1}{\sqrt{2}}\right)\sqrt{\frac{\omega
A}{B^{3}}}\right)$

showing that their zeros are given like the zeros and extrema of $D_{R}$ with
$\beta_{5}^{\backprime}$ (57) and $\varphi_{5}=-\sigma$ in $\alpha_{5}$ (57).

By the condition (25), one has $\sigma<<1$ and $\tau<<1$. The initial phase
between $D$ and $U$ (or $\varPhi$) is $\left(\pi/2\right)-\sigma\approx\pi/2$,
while the initial phase between $U$ (or $\varPhi$) and $D^{\prime}$ is
$\left(\tau-\sigma\right)\approx 0$.

Let us note that, in the physical optics approximation, retaining the term
$S_{1}(R)$ in $D_{R}$ would change the amplitudes of $U$ and $\varPhi$ (by the
addition of decreasing terms in $R$ in front of the sin function) and it would
change the coefficient of $\sqrt{\omega A/B^{3}}$ and add negligible terms in
the argument of the $\arctan$ of $\sigma$ . But the distance ratio of two
successive zeros of $U$ (or $\varPhi$) is unaffected and is still given by
$\beta_{5}^{\backprime}$ (57) in both WKB approximations.

The distances ratio of two successive maxima of $D$ is given by

$\beta_{5}=\left(\beta_{5}^{\backprime}\right)^{2}$ (62)

with

$\displaystyle\beta_{5}$
$\displaystyle=\exp\left(\frac{\pi}{\frac{2}{3}\left(1-\frac{1}{\sqrt{2}}\right)\frac{1}{\sqrt{\omega}}\left(4\pi
G\rho_{c}\right)^{\frac{3}{4}}\frac{r_{c}}{c_{c}}+\frac{3}{4}\sqrt{\frac{\omega}{2}}\left(4\pi
G\rho_{c}\right)^{-\frac{3}{4}}\frac{c_{c}}{cr_{c}}}\right)$
$\displaystyle\approx\exp\left(\frac{3\pi\left(1+\frac{1}{\sqrt{2}}\right)c_{c}\sqrt{\omega}}{r_{c}\left(4\pi
G\rho_{c}\right)^{\frac{3}{4}}}\right)$ (63)

where the approximated value of W (51) is used in (63). The distances ratio
$\beta_{5}$ is a constant, function of the perturbations circular frequency
$\omega$ and of the disc reference characteristics. The period of the small
perturbations must be larger than a minimum value $P_{m}$, which is the
greatest of the two values that can be deduced from the two conditions (25)
and (36) on $\omega$, applied to the whole range of radial distances up to
$R_{max}$, yielding

$P_{m}=\frac{c_{c}}{2G\rho_{c}r_{c}}\,\,\,\text{or}\,\,\,P_{m}=\sqrt{\frac{\pi}{G\rho_{c}}}R_{max}$
(64)

##  5 Conclusions

We have extended the resolution of the wave-like equation of perturbed
specific mass deduced in Paper 1 to two other general cases. The solution for
the "polytropic case" could not be solved generally as one must choose
particular values of $d$ and $s$, fixing the value of the polytropic index
$\gamma$. However, a solution was found in the WKB physical-optics
approximation for an important particular case ($\gamma=3/2$ with $d=-2$ and
$s=-1$).

For the two above cases, conclusions similar to the ones of Paper 1 are
reached concerning the functions $D$, $D^{\prime}$, $U$ and $\varPhi$, namely
that, first, $D$ has a sign opposite to the signs of $D^{\prime}$, $U$ and
$\varPhi$; second, the functions $D^{\prime}$, $U$ and $\varPhi$ are in phase
and have an initial phase difference of approximately $\pi/2$ with respect to
the function $D$; third, the zeros of $U$ corresponds to the extrema of $D$
and vice-versa; and finally, for increasing $R$, the functions $U$ and
$\varPhi$ are positive (respectively negative) between successive minima and
maxima (respectively successive maxima and minima) of $D$. This situation
yields radial outward flows of gas between successive minima and maxima of $D$
and radial inward flows of gas between successive maxima and minima of $D$,
that would eventually form annular structures of gas, with axial radii
corresponding to the distances of maxima of the gas perturbed specific mass.
Furthermore, the maxima of the gas perturbed specific mass are found to be
exponentially spaced for the two cases and their distances ratios are
constants depending on discs characteristics and on the circular frequency of
the perturbations. These results can be applied to protoplanetary and proto-
satellite discs.

###### Acknowledgements.

We thank Dr D. Poelaert for some mathematical advice for Section 3 and Dr T.
Dewandre for suggesting the WKB resolution of Section 4 and for valuable
discussions.

