# Magnetic waves in a stratified medium: dispersion relations and exact
solutions

Bindesh Tripathi1,2 and Dhrubaditya Mitra2
1Department of Physics, St. Xavier’s College, Tribhuvan University, Kathmandu,
Nepal
2Nordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-10691 Stockholm, Sweden
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(, Revision: 1.2 )

###### Abstract

We solve for waves in an isothermal, stratified medium with a magnetic field
that points along a direction perpendicular to that of gravity and varies
exponentially in the direction of gravity. For waves propagating along the
magnetic field, we calculate approximate dispersion relation as a function of
the magnetic field strength. We also find exact solutions for two different
cases: (a) waves propagating along the direction of the magnetic field and (b)
waves propagating along the direction of gravity. In each of these cases, we
find solutions in terms of either confluent hypergeometric functions or Gauss
hypergeometric functions depending on whether the ratio of the scale height of
the magnetic field over the density scale height is equal to two or not.

###### keywords:

MHD - Sun: waves : Helioseismology

##  1 Introduction

Helioseismology has revolutionized solar physics. The technique has allowed us
to probe sub-surface flows in the Sun. In near future it may be generalized to
other stars too. But our knowledge of the subsurface magnetic field in the Sun
remains limited. Several recent works (Schunker et al., 2005; Ilonidis et al.,
2011; Singh et al., 2014, 2015, 2016; Singh et al., 2018) have attempted, in
different ways, to infer, from surface signatures, the sub-surface magnetic
field that is going to emerge at a certain location in the Sun. The problem of
inferring the subsurface magnetic field from observations of the surface is an
inverse problem. The corresponding direct problem is to calculate what the
surface signature of a subsurface magnetic field would be. In this paper we do
this calculation for one particular case. It is noteworthy to mention that
almost all helioseismic calculations have been carried out without considering
the effect of magnetic field on the solar atmospheric waves. Recently, there
has been a growing concern to include the effects of magnetic field in
helioseismic models (Cally, 2007).

For simplicity, consider a Cartesian slab of plasma with constant gravity
acting in the negative $z$ direction as shown in Fig. 1. Assume the
temperature to be a constant. Furthermore, assume periodic boundary conditions
in the horizontal ($x$–$y$) directions. In the absence of any magnetic field,
there is a static solution of the equations of magnetohydrodynamics (MHD); the
density $\rho=\rho_{\rm 0}(z)$ is a function of the vertical coordinate $z$
alone and the velocity is zero everywhere. It is straightforward to apply the
theory of linear waves to this problem. Due to the symmetry of the problem,
the dynamical variables – velocity or Lagrangian displacement and density –
can be written down as $\exp i(k_{x}x+k_{y}y-\omega t)$, multiplied by a
function of the vertical coordinate $z$. In particular, the $z$ component of
the Lagrangian displacement satisfies a second order differential equation in
$z$, (see, e.g., Tolstoy, 1963, section 4). By solving this equation with
proper boundary conditions, we can find out the dispersion relation of the
waves in this setup. How would this dispersion relation change in the presence
of a horizontal magnetic field? For a constant magnetic field acting in the
$x$ direction, with a background density that varies exponentially as a
function of the vertical coordinate, this problem was solved for waves
propagating along the direction of the magnetic field (the $x$ direction) by
Nye & Thomas (1976); Adam (1977),(see also, Thomas, 1983, for a more
comprehensive treatment) and for waves propagating along the direction of
gravity (the $z$ direction) by Campos (1983). In both of these cases, it is
possible to find an exact solution that uses the hypergeometric functions.
Campos & Marta (2015) generalized the latter to the one where the magnetic
field is also an exponential function of the vertical coordinate $z$, with a
characteristic length scale $\ell_{\rm B}$, but its direction remains fixed –
along the $x$ direction. For the particular case where $\ell_{\rm B}$ is twice
the density scale height, $\ell_{\rho}$, Campos & Marta (2015) found an exact
solution for waves propagating along the gravity field in terms of confluent
hypergeometric functions. The problem remains unsolved for any value of
$\ell_{\rm B}/\ell_{\rho}$, not equal to two as well as for waves propagating
along the $x$ direction – the direction of the magnetic field and along the
$y$ direction – the direction orthogonal to both the direction of the magnetic
field and the direction of gravity.

Figure 1: A schematic sketch of the domain we consider. The velocity is zero
everywhere and the background magnetic field, pointing along the
$x$-direction, varies only as a function of height, $z$; the gravity acts
vertically downward. We choose periodic boundary conditions in the horizontal
direction $xy$ plane.

##  2 Model

We do a straightforward calculation of an idealized problem: the properties of
linear waves in a stratified isothermal plasma, permeated with magnetic field.
The equations of motion of the plama is given by the MHD equations (see, e.g.,
Choudhuri, 1998) with uniform gravity, $\bm{g}$, acting in the negative $z$
direction as:

$\displaystyle\partial_{t}\rho+{\bm{\nabla}}\cdot\left(\rho{\bm{U}}\right)$
$\displaystyle=0\/;$ (1a) $\displaystyle\partial_{t}(\rho
U_{\alpha})+\partial_{\beta}\left(\rho
U_{\alpha}U_{\beta}+P\delta_{\alpha\beta}\right)$ $\displaystyle=-\rho
g\delta_{\alpha 3}+\epsilon_{\alpha\beta\gamma}J_{\beta}B_{\gamma}\/;$ (1b)
$\displaystyle\partial_{t}\bm{B}$
$\displaystyle={\bm{\nabla}}\times\left({\bm{U}}\times\bm{B}\right)\/.$ (1c)

Here $\rho$, ${\bm{U}}$, $P$, and $\bm{B}$ are the density, the velocity, the
pressure, and the magnetic field respectively. ${\bm{J}}=(1/\mu_{{\rm
0}}){\bm{\nabla}}\times\bm{B}$ is the current density. Equation (1a), (1b),
and (1c) are respectively the continuity equation, the momentum equation, and
the induction equation. As the equations of MHD are a consequence of Maxwell’s
equations, we also have,

${\bm{\nabla}}\cdot\bm{B}=0.$ (2)

To simplify our notation, we have written the momentum equation in tensor
notation and the other two in vector notation. The second rank tensor
$\delta_{\alpha\beta}$ is the Kroneker delta and the third rank tensor
$\epsilon_{\alpha\beta\gamma}$ is the Levi-Civita tensor. We have used SI
units where $\mu_{{\rm 0}}$ denotes the permittivity of vacuum. We have also
ignored all dissipative effects – viscosity and electrical resistance of the
plasma. These equations must be supplemented by an equation of state, which we
assume is the ideal gas equation of state,

$c_{\rm s}^{2}=\frac{\gamma_{\rm ad}P}{\rho}\/.$ (3)

where $c_{s}$ and $\gamma_{ad}$ are the speed of sound and adiabatic index of
the gas. We study this problem under isothermal condition, i.e., the speed of
sound, $c_{\rm s}$, is a constant.

###  2.1 Static background state

To study linear waves, let us first consider the background state of the
system. Assume that the fluid in the box is in a static state – velocity
${\bm{U}}=0$, with density $\rho=\rho_{\rm 0}$ and magnetic field
$\bm{B}=\bm{B}_{\rm 0}$, which are constant in time. Here, in addition to
density $\rho_{\rm 0}$ being a function of the vertical coordinate ($z$), the
magnetic field is also a function of height. If the magnetic field had been
uniform (or if the magnetic field were absent), it would have played no role
in the stationary state. The background density $\rho_{\rm 0}$ would have been
an exponential function of $z$ with the scale height, $\ell_{\rho}=c_{\rm
s}^{2}/(g\gamma_{\rm ad})$. We consider the simplest possible case where the
magnetic field is not a constant. It points in the $x$ direction and is a
function of the $z$ coordinate. This is motivated by the fact that in a
typical emergence phenomenon, a bipolar active region on the solar surface, is
believed to form from underlying magnetic field lines that are horizontal. The
stationary state of such a system is a solution of

$\displaystyle\frac{c_{\rm s}^{2}}{\gamma_{\rm ad}}{\bm{\nabla}}\rho_{\rm 0}$
$\displaystyle=\rho_{0}{\bm{g}}+\bm{J}_{\rm 0}\times\bm{B}_{\rm 0}\/;$ (4a)
$\displaystyle{\bm{\nabla}}\cdot\bm{B}_{\rm 0}$ $\displaystyle=0\/.$ (4b)

####  2.1.1 Non-dimensionalization

Before proceeding further let us non-dimensionalize our equations. We consider
$\ell_{\rho}$ to be our characteristic length scale, the speed of sound
$c_{\rm s}$ as our characteristic scale of velocity, the density at the bottom
of our domain $\rho_{\rm 00}$ as our characteristic density scale and (the
magnitude of) the magnetic field at the bottom of our domain $B_{\rm 00}$ as
our characteristic scale of magnetic field.

