PROPAGATION OF ACOUSTIC WAVES IN EARTH'S

PROPAGATION OF ACOUSTIC WAVES IN EARTH’S ATMOSPHERE

by MADHU KASHYAP

Presented to the Faculty of the CPGS of The Jain University in Partial Fulfillment of the Requirements for the Degree in

Master of Science

Jain University June 2014

In deep respect, appreciation, and love to my parents Jagadeesh and Sathyavathi

ii

ACKNOWLEDGEMENTS I am heartily grateful to my guide Dr.Swati Routh,who is the g-person behind my thesis.Who corrected all my silly math errors patiently in the domain of calculation.I want to really thank my parents for their constant encouragement and their support in all aspects of my life. My father Jagadeesh for building my spirit of light in the path of wisdom and my mother Sathyavathi for all her love and sacrifices. My little brother Karthik who plays a role of enlightening me with his own quiet way.Also,Special thanks to my inspirational paternal uncle Manjunath and also my missing uncle Ravi Kumar.Since I am born in a cultured Brahmin’s family for all the folks who love god can thank him on behalf of me.Special thanks to my friends who are keeping me alive in reality Manasa,Prashanth,Kiran and V.K.Bhardwaj. Finally wanna thank my heroes who gave me knowledge my middle school teacher Nagendra sir, my college lecturers (NMK),A.P.Jain who constantly scolded me and called me scientist to put me in the right path.All my Joseph’s teachers Narsim Murthy,Sheela Thomas,Dr.Sudha,Dr.Sandra,Dr.Rabbi,J.B and Dr.Melvin.The person who taught me dynamism and not to care for foolishness Stephan Titus.My first ever project guide Madhusudhan Sir who really have all right to trash me.But he really made me know what experimentation in physics is all about.My all time supporter Dr.Shylaja mam director of planetarium ,who is my real unseen hero of my life so special thanks to her.The person who has kept my spirit going in Jain University Dr.Sovan Ghosh.The person with cute English ascent Dr.Dinesh,who has supported in many ways.Also thank all my cheering IAYC friends Anika, Lama, Ero,Maciej and my International guide Aga.I should not forget all the people who are blessing me from the doors of iii

heaven in the loving memory of the following:My grand father Bhyrappa,grand mothers Sitha Lakshmi and with more love Rajamma. My paternal uncle Nagraj,teacher (NMK) May 23, 2014

iv

ABSTRACT

PROPAGATION OF ACOUSTIC WAVES IN EARTH’S ATMOSPHERE Madhu Kashyap, MSc Jain University, 2014

Supervising Professor: Dr.Swati Routh The earth is surrounded by five gaseous layers which is constrained by earth’s gravitational pull. Acoustic waves travel with the speed of sound through a medium.H.Lamb few years ago has derived cutoff frequency for stratified and isothermal medium.In order to find the cutoff frequency for non-isothermal medium many methods were introduced.But, in this thesis we have choosen the turning point frequency method to determine the acoustic cutoff freequency of earth’s troposphere.This turning point frequency method can be applied to various atmospheres like solar atmosphere, stellar atmosphere etc. Basically, in this thesis the earth’s troposphere is chosen for studying the variation of temperature with respect to atmospheric height, these temperature profile’s data is obtained by International Standard Atmosphere(ISA) model. Using the turning point frequency method in non-isothermal medium, I have analytically derived the cutoff frequency for earth’s troposphere.

v

TABLE OF CONTENTS ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

Chapter

Page

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1

Overview of Earth’s Atmosphere . . . . . . . . . . . . . . . . . . . . .

2

1.2

Troposphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Stratosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.4

Mesosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.5

Thermo sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.6

Exosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2. Overview of Propagation of Acoustic Waves . . . . . . . . . . . . . . . . .

5

3.

2.1

Propagation of Acoustic Wave . . . . . . . . . . . . . . . . . . . . . .

5

2.2

Lamb’s acoustic cutoff frequency . . . . . . . . . . . . . . . . . . . . .

7

Cutoff Frequencies for In-Homogeneous Medium . . . . . . . . . . . . . .

9

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

3.2

Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

3.3

Transformed wave equation . . . . . . . . . . . . . . . . . . . . . . .

10

3.4

Oscillation and turning point theorems . . . . . . . . . . . . . . . . .

11

3.5

Euler’s equation with its turning point frequencies . . . . . . . . . . .

12

3.6

Turning-point frequencies

. . . . . . . . . . . . . . . . . . . . . . . .