## Appendix A

The $k$-th complex coefficient of the hyperbolic Bessel functions in the
series of (7) reads

$H_{k}=\frac{\left(-1\right)^{k}\left(2\nu\right)_{k}\left(b-2a\right)_{k}}{k!\,\left(b\right)_{k}}$
(65)

where $\left(x\right)_{k}$ are Pochhammer polynomials. This expression can be
written $H_{k}=H_{Rk}+jH_{Ik}$ by posing $a=a_{R}+ja_{I}$ and by developing
the Pochhammer polynomials, yielding

$H_{Rk}=T_{k}\left(P_{Rk}Q_{Rk}-P_{Ik}Q_{Ik}\right);\,H_{Ik}=T_{k}\left(P_{Rk}Q_{Ik}-P_{Ik}Q_{Rk}\right)$
(66)

with

$\displaystyle P_{Rk}$
$\displaystyle=\sum_{m=0}^{L_{R}}\left(\sum_{q=2m}^{k}\left(-1\right)^{m+q}\left({}_{2m}^{q}\right)S_{k}\left(q\right)a_{R}^{q-2m}a_{I}^{2m}\right)$
(67) $\displaystyle Q_{Rk}$
$\displaystyle=\sum_{m=0}^{L_{R}}\left(-1\right)^{m}\left|S_{k}\left(2m\right)\right|a_{I}^{2m}$
(68) $\displaystyle P_{Ik}$
$\displaystyle=\sum_{m=0}^{L_{I}}\left(\sum_{q=2m+1}^{k}\left(-1\right)^{m+q}\left({}_{2m+1}^{q}\right)S_{k}\left(q\right)a_{R}^{q-\left(2m+1\right)}a_{I}^{2m+1}\right)$
(69) $\displaystyle Q_{Ik}$
$\displaystyle=\sum_{m=0}^{L_{I}}\left(-1\right)^{m}\left|S_{k}\left(2m+1\right)\right|a_{I}^{2m+1}$
(70) $\displaystyle T_{k}$
$\displaystyle=\frac{\left(-1\right)^{k}}{k!\sum_{m=0}^{k}\left|S_{k}\left(m\right)\right|b^{m}}$
(71)

where $\left({}_{2m}^{q}\right)$ are the binomial coefficients,
$\left|S_{k}\left(m\right)\right|$ is the absolute value of the Stirling
numbers of the first kind and $L_{R}=L_{I}=k/2$ for $k$ even and
$L_{R}=\left(k-1\right)/2$, $L_{I}=\left(k+1\right)/2$ for $k$ odd.

## Appendix B

Writing $\nu=\nu_{R}+j\nu_{I}$, with

$\nu_{R}=\frac{3d+2}{2d\gamma}\,\,\,;\,\,\nu_{I}=-y=\frac{-B^{2}}{\omega
Aa\left(\frac{3d+2}{\gamma}\right)-\left(3d+4\right)}$ (72)

the term $G\left(\nu\right)$ in (10) can be written
$G\left(\nu\right)=G_{R}+jG_{I}$ with, for $\nu_{1}$,

$\displaystyle G_{1R}$
$\displaystyle=\sum_{m=0}^{\infty}\frac{1}{m!}\left(\sum_{n=0}^{m}P_{n}+\sum_{n=m+1}^{\infty}\frac{T_{n}}{C_{n}}\right);G_{1I}=\sum_{m=0}^{\infty}\frac{1}{m!}\left(\sum_{n=0}^{m}Q_{n}+\sum_{n=m+1}^{\infty}\frac{W_{n}}{C_{n}}\right)$
(73) $\displaystyle P_{n}$
$\displaystyle=\frac{1}{n!}\sum_{p=0}^{L_{1}}\sum_{q=2p}^{m-n}\left(-1\right)^{p-n}\left({}_{2p}^{q}\right)\left(S_{m-n}\left(q\right)\right)\nu_{R}^{q-2p}\nu_{I}^{2p}$
(74) $\displaystyle Q_{n}$
$\displaystyle=\frac{1}{n!}\sum_{p=0}^{L_{2}}\sum_{q=2p+1}^{m-n}\left(-1\right)^{p-n}\left({}_{2p+1}^{q}\right)\left(S_{m-n}\left(q\right)\right)\nu_{R}^{q-2p-1}\nu_{I}^{2p+1}$
(75) $\displaystyle T_{n}$
$\displaystyle=\sum_{p=0}^{L_{3}}\sum_{q=2p}^{n-m}\left(-1\right)^{m+p+q}\left({}_{2p}^{q}\right)\left(S_{n-m}\left(q\right)\right)\left(\nu_{R}+1\right)^{q-2p}\nu_{I}^{2p}$
(76) $\displaystyle W_{n}$
$\displaystyle=\sum_{p=0}^{L_{4}}\sum_{q=2p+1}^{n-m}\left(-1\right)^{m+p+q+1}\left({}_{2p+1}^{q}\right)\left(S_{n-m}\left(q\right)\right)\left(\nu_{R}+1\right)^{q-2p-1}\nu_{I}^{2p+1}$
(77) $\displaystyle C_{n}$ $\displaystyle=n!\left(T_{n}^{2}+W_{n}^{2}\right)$
(78)

with $L_{1}=L_{2}=-L_{3}=-L_{4}=\left(m-n\right)/2$ for $\left(m-n\right)$
even and $L_{1}=-L_{4}=\left(m-n-1\right)/2$ and
$L_{2}=-L_{3}=\left(m-n+1\right)/2$ for $\left(m-n\right)$ odd.