Table 1: Summary of non-dimensional parameters: $\ell_{\rho}$ is the density scale-height and $\ell_{\rm B}$ is the scale-height of the magnetic field.  Non-dimensional | Comment | Non-dimensional | Comment
---|---|---|---
parameter |  | parameter |
$z^{\ast}=\frac{z}{\ell_{\rho}}$ |  | $\rho_{\rm 0}^{\ast}=\frac{\rho_{0}}{\rho_{\rm 00}}$ |
$D=\ell_{\rho}\frac{d}{dz}$ |  | $B_{\rm 0}^{\ast}=\frac{{B_{0}}}{B_{00}}$ |
$k_{x}^{\ast}=k_{x}\ell_{\rho}$ |  | $\mbox{M}_{\rm A}=\frac{v_{A}^{2}}{c_{s}^{2}}=\frac{{B_{00}}^{2}}{\mu_{{\rm 0}}\rho_{00}c_{s}^{2}}$ | ${\left(\frac{{\cal M}_{s}}{{\cal M}_{A}}\right)}^{2}$
$k_{y}^{\ast}=k_{y}\ell_{\rho}$ |  | $\beta=\frac{P_{\mathrm{gas}}}{P_{\mathrm{mag}}}=\frac{2\mu_{{\rm 0}}P_{g}}{{B_{00}}^{2}}$ | Plasma beta
$\omega^{\ast}=\frac{\omega\ell_{\rho}}{c_{s}}$ |  | $q=\frac{\ell_{\rm B}}{\ell_{\rho}}$ |

Let us now assume that the background magnetic field has the form,

$\bm{B}_{\rm 0}(z)=\hat{x}B_{\rm 00}\exp(-z/\ell_{\rm B}).$ (5)

where $\ell_{\rm B}$ is the scale height of magnetic field. Rewriting this
equation in terms of dimensionless variables we obtain

${\bm{B}}_{\rm 0}^{\ast}=\hat{x}\exp(-z^{\ast}/q)$ (6)

where $q=\ell_{\rm B}/\ell_{\rho}$ is the ratio of the characteristic length
scale of magnetic field variation to the characteristic length scale of
density variation. Equation (4a) rewritten in the non-dimensionalized form is

$D\rho_{\rm 0}^{\ast}+\rho_{\rm 0}^{\ast}+\frac{1}{2}\gamma_{\rm
ad}\mbox{M}_{\rm A}D(B_{\rm 0}^{\ast})^{2}=0\/,$ (7)

where the differential operator $D\equiv\ell_{\rho}(d/dz)$ and $\mbox{M}_{\rm
A}$ is the dimensionless number defined in table 1. We have further assumed
that the background magnetic field and the density are functions of only the
vertical coordinate, $z$. Integrating (7) we obtain a relationship between the
background density and the background magnetic field,

$\rho_{\rm 0}^{\ast}(z^{\ast})=-\left(\frac{\gamma_{\rm ad}\mbox{M}_{\rm
A}}{2}\right)e^{-z^{\ast}}\int e^{z^{\ast}}D(B_{\rm
0}^{\ast})^{2}dz^{\ast}\/.$ (8)

This equation is true, in general, irrespective of whether the background
magnetic field is an exponential function of the vertical coordinate or not.
Substituting (6) in (8) and simplifying, we obtain:

$\displaystyle\begin{split}{}&\rho_{\rm
0}^{\ast}(z^{\ast})=\left(1-\frac{\gamma_{\rm ad}\mbox{M}_{\rm
A}}{q-2}\right)e^{-z^{\ast}}+\frac{\gamma_{\rm ad}\mbox{M}_{\rm
A}}{q-2}e^{-2z^{\ast}/q}\mathrm{\hskip 5.69046ptfor\hskip 5.69046pt}q\neq
2\/;\end{split}$ (9a) $\displaystyle\begin{split}{}&\rho_{\rm
0}^{\ast}(z^{\ast})=\left(1+\frac{\gamma_{\rm ad}\mbox{M}_{\rm
A}}{2}z^{\ast}\right)e^{-z^{\ast}}\mathrm{\hskip 5.69046ptfor\hskip
5.69046pt}q=2.\/\end{split}$ (9b)

This means that the density drops exponentially with height for $q\neq 2$.
However, when $q=2$, it increases linearly with height for $z^{\ast}\ll 1$ and
drops exponentially for $z^{\ast}\gg 1$.

For notational simplicity, in the rest of the paper, we shall drop the
asterisks from the non-dimensional quantities.

###  2.2 Linearized MHD equations

The standard way of studying linear waves is to linearize the MHD equations
about the static background state:

$\left(\rho,{\bm{U}},\bm{B}\right)=\left(\rho_{\rm 0},\bm{0},\bm{B}_{\rm
0}\right)+\varepsilon\left(\rho,{\bm{u}},{\bm{b}}\right)\/,$ (10)

where $\varepsilon$ is a small parameter. We have not included pressure as a
separate variable as both the presssure and its perturbation are related to
density by the equation of state. We substitute (10) in (1), expand it and
keep terms upto first order in $\varepsilon$. After some straightforward
algebraic simplifications we reduce the problem to a second-order-in-time
equation for the perturbed velocity alone:

$\displaystyle\begin{aligned} \frac{\partial^{2}{\bm{u}}}{\partial
t^{2}}=\frac{1}{\rho_{\rm 0}}&\nabla(\rho_{\rm 0}c_{\rm
s}^{2}\nabla\cdot{\bm{u}})+\nabla({\bm{u}}\cdot\bm{g})-\bm{g}(\nabla\cdot{\bm{u}})\\\
&-\frac{1}{\rho_{\rm
0}}\nabla\left[{\bm{u}}\cdot\left\\{\nabla\left(\frac{{B_{\rm
0}}^{2}}{2\mu_{{\rm 0}}}\right)-\frac{1}{\mu_{{\rm 0}}}\bm{B}_{\rm
0}\cdot\nabla\bm{B}_{\rm 0}\right\\}\right]\\\ &+\frac{1}{\mu_{{\rm
0}}\rho_{\rm
0}}\left[\nabla\times\left(\nabla\times\left({\bm{u}}\times\bm{B}_{\rm
0}\right)\right)\times\bm{B}_{\rm 0}\right]\\\ &+\frac{1}{\mu_{{\rm
0}}\rho_{\rm 0}}\left[\left(\nabla\times\bm{B}_{\rm
0}\right)\times\left(\nabla\times\left({\bm{u}}\times\bm{B}_{\rm
0}\right)\right)\right]\end{aligned}$ (11)

This is a standard method of analysis, see, e.g., Chandrasekhar (1961).

##  3 General treatment

We have chosen a domain that is periodic in the horizontal direction but not
in the vertical direction. We now decompose the velocity perturbation as

${\bm{u}}(x,y,z)=\hat{{\bm{u}}}(k_{x},k_{y},z,\omega)\exp
i(xk_{x}+yk_{y}-\omega t)$ (12)

where the amplitude, $\hat{{\bm{u}}}(k_{x},k_{y},z,\omega)$, depends on
height, $z$. Substituting in (11), we obtain three coupled ordinary
differential equations (ODEs) for the three components of velocity
$\hat{u}_{x},\hat{u}_{y},\hat{u}_{z}$. We nondimensionalize them following the
same prescription described in the section 2.1.1 to obtain:

$\displaystyle\left(\omega^{2}-{k_{x}}^{2}\right){}$
$\displaystyle\hat{u}_{x}-k_{x}k_{y}\hat{u}_{y}-ik_{x}\left(\frac{1}{\gamma_{\rm
ad}}-D\right)\hat{u}_{z}=0,\/$ (13a)
$\displaystyle\begin{split}-k_{x}k_{y}\hat{u}_{x}+{}&\left(\omega^{2}-{k_{y}}^{2}-k^{2}\mbox{M}_{\rm
A}\frac{{B_{0}}^{2}}{\rho_{0}}\right)\hat{u}_{y}\\\
&-ik_{y}\left[\frac{1}{\gamma_{\rm
ad}}-\left(1+\frac{{B_{0}}^{2}}{\rho_{0}}\mbox{M}_{\rm
A}\right)D\right]\hat{u}_{z}=0,\end{split}$ (13b) and
$\displaystyle\begin{split}{}&ik_{x}\left[D+\frac{1}{\gamma_{\rm
ad}}-1-\frac{\gamma_{\rm ad}\mbox{M}_{\rm
A}}{2}\frac{D{(B_{0})}^{2}}{\rho_{0}}\right]\hat{u}_{x}\\\
&+ik_{y}\left[D+\frac{1}{\gamma_{\rm ad}}-1+\mbox{M}_{\rm
A}\frac{{B_{0}}^{2}}{\rho_{0}}D+(2-\gamma_{\rm ad})\frac{\mbox{M}_{\rm
A}}{2}\frac{D{(B_{0})}^{2}}{\rho_{0}}\right]\hat{u}_{y}\\\
&+\left[\omega^{2}+D^{2}-D+(2-\gamma_{\rm ad})\frac{\mbox{M}_{\rm
A}}{2}\frac{D{(B_{0})}^{2}}{\rho_{0}}D\right]\hat{u}_{z}\\\ &\hskip
85.35826pt+\mbox{M}_{\rm
A}\frac{{B_{0}}^{2}}{\rho_{0}}(D^{2}-{k_{x}}^{2})\hat{u}_{z}=0.\end{split}$
(13c)

Equation (13a)-(13c) are completely general for the case where the background
magnetic field everywhere points along the $x$ direction and is a function of
the the $z$ coordinate alone. These equations simplifies further if the
magnetic field is a constant in space. Several such cases has already been
studied in detail by Nye & Thomas (1976); Adam (1977); Thomas (1983); Campos
(1983). In this paper we consider the more general case where $\bm{B}_{\rm 0}$
is a function of the $z$ coordinate.

In particular, we list the following cases:

  1. 1.

Waves propagating along the $x$ direction (along the background magnetic
field).

  2. 2.

Waves propagating along the $z$ direction (along the direction of gravity).