13

3.7

The cutoff frequency . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

vi

4. Propagation of Acoustic Waves in troposphere of Earth’s Atmosphere . . .

15

4.1

Acoustic Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . .

15

4.2

Local acoustic cutoff frequency

. . . . . . . . . . . . . . . . . . . . .

16

4.3

Cutoff Frequency for Troposphere . . . . . . . . . . . . . . . . . . . .

18

4.4

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

4.5

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

vii

LIST OF FIGURES Figure

Page

1.1

Layers of Earth Atmosphere . . . . . . . . . . . . . . . . . . . . . . .

2

4.1

Lamb cutoff frequency . . . . . . . . . . . . . . . . . . . . . . . . . .

19

4.2

A plot of cutoff freequecy versus height of the troposphere

20

viii

. . . . . .

CHAPTER 1 Introduction Acoustic-gravity wave is basically an acoustic wave with low frequency which cannot be detected by humans directly. So, it can be measured with a technique having lower and higher cut-off frequencies. Usually, acoustic modes have higher cut-off frequency and gravity modes have lower cut-off frequency. In other words, we can say acoustic waves propagating in the presence of Earths gravitational field for our future discussions in this paper. During the year 1960 and 1980s number of meteor-fireball (large bodies) entered into the earth’s atmosphere creating numerous blast waves during their drag interaction with the air [35].A century before H.Lamb [34] worked out the cut-off frequency for linear acoustic waves propagating in isothermal and stratified medium, which was called as isothermal atmosphere. The cut-off frequency is defined as the ratio of sound speed to twice of its density displacement [34]. Since, Lamb [1] had also worked on non-isothermal medium he has obtained the corresponding cut-off frequency formula analytically. Even though there are many ways to understand Lambs work. In this paper we have chosen to work with the analytical and numerical solutions of acoustic waves [4,6-8,16,17].According, to Lamb the propagating acoustic waves excite atmospheric oscillations with the frequency equal to the natural frequency of the atmosphere [1,3,18].So, this phenomenon can be applied to various stellar and also planetary atmospheres. Therefore, on the basis of the result of local cut-off frequency the required cut-off frequency for earths troposphere is derived and is described graphically. These results are obtained with the help of ISA (International Standard Atmosphere) model [36], which has provided the 1

2 data for variation of temperature and speed of sound with respect to height which is premier to troposphere only.

1.1

Overview of Earth’s Atmosphere

Atmosphere Layers The earth’s atmosphere is divided into several layers namely :Troposphere, Stratosphere, Mesosphere and upper atmospheres.Basically, the atmospheres are classified on the basis of variation of temperature with respect to height.The temperature decreases with increase in height until the troposphere.But, in stratosphere the temperature increases with height..We know that speed of sound is dependent on temperature which involves various factors like pressure , specific heat constant etc ; however these factors will be studied in detailed in the upcoming chapter.

Figure 1.1. Layers of Earth Atmosphere.

3 1.2

Troposphere It is nearest atmosphere to the earth’s surface and its height is measured up

to 12 Km.The troposphere is bounded by a tropopause as shown in the figure.In tropopause the temperature variation in stabilized.The word pause refers to isothermal medium.Whereas troposphere is a non-isothermal medium.

1.3

Stratosphere It is about 12km above from the tropopause .Here, temperature increases w.r.t

height.This is because of increase of ultra-violet absorption from the ozone layer.Similar to tropopause we have stratopause which is located between stratosphere and mesosphere.The stratopuse has a length of around 50 to 55 Km.

1.4

Mesosphere It is the third layer of the earths atmosphere, it extends up to 50 Km from

stratopause to the mesopause around 80-85 Km. The temperature decreases with respect to increase in the height.Note, this mesopause is the coldest part of the earth’s atmosphere with a temperature around -85 deg.

1.5

Thermo sphere It is the penultimate layer of the earth’s atmosphere, it extends up to 80 Km

from mesopause to thermopause around 500-1000 Km.Here, the temperature increases w.r.t height.In the analogy with stratosphere it does not happen due to absorption of ozone layer, this happens due to low density of molecules.

4 1.6

Exosphere It is the final layer of the earth atmosphere , it extends from exobase of the

thermo sphere with an height around 700 Km to 10,000 Km which is the upper limit of the atmosphere.We can observe the satellites moving around in this atmosphere.