Similar relations are found for $\nu_{2}$ replacing $\nu_{R}$ by $-\nu_{R}$.

The complex constants in (12) read
$\varLambda=\left|\varLambda\right|\exp\left(j\lambda\right)$ and
$M=\left|M\right|\exp\left(j\mu\right)$ with

$\displaystyle\left|\varLambda\right|$
$\displaystyle=2^{2\left(\frac{3d+2}{\gamma
d}\right)}\sqrt{\left(\frac{2}{\left|d\right|}\omega
A\right)^{\left(1-\left(\frac{3d+2}{\gamma d}\right)\right)}}\exp\left(-\pi
y\right)\frac{\left|G_{1}\right|}{\left|\nu_{1}\right|}$ (79)
$\displaystyle\lambda$
$\displaystyle=\frac{\pi}{4}\left(1-\left(\frac{3d+2}{\gamma
d}\right)\right)-4y\ln\left(2\right)+\arg\left(G_{1}\right)-\arg\left(\nu_{1}\right)$
(80) $\displaystyle\left|M\right|$
$\displaystyle=2^{-2\left(\frac{3d+2}{\gamma
d}\right)}\sqrt{\left(\frac{2}{\left|d\right|}\omega
A\right)^{\left(1+\left(\frac{3d+2}{\gamma d}\right)\right)}}\exp\left(-\pi
y\right)\frac{\left|G_{2}\right|}{\left|\nu_{2}\right|}$ (81)
$\displaystyle\mu$
$\displaystyle=\frac{\pi}{4}\left(1+\left(\frac{3d+2}{\gamma
d}\right)\right)-4y\ln\left(2\right)+\arg\left(G_{2}\right)-\arg\left(\nu_{2}\right)$
(82)

where $\left|G\right|$ and $\arg\left(G\right)$ are the modulus and the
argument of the complex valued function $G\left(\nu\right)$.

## Appendix C

We verify the conditions (28) of application of the WKB physical optics
approximation for the moduli

$\left|S_{1}\left(R\right)\right|<<\left|\frac{S_{0}\left(R\right)}{\epsilon}\right|\,\,\,;\,\,\left|\epsilon
S_{2}\left(R\right)\right|<<\left|S_{1}\left(R\right)\right|\,\,\,;\,\,\left|\epsilon
S_{2}\left(R\right)\right|<<1$ (83)

with

$\displaystyle\left|\frac{S_{0}\left(R\right)}{\epsilon}\right|$
$\displaystyle=W\ln\left(2\left(1+\frac{1}{\sqrt{2}}\right)\sqrt{\frac{\omega
A}{B}R}\right)+\frac{2}{3}\sqrt{\left(\omega
A\right)^{2}R^{3}+B^{2}R-\frac{3}{4}}$ (84)
$\displaystyle\left|S_{1}\left(R\right)\right|$
$\displaystyle=\frac{1}{4}\sqrt{\pi^{2}+\left(\ln\left(\left(\frac{\omega
A}{RB^{2}}\right)^{2}\left(\left(\omega
A\right)^{2}R^{3}+B^{2}R-\frac{3}{4}\right)\right)\right)^{2}}$ (85)
$\displaystyle\left|\epsilon S_{2}\left(R\right)\right|$
$\displaystyle=\frac{1}{32}\intop^{R}\left(\frac{4\left(-2B^{2}x+\frac{9}{2}\right)\left(\left(\omega
A\right)^{2}x^{3}+B^{2}x-\frac{3}{4}\right)}{x\left(\left(\omega
A\right)^{2}x^{3}+B^{2}x-\frac{3}{4}\right)^{\frac{5}{2}}}\right.$
$\displaystyle+\left.\frac{5\left(\left(\omega
A\right)^{2}x^{3}+B^{2}x\right)^{2}+15\left(\left(\omega
A\right)^{2}x^{3}-B^{2}x+\frac{3}{4}\right)}{x\left(\left(\omega
A\right)^{2}x^{3}+B^{2}x-\frac{3}{4}\right)^{\frac{5}{2}}}\right)dx$ (86)

and $W$ is given by (51). The first condition (83) reads

$\left|\frac{S_{0}\left(R\right)}{\epsilon}\right|-\left|S_{1}\left(R\right)\right|>>0$
(87)

One easily verifies that the left hand side of the inequality tends towards
positive infinity either when taking the limit for $R\rightarrow+\infty$ with
$\epsilon$ constant or when taking the limit for $\epsilon\rightarrow 0$ (or
$\omega\rightarrow 0$) with $R$ constant. The verification of the second and
third conditions implies the solution of the uneasy elliptic integral (86).
One can get some insights into the verification of these two conditions
without solving (86), although, strictly speaking, this method is not exactly
rigorous. Looking at the behaviour of the dominant terms, we take the limit
for $\epsilon\rightarrow 0$ (or $\omega\rightarrow 0$) under the integral
sign, which yields