  3. 3.

Waves propagating along the $y$ direction (orthogonal to the direction of both
the gravity and the background magnetic field).

For waves traveling along the gravity, Campos & Marta (2015) presented an
exact solution for a very special case of $q=2$. In this paper, we consider
the first and the second cases and present exact solutions for arbitrary
values of $q$. For waves propagating along a general direction the problem
becomes too cumbersome. We also point out the potential signatures of
subsurface solar magnetic field by looking at the changes in dispersion
relation for waves.

Clearly a general way to treat these problems is to eliminate two variables
from the three equations, (13a)-(13c); then consider the ratio of the scale-
heights, $q$ to be a small-parameter and study these problems using
perturbation theory. It turns out that in the two cases we consider, this
route is not necessary as exact solutions in terms of hypergeometric functions
exist. Furthermore, in each of these two cases, $q=2$ is special and must be
dealt with separately.

Let us first give a short summary of our calculations. In general, when we
eliminate two of the three variables from the three equations we obtain a
differential equation with coefficients that depend on the background density
and magnetic field. As both of these quantities have exponential dependence on
$z$, we obtain a differential equation with exponential coefficients. Next, we
look at dispersion relation for waves in Section 4 and obtain exact solutions
in Section 5. For the latter, several change of variables is necessary to
convert the differential equation with exponential coefficients to
differential equations with coefficients that depend algrebrically on the
independent variable. This ultimately leads us to the hypergeometric
differential equation.

##  4 Dispersion relation for waves propagating along the magnetic field

We set $k_{y}=0$ and $k_{x}=k$ in Equation (13a) - (13c) to obtain:

$\displaystyle\begin{split}{}&\hskip
48.36958pt\left(\omega^{2}-{k}^{2}\right)\hat{u}_{x}-ik\left(\frac{1}{\gamma_{\rm
ad}}-D\right)\hat{u}_{z}=0,\end{split}$ (14a)
$\displaystyle\begin{split}{}&\hskip
88.2037pt\left(\omega^{2}-k^{2}\mbox{M}_{\rm
A}\frac{{B_{0}}^{2}}{\rho_{0}}\right)\hat{u}_{y}=0,\end{split}$ (14b)
$\displaystyle\begin{split}{}&ik\left[D+\frac{1}{\gamma_{\rm
ad}}-1-\frac{\gamma_{\rm ad}\mbox{M}_{\rm
A}}{2}\frac{D{(B_{0})}^{2}}{\rho_{0}}\right]\hat{u}_{x}\\\ &+\left[\
\left(1+\mbox{M}_{\rm
A}\frac{{B_{0}}^{2}}{\rho_{0}}\right)D^{2}-\left(1+(\gamma_{\rm
ad}-2)\frac{\mbox{M}_{\rm
A}}{2}\frac{D{(B_{0})}^{2}}{\rho_{0}}\right)D\right]\hat{u}_{z}\\\ &\hskip
85.35826pt+\left(\omega^{2}-{k}^{2}\mbox{M}_{\rm
A}\frac{{B_{0}}^{2}}{\rho_{0}}\right)\hat{u}_{z}=0.\end{split}$ (14c)

The $\hat{u}_{y}$ mode in Equation (LABEL:Ecaseiiib) is decoupled from the
other two. We are primarily interested in the $\hat{u}_{z}$ mode, so we
eliminate $\hat{u}_{x}$ betweeen (LABEL:Ecaseiiia) and (LABEL:Ecaseiiic) to
obtain

$\displaystyle\begin{aligned}
&\left[\left(\omega^{2}-k^{2}\right)\mbox{M}_{\rm
A}\frac{{B_{0}}^{2}}{\rho_{0}}+\omega^{2}\right]D^{2}\hat{u}_{z}-\omega^{2}D\hat{u}_{z}\\\
&\hskip
8.5359pt+\left[\left(\omega^{2}-k^{2}\right)\left(\omega^{2}-k^{2}\mbox{M}_{\rm
A}\frac{{B_{0}}^{2}}{\rho_{0}}\right)+\frac{k^{2}}{\gamma_{\rm
ad}^{2}}\left(\gamma_{\rm ad}-1\right)\right]\hat{u}_{z}\\\ &\hskip
8.5359pt+\frac{D\left({B_{0}}^{2}\right)}{\rho_{0}}\mbox{M}_{\rm
A}\left[\left(\omega^{2}-k^{2}-\frac{\omega^{2}\gamma_{\rm
ad}}{2}\right)D\hat{u}_{z}+\frac{k^{2}}{2}\hat{u}_{z}\right]=0.\end{aligned}$
(15)

In the above equation, the expressions for $\rho_{\rm 0}$ and $B_{\rm 0}$ are
given by $\rho_{\rm 0}^{\ast}$ and $B_{\rm 0}^{\ast}$ in (9) and (6). To
simplify our notations we have now renamed our dimensionless variables. If the
background magnetic field is a constant, these equations reduce to the ones
considered by Nye & Thomas (1976).

The above equation, (LABEL:Eworking), can be rewritten as:

$D^{2}\hat{u}_{z}+f(z)D\hat{u}_{z}+r(z)\hat{u}_{z}=0$ (16)

where

$\displaystyle\begin{split}{}&f(z)=\frac{\left[\frac{D\left({B_{0}}^{2}\right)}{\rho_{0}}\mbox{M}_{\rm
A}\left(\omega^{2}-k^{2}-\frac{\omega^{2}\gamma_{\rm
ad}}{2}\right)-\omega^{2}\right]}{\left[\left(\omega^{2}-k^{2}\right)\mbox{M}_{\rm
A}\frac{{B_{0}}^{2}}{\rho_{0}}+\omega^{2}\right]}\\\ \end{split}$ (17a) and
$\displaystyle\begin{split}{}&r(z)=\frac{\left[\left(\omega^{2}-k^{2}\right)\left(\omega^{2}-k^{2}\mbox{M}_{\rm
A}\frac{{B_{0}}^{2}}{\rho_{0}}\right)+\frac{k^{2}}{\gamma_{\rm
ad}^{2}}\left(\gamma_{\rm
ad}-1\right)\right]}{\left[\left(\omega^{2}-k^{2}\right)\mbox{M}_{\rm
A}\frac{{B_{0}}^{2}}{\rho_{0}}+\omega^{2}\right]}\\\ &\hskip
85.35826pt+\frac{\frac{D\left({B_{0}}^{2}\right)}{\rho_{0}}\mbox{M}_{\rm
A}\frac{k^{2}}{2}}{\left[\left(\omega^{2}-k^{2}\right)\mbox{M}_{\rm
A}\frac{{B_{0}}^{2}}{\rho_{0}}+\omega^{2}\right]}.\end{split}$ (17b)

Following Tolstoy (1963), we apply the transformation

$\hat{u}_{z}(z)=\psi(z)\mathrm{\hskip
2.84544ptexp}\left[-\frac{1}{2}\int^{z}f(z)dz\right]$ (18)

in equation (16) from which we obtain,

$D^{2}\psi(z)+\Gamma^{2}(z)\psi(z)=0,$ (19)

with

$\Gamma^{2}(z)=r(z)-\frac{f^{2}(z)}{4}-\frac{Df(z)}{2}.$ (20)

The equation (19) is the usual form of the wave equation. When
$\Gamma^{2}(z)>0$, $\psi(z)$ locally exhibits oscillatory behaviour in $z$ and
when $\Gamma^{2}(z)<0$, exponential or hyberbolic nature of $\psi(z)$ is
realized. Thus $\Gamma$ can be interpreted locally as the vertical component
of the wavenumber (Tolstoy, 1963).

We consider the boundary conditions: $\hat{u}_{z}=0$ at $z=0$ (bottom) and
$\partial_{z}\hat{u}_{z}=0$ at $z=1$ (top). Our ansatz for discretising
$\Gamma$ is analogous to the organ pipes where we have:

$\Gamma=\left(n+\frac{1}{2}\right)\frac{\pi}{L_{z}},\hskip
11.38092ptn=0,1,2,3,...$ (21)

where $n$ represents the number of nodes in the z-direction (radial nodes).
This is valid only when $\Gamma$ does not depend on $z$. Since in this paper,
we are interested in dealing with high-degree (i.e. high-wavenumber)
$p$-modes, which are trapped in a very thin region near the solar surface, we
consider the size of the box to be $L_{z}=\ell_{\rho}$. Under such
circumstance, we find that $\Gamma$ of equation (20) remains approximately
constant for $z\in[0,1]$ and hence, the ansatz (21) is reasonable for
obtaining a qualitative idea of how magnetic field could change the dispersion
relation.

If $\Gamma$ is not a function of $z$, one should use the WKB approximation:

$\int^{z}\Gamma(z)dz=\left(n+\frac{1}{2}\right)\pi,$ (22)

which is a very good approximation for large $n$. If accurate eigenfrequencies
valid for all $n$ are required, an exact quantization scheme for $\Gamma$ is
to be used:

$\int^{z}\Gamma(z)dz=\left(n+\frac{1}{2}\right)\pi+\delta_{c}.$ (23)

where $\delta_{c}$ is the ’quantum correction’, similar to the one proposed by
Ma & Xu (2005). However, we do not employ such approach here.