$\left|\epsilon
S_{2}\right|\approx\frac{1}{8}\int^{R}\left(\frac{-2B^{2}}{\sqrt{\left(B^{2}x-\frac{3}{4}\right)^{3}}}+\frac{3}{4}\frac{1}{x\sqrt{\left(B^{2}x-\frac{3}{4}\right)^{3}}}+\frac{5}{4}\frac{B^{4}x}{\sqrt{\left(B^{2}x-\frac{3}{4}\right)^{5}}}\right)dx$
(88)

which solves easily in

$\left|\epsilon
S_{2}\right|\approx\left|\frac{1}{16}\left(\frac{5}{4}\frac{1}{\sqrt{\left(B^{2}R-\frac{3}{4}\right)^{3}}}+\frac{1}{\sqrt{B^{2}R-\frac{3}{4}}}-\frac{1}{2\sqrt{3}}\arctan\left(\frac{1}{\sqrt{\frac{4}{3}B^{2}R-1}}\right)\right)\right|$
(89)

The second condition, for $\epsilon\rightarrow 0$ (or $\omega\rightarrow 0$),
is always satisfied, provided that $B^{2}R\neq 3/4$ for all $R$, as
$\left|S_{1}\right|\rightarrow+\infty$. The third condition is also satisfied
provided that $B^{2}R>>3/4$ for all $R$, which is the condition (36). Taking
now the limit for $R\rightarrow+\infty$ in (89) yields that $\left|\epsilon
S_{2}\right|\rightarrow 0$, which satisfies both the second and third
conditions (83).

## References

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# Remember the context! ASR slot error correction through Memorization

###### Abstract

Accurate recognition of slot values such as domain specific words or named
entities by automatic speech recognition (ASR) systems forms the core of the
Goal-oriented Dialogue Systems. Although it is a critical step with direct
impact on downstream tasks such as language understanding, many domain
agnostic ASR systems tend to perform poorly on domain specific or long tail
words. They are often supplemented with slot error correcting systems but it
is often hard for any neural model to directly output such rare entity words.
To address this problem, we propose $k$-nearest neighbor ($k$-NN) search that
outputs domain-specific entities from an explicit datastore. We improve error
correction rate by conveniently augmenting a pretrained joint phoneme and text
based transformer sequence to sequence model with $k$-NN search during
inference. We evaluate our proposed approach on five different domains
containing long tail slot entities such as full names, airports, street names,
cities, states. Our best performing error correction model shows a relative
improvement of 7.4% in word error rate (WER) on rare word entities over the
baseline and also achieves a relative WER improvement of 9.8% on an out of
vocabulary (OOV) test set.

Index Terms—  speech recognition, slot error correction, transformer,
$k$-nearest-neighbor search, long tail recognition

##  1 Introduction

Over the recent years, open domain Automatic Speech Recognition (ASR) models
have become very powerful tools pushing the state-of-the-art performance in
dialogue systems [1, 2, 3]. However, these domain agnostic models might rely
on limited vocabulary during training and recognition. When such a model is
used for recognizing domain specific long tail and rare word entities such as
street names, retail business names, email domain names, first names, last
names etc., the outputs gets mapped to the most similar sounding words in the
recognition lexicon (pronunciation dictionaries). When these ASR models are
used as front ends in end to end goal oriented dialogue systems, failure to
recognize slots / entities leads to failure in dialogue state change.

Various methods have been proposed to mitigate this issue, which include using
mixture of domain experts [4], context based interpolation weights [5] and
second-pass rescoring through domain-adapted models [6] to feature based
domain adaptation [7]. In [8, 9], user-provided speech patterns were leveraged
for on-the-fly adaptation. Yet another way to solve this problem is called ASR
error correction. This task is modelled as machine translation problem, where
one maps incorrect ASR outputs to domain words. Since ASR outputs are readily
available together with the ground truth as parallel data, one can directly
train sequence-to-sequence (seq2seq) models [10, 11, 12] and use them as post
transcription editors on top of the domain-general ASR outputs.

In machine translation, [13] showed that contexts which are close in
representation space are more likely to be followed by same target word.
Further, in [14], it was shown that learning similarity between sequence of
texts is easier for a language model than predicting the next word
(particularly the one from the long tail). On a similar note, we identify that
seq2seq models in ASR error correction tasks find it difficult to correct ASR
output to a very rare domain-specific word. We hypothesize that it might be
easier for the model to explicitly memorize the speech and text pattern in an
external datastore and then retrieve it using k-nearest neighbours approach to
boost the softmax scores of such rare words. Following this idea, we introduce
$k$-PAT (k-nearest neighbours based Phone Augmented Transformer) and
demonstrate improvements in word error rate and slot accuracy across four
different domains. We summarize the major contributions of our work as follows

  * •

We propose a novel method to adapt $k$-NN search for error correction task by
memorizing both phone and text based representations that can be used on top
of any error correcting model without the need of any additional tuning or
training.

  * •

We demonstrate the benefits of using such a datastore with significant
improvements in WER and accuracy, particularly for rare words and out of
vocabulary entities, which are critical for downstream goal-oriented dialog
system tasks.

  * •

We explore the effects of using lightweight domain specific datastores for
efficient and fast adaptation of ASR error correction models on-the-fly.