It should be noted that the Duvall’s law, which was used in solar observations
and which led to the beginning of inversion of sound speed with depth in the
Sun, also had a similar functional form (Duvall, 1982):

$\Gamma=\frac{\left(n_{D}+\alpha\right)\pi}{L_{z}},\hskip
11.38092ptn_{D}=1,2,3,...$ (24)

Importantly, $n_{D}=0$ was excluded in Duvall’s analysis as its frequencies
did not fall along a curve where all frequencies corresponding to other values
of $n_{D}$ collapsed. On the other hand, the parameter $\alpha$ was found to
best fit the curve with its value nearly equal to $3/2$. In fact, $\alpha$ is
shown to be a function of frequency (Gough, 1987; Christensen-Dalsgaard,
2003). We note that the equations (21) and (24) are identical if we include
$n_{D}=0$ and parametrize $\alpha=1/2$.

Using equations (20) and (21), we find an approximate dispersion relation:

${\left(n+\frac{1}{2}\right)}^{2}\frac{\pi^{2}}{L_{z}^{2}}=r(z)-\frac{f^{2}(z)}{4}-\frac{Df(z)}{2},$
(25)

which results into a sixth order polynomial equation in $\omega$. They
represent a pair of acoustic, Alfven, and internal gravity waves, each
modified by the other ones. This equation, (25), is the dispersion relation
valid for waves that propagate along the magnetic field with the background
horizontal magnetic field varying vertically in any arbitrary manner. We now
choose the profile of the magnetic field in two different cases – uniform
field and exponentially decaying field along the vertical direction.

###  4.1 Uniform background magnetic field

From equation (7), we find that the background density varies exponentially
along the direction of gravity when the background magnetic field is uniform
in space. So, the non-dimensionalised Alfven speed becomes:

${v_{A}}^{2}(z)=\frac{B_{0}^{2}}{\rho_{0}}=e^{z}.$ (26)

Hence, the expressions for $f(z)$ and $r(z)$, utilizing equations (17a) and
(17b), are:

$\displaystyle\begin{split}{}&f(z)=\frac{-\omega^{2}}{\left[\left(\omega^{2}-k^{2}\right)\mbox{M}_{\rm
A}e^{z}+\omega^{2}\right]}\\\ \end{split}$ (27a) and
$\displaystyle\begin{split}{}&r(z)=\frac{\left(\omega^{2}-k^{2}\right)\left(\omega^{2}-k^{2}\mbox{M}_{\rm
A}e^{z}\right)+k^{2}N^{2}}{\left[\left(\omega^{2}-k^{2}\right)\mbox{M}_{\rm
A}e^{z}+\omega^{2}\right]}\end{split}$ (27b)

where the non-dimensionalised Brunt-Väisälä frequency, $N$, is defined as

$N=\sqrt{\left(1/\gamma_{\rm ad}-1/\gamma_{\rm ad}^{2}\right)}.$ (28)

This frequency, $N$, of course, changes in the presence of magnetic field. The
first term inside the radical of $N$ accounts for density stratification and
the other captures the effect of adiabatic expansion.

Substituting these expressions for $f(z)$ and $r(z)$ in equation (20), we get,

$\begin{split}{}&4\Gamma^{2}{\left[\left(\omega^{2}-k^{2}\right)\mbox{M}_{\rm
A}e^{z}+\omega^{2}\right]}^{2}\\\
&=4\left[\left(\omega^{2}-k^{2}\right)\left(\omega^{2}-k^{2}\mbox{M}_{\rm
A}e^{z}\right)+k^{2}N^{2}\right]\cdot\\\ &\hskip
85.35826pt\left[\left(\omega^{2}-k^{2}\right)\mbox{M}_{\rm
A}e^{z}+\omega^{2}\right]\\\ &\hskip
9.95863pt-2\omega^{2}\left(\omega^{2}-k^{2}\right)\mbox{M}_{\rm
A}e^{z}-\omega^{4}\end{split}$ (29)

This equation is cubic in $\omega^{2}$, representing three pairs of magneto-
acoustic-gravity waves. We solve for the root of this equation that
corresponds to the pressure mode.

The frequencies for pure acoustic and pure internal gravity waves in a
stratified medium are respectively given as (Tolstoy, 1963):

$\displaystyle\begin{split}{}&\omega_{A}^{2}=k^{2}+\Gamma^{2}+1/4,\end{split}$
(30a)
$\displaystyle\begin{split}{}&\omega_{I}^{2}=\frac{k^{2}N^{2}}{k^{2}+\Gamma^{2}+1/4},\end{split}$
(30b)

where $\Gamma$ is defined in the equation (21).

In a compressible fluid with gravity, these frequencies become:

$\omega_{\pm}^{2}=\frac{1}{2}\omega_{A}^{2}\left[1\pm{\sqrt{1-4\left(\frac{\omega_{I}^{2}}{\omega_{A}^{2}}\right)}}\hskip
2.84544pt\right]$ (31)

where $\omega_{\texttt{+}}$ and $\omega_{\texttt{-}}$ stand for the high-
frequency acoustic wave and the low-frequency internal gravity wave, each
modified by the other one. This gravity-modified acoustic wave frequency
($\omega_{\texttt{+}}$) is compared in Fig. 2 with the its further modified
wave frequency due to the presence of magnetic field, which is computed using
the equation (29).

We find that the presence of a horizontal magnetic field increases the
frequencies of $p$-modes. Fig. 2 demonstrates that the ridges in the
$k_{x}-\omega$ diagram shift only when the Alfven speed is significantly close
to (or more than) the sound speed. It can, therefore, be concluded that the
ring-diagram analysis applied for the quite Sun is unaffected by the
ubiquitous small-scale magnetic field. However, when the magnetic field
becomes stronger, frequency shifts in the dispersion relation are observed.

Figure 2: Dispersion relation for acoustic waves propagating along the uniform
background magnetic field for three cases: in the absence of magnetic field,
in the presence of magnetic field whose strengths are measured as
$\mbox{M}_{\rm A}=0.01$ and $\mbox{M}_{\rm A}=1$. This non-dimensionalised
parameter $\mbox{M}_{\rm A}$ is defined in Table 1. $p_{0}$, $p_{1}$, $p_{2}$,
and $p_{3}$ represent the fundamental pressure mode and the first, second and
third overtones respectively.

###  4.2 Non-uniform background magnetic field

Now we need to consider two different cases, the more general $q\neq 2$ and
the particular $q=2$.

####  4.2.1 Ratio of scale-heights $q\neq 2$

For this case, the density profile is given by equation (LABEL:eq:rhoqneq2).

Let’s calculate the non-dimensionalized Alfven speed as a function of height,

${v_{A}}^{2}(z)=\frac{B_{\rm 0}^{2}}{\rho_{\rm 0}}=\left[\frac{\gamma_{\rm
ad}\mbox{M}_{\rm A}}{q-2}+\left(1-\frac{\gamma_{\rm ad}\mbox{M}_{\rm
A}}{q-2}\right)e^{-(1-2/q)z}\right]^{-1}\/.$ (32)

Straighforward algebra show that, in equation (32), we can write

$\gamma_{\rm ad}\mbox{M}_{\rm A}=2/\beta$ (33)

where $\beta$ is the plasma-beta, the ratio of the gas-pressure to the
magnetic pressure at the bottom of our domain.

In order to obtain the dispersion relation for this case, we evaluate the
expressions for $f(z)$ and $r(z)$ of equation (20) using the equations (17a),
(17b), and (32). As in the previous section, 4.1, we get three pairs of waves,
out of which the pressure-modes are shown in Fig. 3. We find that the
frequency shift due to the presence of a horizontal magnetic field on the
$k_{x}-\omega$ diagram is less when the magnetic field drops with height, but
is still significant for $q=10$.

Figure 3: Dispersion relation for acoustic wave (the fundamental pressure
mode, $p_{0}$) propagating along the magnetic field ($\mbox{M}_{\rm A}=1$)
where the field drops exponentially in vertical direction. The ratio of scale
heights of magnetic field and density is $q$; $q=\infty$ is the case of
uniform background magnetic field.

####  4.2.2 Ratio of scale-heights $q=2$

The Alfven speed, in this case, is given as:

${v_{A}}^{2}(z)=\frac{B_{0}^{2}}{\rho_{0}}={\left[1+\frac{\gamma_{\rm
ad}\mbox{M}_{\rm A}}{2}z\right]}^{-1}.$ (34)

The dispersion relation for acoustic wave for $\mbox{M}_{\rm A}=1$ is
presented in the Fig. 3.

##  5 Exact solutions

###  5.1 Waves propagating along the magnetic field

We obtain an exact solution for waves that propagate along the magnetic field.
The magnetic field with its scale height twice the density scale height turns
out to be a special case and is, therefore, dealt with separately later.

####  5.1.1 Ratio of scale-heights $q\neq 2$

We substitute (32) in (LABEL:Eworking), simplify it, and perform the following
change of variable

$s=\left(1-\frac{2}{q}\right)z\/.$ (35)

Equation (LABEL:Eworking) can then be written in the following form:

$\displaystyle\begin{aligned}
\left[A_{2}e^{s}+B_{2}\right]\frac{d^{2}\hat{u}_{z}}{ds^{2}}&+\left[A_{1}e^{s}+B_{1}\right]\frac{d\hat{u}_{z}}{ds}\\\
&+\left[A_{0}e^{s}+B_{0}\right]\hat{u}_{z}=0\end{aligned}$ (36)

where the expressions for $A_{0},A_{1},A_{2}$ and $B_{0},B_{1},B_{2}$ are
given in appendix A.