Fig. 1: Proposed PAT model with $k$-NN datastore augmentation. The left half
of the figure is the PAT model and the botton right half is the $k$-NN
datastore creation. The top half of the model shows the inference logic.

##  2 Related Work

ASR error correction task is a well studied problem in literature and usually,
it has been treated as a post processing task along with other tasks like
punctuation prediction [15, 16] and inverse text normalization [17]. The prior
works have explored the problem using a variety of subtasks including grammar
error correction, improving human readability [18], entity retrieval [19] etc.
In [20], the authors use an RNN based external language model along with a
stacked RNN based seq2seq spelling correction model for improving a baseline
Listen Attend and Spell (LAS) based ASR system. Recently, pre-trained language
models such as BERT, Transformer-XL and RoBERTa have been applied to the task
and have been shown to be effective in making significant gains in WER [12,
18]. The work in [10] adapts an open domain google ASR model to medical domain
using a Transformer based seq2seq model. All these works also show qualitative
improvements in readability, grammatical and semantic correctness of corrected
sentences to support the quantitative metrics. However, all the above
approaches model ASR error correction task as a text only approach. ASR also
consists of phonetic information which can be used to disambiguate and gain
further improvements.

In [19], an Augmented Transformer model is proposed which leverages phonetic
along with text for correcting ASR outputs. They show that jointly encoding
both phoneme and text information helps in improving entity retrieval compared
to a vanilla text transformer. In this work, we use this as our base model and
augment $k$-NN datastore creation and inference techniques [13, 14] for
improving the model without any further training. To this end, we generalize
$k$-NN datastore creation idea [14] from storing just context based text
patterns to using joint text and speech contextual information that provides
us an additional leverage in task-oriented dialogue systems. In addition to
that, we bring in a novel idea and experimentation on smaller domain specific
datastores which none of the existing works cover.

##  3 Models

###  3.1 Phone Augmented Transformer

Following [19], we implement a Phone Augmented Transformer (PAT) which
consists of phonetic and text encoders and a joint decoder. The encoders and
the decoder are made of stacked multi headed self attention layers [21]. The
input of the transformer model is an ASR output word sequence
$U=\\{u_{1},u_{2},...u_{N}\\}$ tokenized by sentence-piece tokenization[22]
and a phoneme sequence $P=\\{p_{1},p_{2},...p_{M}\\}$ generated from the ASR
output words using the ASR lexicon. The output of the model is the rewritten
ASR output represented by a new sequence of words
$W=\\{w_{1},w_{2},...w_{O}\\}$.

The Phonetic encoder transforms the phone sequence $P$ into a sequence of
hidden representations
$H_{p}=\\{h_{p_{1}},h_{p_{2}},...h_{p_{M}}\\}$ and the text encoder encodes
the sequence of tokenized words $U$ into hidden representations
$H_{u}=\\{h_{u_{1}},h_{u_{2}},...h_{u_{N}}\\}$

The decoder also consists of stacked multi-headed self attention layers. Each
decoder self attention layer has a decoder self attention, and two encoder-
decoder self attentions. The two encoder-decoder attention layers are used to
fuse both input text and phonetic information. The encoder-decoder attention
layers are sequentially stacked on top of the decoder attention layer. First,
the text encoder-decoder attention $A_{u}$ between encoder output $H_{u}$ and
the decoder attention output $H_{w}$ is computed using:

$\displaystyle
A_{u}=softmax(\frac{(Q_{u}H_{u})(K_{u}H_{u})}{(T\sqrt{d_{k}})})(V_{u}H_{w})$
(1)

where $Q_{u},K_{u},V_{u}\epsilon R^{d_{k}\texttimes d_{k}}$ are the parameters
of the attention layer, and $d_{k}$ is the dimension of the output from the
self-attention layer

We then fuse the phone encoder output $H_{p}$ with the previous attention
layer output $A_{u}$ using:

$\displaystyle
A_{p}=softmax(\frac{(Q_{p}H_{p})(K_{p}H_{p})}{(T\sqrt{d_{k}})})(V_{p}A_{u})$
(2)

where $Q_{p},K_{p},V_{p}\epsilon R^{d_{k}\texttimes d_{k}}$ are the parameters
of the attention layer.

The attention output is then passed through a feed forward layer and a layer
normalization layer to generate the decoder outputs. These outputs can be fed
to another layer or be used as the final output representations for predicting
the output word sequence. The final decoder output consists of a sequence of
hidden representations $D_{w}=\\{d_{w_{1}},d_{w_{2}},...d_{w_{O}}\\}$ which is
passed through a feed forward layer followed by a softmax layer to predict the
tokenized target words $W=\\{w_{1},w_{2},...w_{O}\\}$. For training the
Transformer model, we follow vanilla Transformer training techniques except in
our case, we also have an additional encoder. Additionally, we use the text
input dropout and clean input techniques mentioned in [19].