Next, following Campos & Marta (2015), we do two more change of variables,

$\displaystyle\xi$ $\displaystyle=-\frac{B_{2}}{A_{2}}e^{-s}\/,$ (37a)
$\displaystyle W$ $\displaystyle=\hat{u}_{z}e^{-\theta s}\/,$ (37b)

where $\theta$ is yet undetermined constant. After some straightforward
simplifications we obtain:

$\displaystyle\begin{aligned} &\xi(1-\xi)\frac{d^{2}W}{d\xi^{2}}\\\
&+\left[\left(2\theta+1-\frac{A_{1}}{A_{2}}\right)-\left(2\theta+1-\frac{B_{1}}{B_{2}}\right)\xi\right]\frac{dW}{d\xi}\\\
&-\left(\theta^{2}-\frac{B_{1}}{B_{2}}\theta+\frac{B_{0}}{B_{2}}\right)W+\left(\theta^{2}-\frac{A_{1}}{A_{2}}\theta+\frac{A_{0}}{A_{2}}\right)\frac{W}{\xi}=0\end{aligned}$
(38)

Now, we choose $\theta$ such that the last term in the above equation
vanishes, i.e.

$\theta^{2}-\frac{A_{1}}{A_{2}}\theta+\frac{A_{0}}{A_{2}}=0\/.$ (39)

Consequently, Equation (38) turns into the hypergeometric equation:

$\xi(1-\xi)\frac{d^{2}W}{d\xi^{2}}+\left[C-\left(A+B+1\right)\xi\right]\frac{dW}{d\xi}-ABW=0\/.$
(40)

where the parameters $A$, $B$, and $C$ are given by

$\displaystyle C$ $\displaystyle=2\theta+1-\frac{A_{1}}{A_{2}}$ (41a)
$\displaystyle A+B+1$ $\displaystyle=2\theta+1-\frac{B_{1}}{B_{2}},$ (41b)
$\displaystyle AB$
$\displaystyle=\theta^{2}-\frac{B_{1}}{B_{2}}\theta+\frac{B_{0}}{B_{2}}$ (41c)

and $\theta$ is given by the equation (39). This completes the first stage of
our calculations.

The solution to the Gauss hypergeometric differential equation (40) may be
expressed in terms of hypergeometric functions (for $|\xi|<1$) as:

$W\left(A,B;C;\xi\right)=\frac{\Gamma\left(C\right)}{\Gamma\left(A\right)\Gamma\left(B\right)}\sum_{n=0}^{\infty}\frac{\Gamma(A+n)\Gamma(B+n)}{\Gamma(C+n)}\frac{\xi^{n}}{n!}$
(42)

Transforming back to the original variables, the solution can be written down
as

$\displaystyle\hat{u}_{z}(z)$
$\displaystyle=D_{1}e^{-z\theta(1-2/q)}{}_{2}{F}_{1}\left(A,B;C;-\frac{B_{2}}{A_{2}}e^{-z(1-2/q)}\right)$
(43)
$\displaystyle+D_{2}e^{-z\theta(1-2/q)(A_{1}/A_{2}\theta-1)}{\left[\frac{-B_{2}}{A_{2}}\right]}^{1-C}$
$\displaystyle{}_{2}F_{1}\left(A-C+1,B-C+1;2-C;-\frac{B_{2}}{A_{2}}e^{-z(1-2/q)}\right)\/.$

where $D_{1}$ and $D_{2}$ are two constants that must be determined by the
boundary conditions and ${}_{2}{F}_{1}$ is the Gauss hypergeometric function.
Our result holds true for $\lvert{\xi}\rvert<1$. All constants – $A$, $B$,
$C$, $A_{0}$, $A_{1}$, $A_{2}$, $B_{0}$, $B_{1}$, $B_{2}$ and $\theta$ are
functions of $\omega$, $k$, $\gamma_{\rm ad}$, $\beta$, and $q$.

The eigenfunctions, ${\hat{u}_{z}}$, for the first three $p$-modes of
oscillations, using equation (43), are shown in Fig. 4. The values of
eigenfrequencies, $\omega$, were choosen after computing them for $n=0,1,2$
using the equation (20) for $k_{x}=5$. All eigenfunctions satisfy the boundary
conditions of $\hat{u}_{z}=0$ at $z=0$ (bottom) and
$\partial_{z}\hat{u}_{z}=0$ at $z=1$ (top).

Figure 4: (Normalized) Eigenfunctions of the first three $p$-modes ($p_{0}$,
$p_{1}$, and $p_{2}$ at respective nondimensionalised frequencies
$\omega=5.25$, $6.92$, $9.41$ for $k_{x}=5$). Other parameters choosen are
$\mbox{M}_{\rm A}=10^{-2}$, and $q=10$.

####  5.1.2 Ratio of scale-heights $q=2$

In this case we proceed in a very similar fashion. We do not repeat the
details of our calculations but merely write down the solution. The solution
$\hat{u}_{z}$ can, in general, be written down as

$\displaystyle\hat{u}_{z}(z,\omega,k)=A_{\texttt{+}}$
$\displaystyle{}_{1}F_{1}\left(m_{\texttt{+}};1;\texttt{-}\eta_{5}\left(\eta_{1}+z\right)\right)e^{\left(\frac{1\texttt{+}\eta_{5}}{2}\right)\left(\eta_{1}+z\right)}$
(44)
$\displaystyle+A_{\texttt{-}}{}_{1}F_{1}\left(m_{\texttt{-}};1;\eta_{5}\left(\eta_{1}+z\right)\right)e^{\left(\frac{1\texttt{-}\eta_{5}}{2}\right)\left(\eta_{1}+z\right)}$

where the function ${}_{1}F_{1}$ is the confluent hypergeometric function of
the first kind. Here the constants, $m_{\texttt{+}}$ and $m_{\texttt{-}}$ are
given by

$\displaystyle\begin{aligned} n&=1\\\
m_{\pm}&=\frac{1}{2}\left(1\pm\frac{1}{\eta_{5}}\right)\pm\frac{\eta_{3}}{\eta_{5}}\left(\eta_{4}-\eta_{1}\right)\end{aligned}$
(45)

All $\eta$’s are functions of $\omega$, $k$, $\gamma_{\rm ad}$, and/or $\beta$
as given in appendix B. The constants $A_{\texttt{+}}$ and $A_{\texttt{-}}$
must be determined by the choice of boundary conditions.

####  5.1.3 The special case of a constant $\bm{B}_{\rm 0}$

In the special case where $\bm{B}_{\rm 0}$ is a constant in space we take the
limit $q\to\infty$ to obtain:

$\displaystyle\begin{aligned} A_{2}&=a_{2}=(\omega^{2}-k^{2})M\\\
A_{1}&=a_{1}=0\\\ A_{0}&=a_{0}=-k^{2}M(\omega^{2}-k^{2}).\end{aligned}$ (46)

The equation (39) reduces to

$\theta=\pm k\/.$ (47)

We, thus, reproduce the results of Nye & Thomas (1976).

###  5.2 Waves propagating along the direction of gravity.

Next we consider the problem for waves propagating perpendicular to the
magnetic field, along the direction of gravity. Again we need to consider two
cases $q\neq 2$ and $q=2$. The latter has already been solved exactly by
Campos & Marta (2015). Below we present the solution for the former.

As we consider waves propagating along the $z$ direction we substitute
$k_{x}=0=k_{y}$ in equations (13a)-(13c) to obtain:

$\displaystyle\begin{split}\hskip 102.43008pt\omega^{2}\hat{u}_{x}=0&,\\\
\end{split}$ (48a) $\displaystyle\begin{split}\hskip
102.43008pt\omega^{2}\hat{u}_{y}=0&,\\\ \end{split}$ (48b)
$\displaystyle\begin{split}\left[\omega^{2}+D^{2}\left(1+\mbox{M}_{\rm
A}\frac{{B_{0}}^{2}}{\rho_{0}}\right)-D+(2-\gamma_{\rm ad})\frac{\mbox{M}_{\rm
A}}{2}\frac{D{(B_{0})}^{2}}{\rho_{0}}D\right]&\hat{u}_{z}\\\
&=0\/.\end{split}$ (48c)

Clearly the solution to the differential equation (48c) can be obtained by
substituting $k_{x}=k=0$ in the solution presented in section 5.1.1:

$\displaystyle\hat{u}_{z}(z)$
$\displaystyle=D_{1}e^{-z\theta(1-2/q)}{}_{2}{F}_{1}\left(A,B;C;-\frac{B_{2}}{A_{2}}e^{-z(1-2/q)}\right)$
(49)
$\displaystyle+D_{2}e^{-z\theta(1-2/q)(A_{1}/A_{2}\theta-1)}{\left[\frac{-B_{2}}{A_{2}}\right]}^{1-C}$
$\displaystyle{}_{2}F_{1}\left(A-C+1,B-C+1;2-C;-\frac{B_{2}}{A_{2}}e^{-z(1-2/q)}\right).$

where the parameters $A$, $B$, $C$, $\theta$, $A_{2}$, and $B_{2}$ are
provided in the Appendix C, using $k=0$.

The solution for $q=2$, given by Campos & Marta (2015) can be reproduced from
our results in section 5.1.2 if we substitute $k=0$ in Equation (44).

##  6 Conclusions

To summarise, we investigate the detectable signatures of subsurface solar
magnetic fields by looking at changes in the dispersion relation of waves in
the presence of uniform and non-uniform magnetic fields. We find the changes
are significant for a strong subsurface magnetic field.

We find exact solutions for waves in an isothermal, stratified medium with a
magnetic field that points along a direction perpendicular to that of gravity
and varies exponentially in the direction of gravity. Our exact solutions are
for two different cases: (a) waves propagating along the direction of magnetic
field, (b) waves propagating along the direction of gravity. In all of these
cases we find solutions in terms of either Gauss hypergeometric or confluent
hypergeometric functions. For waves propagating along a general direction in
the horizontal plane – the plane perpendicular to gravity – no exact solution
has been found so far. The case (a) above can also be considered as Alfven
waves in a non-uniform magnetic field modified by the presence of
stratification. Two earlier results by Nye & Thomas (1976) and Campos & Marta
(2015), emerges as special cases of our solutions.

We have considered the background magnetic field to be constant in time. This
is physically justifiable as the time scale of emergence of magnetic flux and
formation of active regions on the solar surface is very slow compared to
characteristic time scales of helioseismic waves. Hence it is a good enough
approximation to consider the background magnetic field to be static. Typical
emergence phenomenon is the formation of a bipolar region, which is expected
to form from an underlying magnetic field lines that are horizontal. Hence
starting with a horizontal magnetic field can also be taken as a reasonable
approximation. But, the assumptions that the gas is ideal and the atmosphere
is isothermal cannot be used (Ulrich, 1970). Hence our result cannot directly
be compared with solar observations. However, certain qualitative results may
still hold. For example, it is clear that a subsurface magnetic field will
give rise to vertical oscillation propagating along horizontal directions on
the surface. In other words, subsurface magnetic fields have surface
signatures that may be detectable. Furthermore, we expect the dispersion
relation to depend on the direction of propagation of the wave with respect to
the direction of the sub-surface magnetic field. In addition, the exact
solution that we provide can be used to verify the accuracy of numerical
codes, which in turn may be more useful in studying the Sun.

## Acknowledgments

BT acknowledges Nordic Institute for Theoretical Physics (NORDITA) for
providing a Visiting Fellowship. DM is supported by Swedish Research Council
Grant no. 638-2013-9243. DM gratefully acknowledge Praphull Kumar for his help
in earlier stages of these calculations.

## References

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## Appendix A: Expressions for $A$ and $B$

We present here the expressions for constants introduced in equation (36) of
section 5.1.1.

$\displaystyle\begin{aligned}
A_{2}&=\left[\left(\omega^{2}-k^{2}\right)\mbox{M}_{\rm
A}+\omega^{2}\frac{\gamma_{\rm ad}\mbox{M}_{\rm
A}}{q-2}\right]{\left(1-\frac{2}{q}\right)}^{2}\\\
B_{2}&=\omega^{2}\left(1-\frac{\gamma_{\rm ad}\mbox{M}_{\rm
A}}{q-2}\right){\left(1-\frac{2}{q}\right)}^{2}\\\
A_{1}&=-\left[\frac{2\mbox{M}_{\rm
A}}{q}\left(\omega^{2}-k^{2}-\frac{\omega^{2}\gamma_{\rm
ad}}{2}\right)-\omega^{2}\frac{\gamma_{\rm ad}\mbox{M}_{\rm
A}}{q-2}\right]{\left(1-\frac{2}{q}\right)}\\\
B_{1}&=-\omega^{2}\left(1-\frac{\gamma_{\rm ad}\mbox{M}_{\rm
A}}{q-2}\right){\left(1-\frac{2}{q}\right)}\\\
A_{0}&=\left(\omega^{2}-k^{2}\right)\left[\omega^{2}\frac{\gamma_{\rm
ad}\mbox{M}_{\rm A}}{q-2}-k^{2}\mbox{M}_{\rm A}\right]\\\ &\hskip
71.13188pt+\frac{k^{2}}{\gamma_{\rm ad}^{2}}\left(\gamma_{\rm
ad}-1\right)\frac{\gamma_{\rm ad}\mbox{M}_{\rm A}}{q-2}-\frac{\mbox{M}_{\rm
A}k^{2}}{q}\\\
B_{0}&=\omega^{2}\left(\omega^{2}-k^{2}\right)\left(1-\frac{\gamma_{\rm
ad}\mbox{M}_{\rm A}}{q-2}\right)\\\ &\hskip
71.13188pt+\frac{k^{2}}{\gamma_{\rm ad}^{2}}\left(\gamma_{\rm
ad}-1\right)\left(1-\frac{\gamma_{\rm ad}\mbox{M}_{\rm
A}}{q-2}\right)\end{aligned}$

## Appendix B: Expressions for $\eta$

The expressions for $\eta$’s used in equation (45) of section 5.1.2 are listed
here:

$\displaystyle\begin{aligned} \eta_{1}&=\frac{2}{\gamma_{\rm
ad}}\left(1-\frac{k^{2}}{\omega^{2}}\right)+\beta\\\
\eta_{2}&=-\frac{2}{\gamma_{\rm
ad}}\left(1-\frac{k^{2}}{\omega^{2}}-\frac{\gamma_{\rm ad}}{2}\right)-\beta\\\
\eta_{3}&=\frac{k^{2}}{\omega^{2}}\frac{\left(\gamma_{\rm
ad}-1\right)}{\gamma_{\rm ad}^{2}}+\left(\omega^{2}-k^{2}\right)\\\
\eta_{4}&=\frac{\left(1-\frac{k^{2}}{\omega^{2}}\right)\left(\omega^{2}\beta-\frac{2k^{2}}{\gamma_{\rm
ad}}\right)+\frac{k^{2}}{\omega^{2}}\frac{\left(\gamma_{\rm
ad}-1\right)}{\gamma_{\rm ad}^{2}}\beta-\frac{k^{2}}{\omega^{2}}\gamma_{\rm
ad}}{\frac{k^{2}}{\omega^{2}}\frac{\left(\gamma_{\rm ad}-1\right)}{\gamma_{\rm
ad}^{2}}+\left(\omega^{2}-k^{2}\right)}\\\
\eta_{5}&={\left(1-4\eta_{3}\right)}^{1/2}\end{aligned}$

## Appendix C: Expressions for $A$,$B$, and $C$

We provide here the parameters utilized in equation (49) of section 5.2

$\displaystyle\begin{aligned} A_{2}&=\omega^{2}M\left(1+\frac{\gamma_{\rm
ad}}{q-2}\right){\left(1-\frac{2}{q}\right)}^{2}\\\
B_{2}&=\omega^{2}\left(1-\frac{\gamma_{\rm
ad}M}{q-2}\right){\left(1-\frac{2}{q}\right)}^{2}\\\
A_{1}&=-\omega^{2}M\left(\frac{2-\gamma_{\rm ad}}{q}+\frac{\gamma_{\rm
ad}}{q-2}\right){\left(1-\frac{2}{q}\right)}\\\
B_{1}&=-\omega^{2}\left(1-\frac{\gamma_{\rm
ad}M}{q-2}\right){\left(1-\frac{2}{q}\right)}\\\
A_{0}&=\omega^{4}\frac{\gamma_{\rm ad}M}{q-2}\\\
B_{0}&=\omega^{4}\left(1-\frac{\gamma_{\rm ad}M}{q-2}\right).\end{aligned}$

Now, $\theta$ can be calculated using the equation (39) using the following
set of equations:

$\theta=\frac{1}{2}\left(\frac{A_{1}}{A_{2}}\pm\sqrt{{\left(\frac{A_{1}}{A_{2}}\right)}^{2}-\frac{4A_{0}}{A_{2}}}\right).$

Lastly, the parameters $A$, $B$, and $C$ in equation (49) are evaluated as:

$\displaystyle\begin{aligned} C&=2\theta+1-\frac{A_{1}}{A_{2}}\\\
A+B+1&=2\theta+1-\frac{B_{1}}{B_{2}},\\\
AB&=\theta^{2}-\frac{B_{1}}{B_{2}}\theta+\frac{B_{0}}{B_{2}}.\end{aligned}$

This completes the specification for obtaining the Gauss hypergeometric
funtion as the solution for vertical propagation of magneto-acoustic-gravity
waves.


# The impact of non-target events in synthetic soundscapes for sound event
detection

###### Abstract

Detection and Classification Acoustic Scene and Events Challenge 2021 Task 4
uses a heterogeneous dataset that includes both recorded and synthetic
soundscapes. Until recently only target sound events were considered when
synthesizing the soundscapes. However, recorded soundscapes often contain a
substantial amount of non-target events that may affect the performance. In
this paper, we focus on the impact of these non-target events in the synthetic
soundscapes. Firstly, we investigate to what extent using non-target events
alternatively during the training or validation phase (or none of them) helps
the system to correctly detect target events. Secondly, we analyze to what
extend adjusting the signal-to-noise ratio between target and non-target
events at training improves the sound event detection performance. The results
show that using both target and non-target events for only one of the phases
(validation or training) helps the system to properly detect sound events,
outperforming the baseline (which uses non-target events in both phases). The
paper also reports the results of a preliminary study on evaluating the system
on clips that contain only non-target events. This opens questions for future
work on non-target subset and acoustic similarity between target and non-
target events which might confuse the system.

Index Terms—  Sound event detection, synthetic soundscapes, open-source
datasets, deep learning

##  1 Introduction

††This work was made with the support of the French National Research Agency,
in the framework of the project LEAUDS Learning to understand audio scenes
(ANR-18-CE23-0020), the project CPS4EU Cyber Physical Systems for Europe
(Grant Agreement number: 826276) and the French region Grand-Est. Experiments
presented in this paper were carried out using the Grid5000 testbed, supported
by a scientific interest group hosted by Inria and including CNRS, RENATER and
several Universities as well as other organizations (see
https://www.grid5000).