###  3.2 Phone Augmented Transformer with k-NN Search

In this section, we propose the $k$-nearest-neighbors ($k$-NN) retrieval
augmentation to the PAT model calling it $k$-PAT. The $k$-PAT model is an
inference augmentation to the PAT model based on the works of $k$-NN language
model [14] and $k$-NN Machine Translation [13]. Figure 1 shows a schematic
overview of the proposed model. The crux of the model is in a memorization
step before inference and a score interpolation step during inference. The
decoder output representations $D_{w}$ of all training data are memorized into
an external datastore during memorization step. During inference step, the $k$
nearest neighbors of the decoder output representations are retrieved from the
cached datastore. A softmax distribution computed over these retrieved
neighbors is interpolated with the decoder softmax scores as show in figure 1.

Interpolating the distribution from a large parametric neural error correction
model, which inherently encodes short term memory, with a non-parametric
episodic memory component like $k$-NN memory cache allows for an effective way
to adapt the final probability distribution towards domain-specific rare words
without having to retrain the underlying model.

####  3.2.1 Memorization - Datastore creation

The datastore creation step involves generating key value pairs $(k,v)$ for
all training data and storing them in an external memory. Let
$D_{w}=\\{d_{w_{1}},d_{w_{2}},...d_{w_{i-1}},...d_{w_{O}}\\}$ be the sequence
of decoder output representations and $W=\\{w_{1},w_{2},...w_{i},...w_{O}\\}$
the sequence of target words. Let $f$ be a function which maps $d_{w_{i-1}}$
to a fixed vector $e_{i-1}$. Then, given the target word $w_{i}$, we store
$(e_{i-1},w_{i})$ as the key value pair $(k_{j},v_{j})$. These pairs are
generated for all $d_{w_{i}}$ in $D_{w}$ and for all $D_{w}$ generated with
the training set. This datastore we create maps the contextual information
(input phonetic and text) implicitly encoded in the hidden state outputs of
the decoder to the target word in the sequence.

In our work we use a dot product self-attention based pooling described in
equation 3 for creating the key vector $e_{i-1}$ for the decoder output
$d_{i-1}$.

$\displaystyle S=softmax(D_{w}\cdot D_{w}^{T})$ (3) $\displaystyle
e_{i-1}=f(d_{w_{i-1}})=s_{i-1}\cdot D_{w}$
$\displaystyle(k_{j},v_{j})=(e_{i-1},w_{i})$

where $S=\\{s_{1},s_{2},...s_{O-1}\\}$ and $i\epsilon(2,O)$ is a sequence of
dot product attention weights. Generally speaking, any hidden layer from the
decoder can be used for generating the keys. In our experiments, we found that
the key vector representation obtained from equation 3 works best. This self-
attention computation allows for look ahead context for each key
representation $e_{i-1}$. The look ahead helps learning information about the
future decoded word. Therefore, the $k$-NN datastore created from these
representations has implicit information about the word to be decoded. This
form of key value generation differs from the original paper [13] where the
key representation only has information about the past. During inference
however, we do not use the look ahead in the self attention.

####  3.2.2 Inference - $k$-NN Retrieval

At inference time, given a source ASR output :

$u=\\{u_{1},u_{2},...u_{N}\\}$

and the phonetic sequence :

$p=\\{p_{1},p_{2},...p_{M}\\}$

the PAT model produces a distribution over vocabulary as :

$p_{PAT}(y_{i}|u,p,\hat{y}_{1:i-1})$

where $y_{i}$ is the target token at every time step and $\hat{y}$ represents
the corrected output. In parallel, the model also generates the output
representation $d_{w_{i-1}}$, which is used to generate the search key :
$f(d_{w_{i-1}})$. During decoding, we use this key to retrieve the $k$ nearest
neighbor keys from the datastore and the corresponding values. In our
experiments, we use $L2$ distance for choosing the neighbors. Once the nearest
neighbors are retrieved, we use the technique described in [13] for assigning
the probability scores to the retrieved values and interpolate them with the
decoder softmax scores using equation 4. For fast retrieval of key, value
pairs during inference, we use a datastore index generated using FAISS [23,
14, 13].

$\displaystyle\begin{split}p_{kPAT}(y_{i}|u,p,\hat{y}_{1:i-1})=(1-\lambda)p_{PAT}(y_{i}|u,p,\hat{y}_{1:i-1})\\\
+\lambda p_{kNN}(y_{i}|u,p,\hat{y}_{1:i-1})\end{split}$ (4)

We set $\lambda$ to 0.5 in our experiments to obtain the final interpolation
weights.