The main goal of ambient sound and scene analysis is to automatically extract
information from sounds that surround us and analyze them for different
purposes and applications. Between the different area of interest, ambient
sound analysis have a considerable impact on applications such as noise
monitoring in smart cities [1, 2], domestic applications such as smart homes
and home security solutions [3, 4], health monitoring systems [5], multimedia
information retrieval [6] and bioacoustics domain [7]. Sound Event Detection
(SED) aims to identify the onset and offset of the sound events present in a
soundscape and to correctly classify them, labeling the events according to
the target sound classes that they belong to. Nowadays, deep learning is the
main method used to approach the problem. However, one of the main limitations
of deep learning models is the requirement of large amounts of labeled
training data to reach good performance. The process of labeling data is time-
consuming and bias-prone mainly due to human errors and disagreement given the
subjectivity in the perception of some sound event onsets and offsets [8]. To
overcome these limitations, recent works are investigating alternatives to
train deep neural networks with a small amount of labeled data together with a
bigger set of unlabeled data [3, 9, 10, 8, 11]. Among them, Detection and
Classification Acoustic Scenes and Events Challenge (DCASE) 2021 Task 4 uses
an heterogeneous dataset that includes both recorded and synthetic soundscapes
[8]. This latter soundscapes provide a cheap way to obtain strongly labeled
data. Until recently, synthesized soundscapes were generated considering only
target sound events. However, recorded soundscapes also contain a considerable
amount of non-target events that might influence the performance of the
system.

The purpose of this paper is to focus on the impact on the system’s
performance when non-target events are included in the synthetic soundscapes
of the training dataset. The study has been mainly divided into three stages.
Firstly, we investigate to what extent using non-target events alternatively
during training or validation helps the system to correctly detect the target
sound events. Mainly motivated from the results of the first experiment, in
the second part of the study, we focus on understanding to what extend
adjusting the target to non-target signal-to-noise ratio (TNTSNR) at training
improves the sound event detection performance. Results regarding a
preliminary study on the evaluation of the system using clips containing only
non-target events are also reported, opening questions for future studies on
possible acoustic similarity between target and non-target sound events which
might confuse the SED system. 111To promote reproducibility, the code,
https://github.com/DCASE-REPO/DESED_task, and pre-trained models
https://zenodo.org/record/5529692, are made available under an open-source
license..

##  2 PROBLEM DEFINITION AND DATASET GENERATION

###  2.1 Problem definition

The primary goal of the DCASE 2021 Challenge Task 4 is the development of a
semi-supervised system for SED, exploiting an heterogeneous and unbalanced
training dataset. The goal of the system is to correctly classify the sound
event classes and to localize the different target sound events present in an
audio clip in terms of timing. Each audio recording can contain more than one
event. Some of those could also be overlapped. The use of a larger amount of
unlabeled recorded clips is motivated by the limitations related to annotating
a SED dataset (human-error-prone and time-consuming). Alternatively,
synthesized soundscapes are an easy way to have strongly annotated data. In
fact, the user can easily generate the soundscapes starting from isolated
sound events. On the other hand, in most of the recorded soundscapes the
target sound classes are almost never present alone. For this reason, one of
the main novelties of the DCASE 2021 Challenge Task 4 is the introduction of
non-target isolated events in the synthetic
soundscapes222http://dcase.community/challenge2021. This paper explores the
impact of the non-target sound events on the baseline system performance, with
the final goal of understanding and highlighting how to correctly exploit them
to generate realistic soundscapes.

###  2.2 Dataset generation

The dataset used in this paper is the DESED
dataset333https://project.inria.fr/desed/ [12, 13], which is the same provided
for the DCASE 2021 Challenge Task 4. It is composed of 10 seconds length audio
clips either recorded in a domestic environment or synthesized to reproduce
such an environment444For a detailed description of the DESED dataset and how
it is generated the reader is referred to the original DESED article [13] and
DCASE 2021 task 4 webpage: http://dcase.community/challenge2021. The synthetic
part of the dataset is generated with Scaper [14], a Python library for
soundscape synthesis and augmentation, which allows to control audio
parameters. The recorded soundscapes are taken from AudioSet [15]. The
foreground events (both target and non-target) are obtained from the Freesound
Dataset (FSD50k) [16], while the background sounds are obtained from the SINS
dataset (activity class “other”) [17] and TUT scenes 2016 development dataset
[18]. In particular, non-target events are the intersection of FUSS dataset
[19] and FSD50k dataset in order to have compatibilty with the source
separation baseline system.

In this article, we modify only the synthetic subset of the dataset. Starting
from the synthetic part of the DESED dataset, we generated different versions
of it in order to investigate how non-target events impact the system
performance and to what extent their relationship with the target events
affects the training phase of the system. The following subsections describe
the different subsets used for the experiments, which have been generated
using Scaper.

####  2.2.1 Synthetic training set

The synthetic training set is the same set of data released for the DCASE 2021
Challenge Task 4. It includes 10000 audio clips where both target and non-
target sound events could be present in each clip. The distribution of the
sound events among the files have been determined considering the co-
occurrences between the different sound events. The co-occurrences have been
calculate considering the strong annotations released for the AudioSet dataset
[20]555The co-occurrences distribution and the code used to compute them will
be distributed.. A second version of this dataset has been generated where
only target events are present. The datasets will be hereafter referred as
synth_tg_ntg (used by the official baseline system) and synth_tg for the
synthetic subset including target and non-target events and the synthetic
subset including only target events, respectively.

####  2.2.2 Synthetic validation set

The synthetic validation set is the same as the synthetic validation dataset
supplied for the DCASE 2021 Challenge Task 4. It includes 3000 audio clips
including target and non-target events, which distribution has been defined
calculating the co-occurrences between sound events. We generated a second
version of the dataset containing only target events. The datasets will be
referred to as synth_tg_ntg_val (used by the baseline system) and synth_tg_val
(only target sound events).

####  2.2.3 Synthetic evaluation set

The synthetic 2021 evaluation set is composed by 1000 audio clips. In the
context of the challenge, this subset is used for analysis purposes. We will
refer to it as synth_tg_ntg_eval. It contains target and non-target events
distributed between the different audio clips according to the pre-calculated
co-occurrences. Two different versions of the synth_tg_ntg_eval set have been
generated, synth_tg_eval (only target sound events) and synth_ntg_eval (only
non-target sound events).

####  2.2.4 Varying TNTSNR training and validation set

With the aim of studying what would be the impact of varying the TNTSNR on the
system performance, different versions of synth_tg_ntg and synth_tg_ntg_val
have been generated. In particular, for each of them, three versions have been
created. The SNR of the non-target events have been decreased by 5 dB, 10 dB
and 15 dB compared to their original value. The original SNR of the sound
events is randomly selected between 6 dB and 30 dB, so the more we decrease
the SNR, the less the sound will be audible, with some of the events that will
not be audible at all. These subsets will be subsequently referred to as
synth_5dB, synth_10dB, synth_15dB for the training subsets and synth_5dB_val,
synth_10dB_val, synth_15dB_val for the validation subsets.

####  2.2.5 Public evaluation set

The public evaluation set is composed of recorded audio clips extracted from
Youtube videos that are under creative common licenses. This is part of the
evaluation dataset released for the evaluation phase of the DCASE 2021
Challenge Task 4 and considered for ranking. The set will be referred to as
public.

##  3 EXPERIMENTS TASK SETUP

In order to compare the results with the official baseline, we used the same
SED mean-teacher system released for this year challenge. More information
regarding the system can be found at Turpault et al. [8] and on the official
webpage of the DCASE Challenge Task 4. All the different models have been
trained 5 times. This paper reports the average of the scores and the
confidence intervals related to those. Only for the baseline model we do no
report the confidence intervals because we have considered the results using
the checkpoint made available for it 666https://zenodo.org/record/4639817. The
metrics considered for the study are the two polyphonic sound detection score
(PSDS) [21] scenarios defined for the DCASE 2021 Challenge Task 4, since these
are the official metrics used in the challenge.

The scope of these experiments is twofold: understand the impact of non-target
events on the system performance and investigate to what extend the TNTSNR
helps the network to correctly predict the sound events in both matched and
mismatched conditions. In order to do so, we divided the experiment into three
stages. The first part of the study is focused on understanding the influence
of training the system with non-target events. This experiment is described
and discussed in Section 4. Section 5 reports the results and the relative
discussion of the second part of the experiment where we investigate if a
mismatch in terms of TNTSNR between datasets could have an impact on the
output of the system. Section 6 reports preliminary results of the last stage
of the experiment, regarding the evaluation of the system on the
synth_ntg_eval dataset, formed by only non-target sound events, in order to
investigate if some classes could get acoustically confused at training,
having a negative impact on the performance. The last stage has been motivated
by the results of the second part of the experiment.

##  4 Using TARGET/NON-TARGET at training

Non-target | PSDS1 | PSDS2
---|---|---
Train | Val
|  | 33.81 (0.36) | 52.62 (0.19)
|  | 35.92 (0.49) | 54.85 (0.29)
|  | 34.90 (0.82) | 53.07 (1.22)
|  | 36.40 | 58.00
Table 1: Evaluation results for the public set, considering the different
combinations of using target and non-target sound events at training and
validation.

In the first experiment we concentrate on training the system with different
combinations of the training dataset. Table 1 reports the results of the
experiment evaluating the system on the public set. We check-marked the
columns NT Train or/and NT Val according to if the non-target sound events are
present or not in the synthetic sounscapes. From the results it is possible to
observe that using non-target sound events during training and validation
improves the performance by a large margin with relaxed segmentation
constraints (PSDS2) but only marginally with strict segmentation constraints
(PSDS1). In this latter case what matters the most is the use of non-target
sound events during the validation. A possible explanation is that synthetic
soundscapes with non-target sound events are actually too difficult and
confuse the systems when used during the training but they still help reducing
the mismatch with recorded soundscapes during model selection (validation).

Non-target | Eval set | PSDS1 | PSDS2
---|---|---|---
Train | Val
|  | synth_tg_ntg_eval | 23.22 (1.33) | 36.44 (2.62)
|  | synth_tg_ntg_eval | 20.08 (0.39) | 31.33 (1.29)
|  | synth_tg_ntg_eval | 20.13 (0.35) | 30.99 (1.07)
|  | synth_tg_ntg_eval | 25.14 | 40.12
|  | synth_tg_eval | 42.82 (2.42) | 58.26 (2.08)
|  | synth_tg_eval | 46.92 (1.02) | 62.79 (0.55)
|  | synth_tg_eval | 47.73 (0.33) | 62.54 (1.00)
|  | synth_tg_eval | 43.22 | 61.09
Table 2: Evaluation results for the synth_tg_ntg_eval set and synth_tg_eval
set, considering the different combination of using target and non-target
sound events at training and validation.

Table 2 reports the results considering the synth_tg_ntg_eval and
synth_tg_eval evaluation sets. In all cases the best performance is obtained
in matched training/evaluation conditions. The performance obtained on
synth_tg_ntg_eval are lower than the performance obtained on synth_tg_eval
even in matched conditions. Not surprisingly, this confirm that including non-
target sound events makes the SED task more difficult. Interestingly, as
opposed to the previous experiment, the most important here is to have matched
conditions during training and to a lesser extent during validation. In order
to verify the low impact of non-target sound events at training when
evaluating on recorded soundscapes, in the next experiment we investigate a
possible mismatch in terms in TNTSNR.

##  5 Varying TNTSNR at training

Non-target | PSDS1 | PSDS2
---|---|---
Train | Val
Original | 5 dB | 35.57 (0.28) | 56.68 (1.77)
5 dB | Original | 36.25 (1.26) | 57.53 (1.06)
5 dB | 5 dB | 35.46 (0.46) |  58.09 (0.74)
Original | Original | 36.40 | 58.00
Table 3: Evaluation results for the second part of the experiment, varying TNTSNR by 5 dB (synth_5dB and synth_5dB_val). Evaluating with public set. Non-target | PSDS1 | PSDS2
---|---|---
Train | Val
Original | 10 db | 36.23 (1.11) | 57.82 (1.37)
10 db | Original | 36.42 (0.77) | 58.94 (0.89)
10 db | 10 db | 36.20 (1.14) | 57.92 (1.04)
Original | Original | 36.40 | 58.00
Table 4: Evaluation results for the second part of the experiment, varying TNTSNR by 10 dB (synth_10dB and synth_10dB_val). Evaluating with public set. Non-target | PSDS1 | PSDS2
---|---|---
Train | Val
Original | 15 dB | 36.08 (1.13) | 57.78 (1.33)
15 dB | Original | 37.37 (0.70) | 58.64 (1.34)
15 dB | 15 dB | 36.10 (0.50) | 57.36 (0.89)
Original | Original | 36.40 | 58.00
Table 5: Evaluation results for the second part of the experiment, varying TNTSNR by 15 dB (synth_15dB and synth_15dB_val). Evaluating with public set. Validation set | PSDS1 | PSDS2
---|---|---
synth_5dB_val | 38.68 (1.07) | 60.57 (0.78)
synth_10dB_val | 39.07 (0.75) | 60.75 (0.80)
synth_15dB_val | 37.95 (0.53) | 59.99 (1.14)
Table 6: Evaluation results of the SED system, training with synth_tg,
validating with varying TNTNSNR set and evaluating with public set.

The second part of the study focuses on understanding the impact of varying
the TNTSNR at training and validation aiming at finding a TNTSNR condition
that could match better the recorded soundscapes. For each TNTSNR, we use
similar combinations as the ones used in Section 4, replacing the set without
non-target sound events by a set with adjusted TNTSNR. For example,
considering the 5 dB case, the combinations considered would be:

  * •

training using the synth_tg_ntg set and validating with synth_5dB_val;

  * •

training with synth_5dB and validating with synth_tg_ntg_val;

  * •

training and validating with synth_5dB and synth_5dB_val.

The fourth combination is the official DCASE Task 4 baseline. Repeating the
experiment with all the varying TNTSNR, allow us to analyse to what extend the
loudness of the non-target events helps matching the evaluation conditions on
recorded clips. Table 3, 4 and 5 report the performance on the public set when
using a TNTSNR of 5 dB, 10 dB and 15 dB, respectively. When the TNTSNR is 5 dB
or 10 dB, the performance changes only marginally between configurations.
Increasing the TNTSNR to 15 dB leads to a behaviour more similar to the one
obtained in Table 1. The best performance is obtained when training with
TNTSNR is 15 dB and validating on synth_tg_ntg_val. This could be explained by
the fact TNTSNR 15 dB is a condition closer to that of the recorded
soundscapes and the fact that it allows for selecting models that will be more
robust towards non-target events at test time.

In the last experiment, we investigate the impact of varying the TNTSNR during
validation phase, while using the synt_tg for training. Results are reported
on Table 6, where it is possible to observe that all of them overcome the
baseline or are comparable with it, with the best performance obtained for 10
dB TNTSNR. These experiments could indicate that recorded soundscapes in
public in general have a TNTSNR of about 10 – 15 dB which should be confirmed
by complementary experiments.

##  6 EVALUATING ON NON-TARGET EVENTS ONLY

Based on the previous experiments, TNTSNR could be one reason of mismatch
between the synthetic soundscapes and the recorded soundscapes. But this could
not explain all the performance differences observed here. In particular why
in general having lower TNTSNR during training is decreasing the performance
regardless of the validation. One possibility is that the system gets
acoustically confused by a possible similarity in sound between events when
soundscapes tend to be less dominated by target events. So we evaluated the
system using the synth_ntg_eval, where only non-target events are considered,
to see for which classes the system would output false positives. We evaluated
the system on the public set; considering the systems trained for the first
experiment (see Table 1). Results show that some sound events are detected
more than others. For some classes as Speech, this could be explained by the
original event distribution (indicated in the first column) but for some other
classes as Dishes there is a discrepancy between the original distribution and
the amount of false alarms. Interestingly the amount of false alarms is
decreased sensibly for most of the classes when including non-target sound
events during training.

| Nref | Nsys
---|---|---
Classes | A | B | C | Base
Dog | 197 | 135 | 126 | 146 | 79
Vacuum_cleaner | 127 | 31 | 42 | 44 | 47
Alarm_bell | 191 | 47 | 50 | 52 | 59
Running_water | 116 | 34 | 41 | 61 | 30
Dishes | 405 | 1478 | 395 | 1270 | 305
Blender | 100 | 63 | 32 | 55 | 19
Frying | 156 | 70 | 41 | 60 | 33
Speech | 1686 | 206 | 181 | 180 | 201
Cat | 141 | 99 | 103 | 98 | 73
Electric_shaver | 103 | 21 | 18 | 18 | 7
Table 7: Preliminary evaluation results by classes, evaluating the system with
synth_ntg_eval. Nsys (A): training with synth_tg, validating with
synth_tg_val; Nsys (B): training with synth_tg_ntg, validating with
synth_tg_val; Nsys (C): training with synth_tg, validating with
synth_tg_ntg_val; Base: baseline using target and non-target events for
training and validation.

##  7 CONCLUSIONS AND FUTURE WORK

This paper analyzes the impact of including non-target sound events in the
synthetic soundscapes of the training dataset for SED systems trained on
heterogeneous dataset. In particular, the experiments are divided into three
stages: in the first part, we explore to what extend using non-target sound
events at training has an impact on the system’s performance, secondly we
investigate the impact of varying TNTSNR and we conclude the study by
analyzing a possible confusion of the SED model in case of false alarms
triggered by non-target sound events.

From the results reported on this paper, we can conclude that using non-target
sound events can help the SED system to better detect the target sound events,
but it is not clear to what extend and what would be the best way to generate
the soundscapes. Results show that the final SED performance could depend on
mismatches between synthetic and recorded soundscapes, part of which could be
due to the TNTSNR but not only. Results on the last experiment show that using
non-target events at training decreases the amount of false alarms at test but
from this experiment it is not possible to conclude on the impact of non-
target sound events on the confusion between the target sound events. This is
a first track for future investigation on the topic. Additionally, the impact
of the non-target sound events at training on the ability of the system to
better segment the target sound events in noisy soundscapes would have to be
investigated. A final open question is the impact of the per class
distribution of the sound events (both target and non-target) and their co-
occurrence distribution on the SED performance.

##  8 ACKNOWLEDGEMENTS

We would like to thank all the other organizers of DCASE 2021 Challenge Task
4\. In particular, we thank Eduardo Fonseca and Daniel P. W. Ellis for their
help with the strong labels of the AudioSet dataset used to compute the events
co-occurrences, and Justin Salamon and Prem Seetharaman for their help with
Scaper.

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