| Domain | PAT |  $k$-PAT
---|---|---|---
|  | WER | Recall | Accuracy | WER (WERR) | Recall | Accuracy
1 | Combined | 10.7 | 0.70 | 82.0 | 9.9 (7.5) | 0.71 | 82.7
2 | Full Names | 16.5 | 0.77 | 81.3 | 15.9 (3.6) | 0.78 | 82.1
3 | Airports | 8.6 | 0.70 | 86.8 | 7.4 (13.9) | 0.74 | 89.0
4 | Street Names | 10.2 | 0.68 | 78.0 | 9.9 (2.9) | 0.69 | 79.4
5 | Cities & States | 8.5 | 0.70 | 89.7 | 8.0 (5.9) | 0.72 | 90.3
6 | OOV | 34.7 | 0.21 | 46.5 | 31.3 (9.8) | 0.30 | 52.6
Table 1: Comparison of models on different slot types. WER - Word Error Rate, Recall - Recall on slot entities, Accuracy - Slot accuracy, OOV - Out-of-vocabulary | Datastore | WER | Recall | Accuracy
---|---|---|---|---
1 | Train | 34.7 | 0.20 | 46.0
2 | Train + OOV | 33.6 | 0.28 | 51.0
3 | OOV | 31.3 | 0.30 | 52.6
Table 2: Performance of $k$-PAT model on OOV data set with different datastores  | Domain | WER | Recall | Accuracy
---|---|---|---|---
1 | Full Names | 15.9 | 0.78 | 82.1
2 | Airports | 7.2 | 0.74 | 89.0
3 | Street Names | 9.5 | 0.71 | 80.3
4 | Cities & States | 7.8 | 0.74 | 90.7
Table 3: WER, Recall and Accuracy of $k$-PAT Model with slot-specific data stores | Good Rewrite | Ground Truth | ASR Output | PAT |  $k$-PAT
---|---|---|---|---|---
1 | Full Names | my name is janie burdick | my name is jamie burger | my name is jami barker |  my name is janie burdick
2 | Airports | i’m at prudhoe | i dont product | i’m at crowdus |  i’m at prudhoe
3 | Street Names | i stay in bedford | i stay in that for | i stay in nagpur |  i stay in bedford
4 | Cities & States | my house is in fairhope | my house and pharaoh | my house is in fairgrove |  my house is in fairhope
5 | Long Tail | i’m at lafon | i’m at a fun | i’m at vaughan |  i’m at lafon
6 | Long Tail | i work in debrecen | i work in the back | i work in napa |  i work in debrecen
7 | OOV | i’m from lauramie | i am from lauren i | i’m from wharton |  i’m from lauramie
8 | OOV | this is anderson driscoll | this is anderson driskell | this is andy driscoll |  this is anderson driscoll
9 | OOV | book a car to zhangxiao | book a car to saying south | book a car to zanesville |  book a car to zhangxiao
| Bad Rewrite |  |  |  |
1 | Long Tail | my home is in subang | my home is in salon | my home is in salon |  my home is in solvang
2 | OOV | my name’s jaylin duran | my name is jaylon joanne | my name’s jaylin ceja |  my name’s jaylin durant
3 | OOV | it’s on leralynn | it’s on there i land | it’s on neron |  it’s on lerawood
Table 4: Qualitative examples from test sets generated with PAT, $k$-PAT
models

##  4 Experiments

###  4.1 Data

We generate synthetic data which comprises of named entities from four domain
data collection [24] namely Airports, Cities, States, Street names. For
synthetic data, we augment the entities with most likely carrier phrases as
prefixes. They are then passed through a Text To Speech (TTS) system followed
an ASR model for simulating ASR errors. Additionally, we have vendor collected
audio datasets comprising common US first and last name entities. The sampled
utterances are then randomly split into train, dev and test sets with a
80:20:20 split.

We also train a sentence piece model [22] using the training split. Both our
transformer and $k$-NN datastore are sub-word based models with a vocabulary
size of 32,000. In addition, we also use an internal pronunciation lexicon
dictionary, same as the one used in the ASR model, to store the pronunciation
of all words in the training data. This dictionary has close to 100k
pronunciations.

####  4.1.1 Error simulation and synthetic data augmentation

We use TTS for synthesizing audios comprising entities from the above
mentioned slots. Each of the synthetically generated text samples are passed
through three randomly selected neural voices for generating the synthetic
utterances. Once the utterances are generated, we pass them through an ASR
model for generating the outputs. We use the n-best lists generated by the ASR
model for augmenting the training data.

Fig. 2: Slot accuracy vs the frequency of slots seen during training time. The
biggest gain with $k$-PAT is on long-tail frequencies.

Fig. 3: Word Error Rates (WER) with PAT and $k$-PAT for different ranges of
slot frequencies seen during training time

###  4.2 Model configurations

We use neural version of AWS Polly for generating synthetic 8kHz audios. For
decoding these synthetic audios, we use a hybrid ASR system with a standard
4-gram lm based first pass decoding.

For all experiments, we use a single PAT model which consists of 4-layer
encoders and a 4-layer decoder. We use 128 dimensions for the hidden
representations with 8 heads for the multi headed attention in both the
encoder and the decoder. The feed forward layer after the decoder has 512
dimensions. For training the model, we apply the same optimizer as the
original transformer [21] with 4000 warm-up steps. We use a batch size of 512
and train the model for 40 epochs. In addition to this, we follow the dropout
and error free token swap-in techniques used in [19].

The $k$-NN datastore key has 128 dimensions, same as the decoder output hidden
representation size. We build the FAISS indexing [23] using the same
techniques mentioned in [14, 13]. The number of cluster centroids for
datastore creation and quantization are the same as those mentioned in [14].
Compared to the original paper, our data store is relatively smaller with
roughly 5 million key, value pairs and 5GB in memory. In addition to a
combined data store, we also create slot specific data stores which have
roughly 1 million key value pairs and 1GB in memory. Our experiments show that
creating individual datastores leads to further improvements over a combined
datastore. In all our experiments, we set k=10 for $k$-NN retrieval during
inference.

##  5 Results and Discussion

Performance on different slot types: We compare the performance of $k$-PAT and
PAT model in terms of Word Error Rate (WER) and Relative Word Error Rate
(WERR). We also calculate recall and accuracy on the slot words of the
corrected outputs to show that the $k$-PAT model is actually retrieving slot
words and not just correcting the carrier phrases. For evaluating on the slot
words, we remove all the stop words (commonly used function words, such as
conjunctions and preposition) from the transcriptions. Table 1 summarizes
these metrics across all our data sets of four different slot types. On the
combined dataset, we observe that our $k$-PAT model achieves a 7.5% WERR over
the PAT model without any additional training. When tested on the domain
specific test sets, $k$-PAT consistently outperforms PAT model across all
domains in terms in WER.

Additionally, we measure slot word recognition performance by tagging slot
words in the reference text for calculating recall and accuracy. From the
results shown in Table 1, it is clear that $k$-PAT is able to resolve slots
better in the incorrect ASR output. This clearly indicates that the $k$-PAT
model is effectively memorizing the context during the datastore creation step
and retrieving the domain relevant neighbors during the inference step.
Qualitative examples in Table 4 show $k$-PAT’s ability to pick the correct
word among similar sounding neighbors.

Effectiveness on Long Tail Entities: For evaluating the performance of the
$k$-PAT model on long tail words, we create separate test sets from the
combined test set with slot words occurring at different frequencies in the
training data. We plot the results of slot accuracy and WER against slot word
frequency in figures 3 and 3 respectively. The figures indicate higher
relative gains in slot accuracy and WER on long tail (low frequency) words
with the $k$-PAT model. For example, at the lowest frequency of 10, we see an
absolute gain of 6 percentage points in slot accuracy and an absolute
reduction of 3 points in WER. This shows that $k$-PAT is effectively
memorizing and correcting long tail words. Additionally, from our experiments,
we observe that at higher frequencies, the gains in WER and slot-accuracy from
$k$-PAT tend to be narrower and close to zero after a certain frequency
threshold. This is expected as, at higher frequencies, the PAT model’s softmax
distribution is already biased to the high frequency word and the $k$-NN score
interpolation doesn’t make any additional contributions.

Performance on OOV Data: We evaluate the effectiveness of our proposed
approach on OOV words or unseen slots during training. For this purpose, we
create an out of vocabulary (OOV) dataset spanning all four domains: airports,
names, streetnames, and cities, states. From Table 1, we see that our $k$-PAT
model performs much better on OOV data with a WERR improvement of 9.8% over
the PAT model. By just memorizing unseen data, the model was able to improve
on slot recovery without any additional tuning.

Additionally, we ablate on the types of datastore that can be used for OOV
prediction. From Table 2, we observe that using training datastore results in
negligible reduction in performance. This proves that the model doesn’t
retrieve any neighbors for OOV data. Additionally, we use a datastore created
from OOV data alone, which gave significant improvements in both WER and slot
accuracy. Furthermore, we create another datastore by combining the oov
datastore with the training datastore which only gave slight improvements in
WER and slot accuracy compared to the oov specific datastore. This is because
the $k$-NN retrieval has more neighbors outside of OOV data to choose from
which leads to higher confusion. We have shown some qualitative examples
comparing PAT and $k$-PAT model in Table 4 which show $k$-PAT retrieving the
slots accurately. We also show some cases where $k$-PAT model is unable to
retrieve the right neighbors. However, even in these cases, we can see that
$k$-PAT model is able to retrieve retrieve similar sounding words. We intend
to improve OOV performance in our future work.

Ablation study on Slot-specific Datastores: We evaluate the performance of the
proposed approach using individual domain specific data stores. In order to
retrieve neigbors from domain specific datastores, we create individual
datastores for airports, names, streetnames, and cities, states domains. Table
3 summarizes our results of evaluating slot specific test sets with individual
data stores. Using this technique, we have observed additional performance
gains in airports, street names and cities. These gains can be attributed to
using a smaller $k$-NN search space and retrieval of neighbors which belong to
the relevant slot being decoded. This method of storing individual datastores
is very effective during run time in chat bot systems. Using the dialog state
context we can retrieve domain specific data stores for correcting asr outputs
in each turn.

##  6 Conclusion

In this work we have proposed an inference technique for using $k$-nearest
neighbour search with a pre-trained Transformer for the task of ASR error
correction. Our experiments have shown that interpolating $k$-NN scores with
the decoder softmax scores provides significant gains in WER performance and
slot word retrieval. We have also shown that using $k$-NN retrieval with
domain specific datastores provides further gains and have an advantage over
the combined datastore during runtime. This method also has the potential to
dynamically augment the vocabulary of any pre-trained error correction model
without retraining. In future works we will explore the impact of this method
on end-to-end models and explore few shot domain adaptation.

### Acknowledgements

Haoyu Wang (Amazon) for providing insights into their PAT work, Saket
Dingliwal (Amazon) for thoughtful discussions and providing feedback on the
paper.

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