CHAPTER 2 Overview of Propagation of Acoustic Waves 2.1

Propagation of Acoustic Wave Previously a century ago, Lamb [34] derived a cutoff frequency for linear acoustic

waves propagating in a isothermal and stratified medium, which was called as isothermal atmosphere. The cutoff frequency, is defined as the ratio of speed of sound to two times the pressure scale height.Which was obtained by solving the acoustic wave equation for the vertical displacement. Since we know that the background medium is isothermal .i.e, with same temperature the cutoff frequency becomes a global quantity for the whole medium. Hence ,the present cutoff is known as the acoustic (or Lamb’s) cutoff frequency. The physical phenomenon of this cutoff is that when the wave propagation is altered by the density gradient ,then the wavelength is either equal to or longer than that of the density scale height. Otherwise, the waves propagate freely in the medium due the cutoff frequency being a global quantity. Therefore, the effect on the wave propagation is same in every given atmospheric height. Lamb [34] also showed that the waves are propagating only when their frequencies are higher than that of the cutoff, otherwise they would be evanescent. Note that the cutoff is the natural frequency of the atmosphere [34]; the latter simply means that any acoustic disturbance imposed on any atmosphere would trigger an atmospheric response at the cutoff frequency [34]. Now consider the Lamb’s [34] work on the non-isothermal atmosphere, which consists of decreasing temperature linearly with respect to height.Which is being used to study the effects of uniform temperature gradient with the acoustic cutoff 5

6 frequency. He also derived the analytical solutions to determine the conditions for the acoustic wave propagation in the atmosphere.In his work Lamb considered the acoustic waves have two dimensions namely vertical and horizontal, with a uniform vertical temperature gradient. The final analytical solutions of this model was used to determine the range for frequencies related to propagation of acoustic waves. The different methods (on the basis of Lamb’s work)that can be used to determine the propagation of acoustic waves are the following; global and local dispersion relations, WKB approximation, finding analytical or numerical solutions of acoustic wave equations The global dispersion relation for acoustic waves can be studied with the help of two quantities that is: when the background medium is homogeneous [34] or when gradients of the physical parameters of the medium doesn’t directly affect the speed of sound, as per Lamb’s isothermal atmosphere (Summers 1976; Thomas 1983; Morse and Ingard 1986; Salomons 2002). The attempts to justify local dispersion relation approach, we need the acoustic wavelength to be shorter than the characteristic scales over which the basic physical parameters in the medium is varied (e.g., Whitman 1974; Thomas 1983). The another requirement is the WKB approximation and many studies of acoustic (and other) waves were performed on the basis of this approximation.

In this thesis, the analytical and numerical solutions of acoustic wave equations play a major role. So, these solutions can be applied to various propagation of acoustic in an isothermal medium (Whitman 1974;Thomas 1983; Campos 1986). The above propagation condition for isothermal medium was done by H.Lamb (1908, 1932). He also showed that the cutoff is the natural frequency of the atmosphere,i.e, the propagating acoustic waves can excite atmospheric oscillations with a frequency equal to that of the natural frequency (e.g., Schmitz and Fleck 1992, 1998).

7 A method for determining the cutoff frequency for linear and adiabatic acoustic waves propagating in non-isothermal media without gravity was developed by Musielak et al. (2006). The method has transformations of wave variables which leads to standard wave equation.Basically, it uses the oscillation theorem to determine the turning point frequencies. Then the physical arguments are used to choose the largest of these turning point frequencies as the acoustic cutoff frequency.The physics arguments could be based on pressure or velocity.

2.2

Lamb’s acoustic cutoff frequency Using original Lamb’s work[1908,1910,1930], consider acoustic waves propa-

gating in the z-direction in the background homogeneous medium with the gravity ~g = −gˆ z .The density gradient ρ0 (z) = ρ00 exp(−z/2H), where ρ00 is the gas density at the height z = 0 and H = c2s γg is the density scale height, with γ being the specific heat ratio and cs is the speed of sound. In Lamb’s model, the background gas pressure p0 varies with respect to height z, however, the temperature T0 remains constant. We get, H = const and cs = const. This stratified isothermal atmosphere can be applied to the solar , stellar and Earth atmospheres. The waves are described by the following variables: velocity u1 (t, z), pressure p1 (t, z) and density ρ1 (t, z). Applying these assumptions to the standard set of 1-D hydrodynamic equations [Morse and Ingard 1986], we get the continuity, momentum and energy equations as ∂ρ1 ∂(ρ0 u1 ) + = 0, ∂t ∂z ∂u1 ∂p1 ρ0 + = 0, ∂t ∂z ∂p1 dp0 ∂ρ1 dρ0 2 +u − cs +u = 0, ∂t dz ∂t dz

(2.1) (2.2) (2.3)

where, the speed of sound is cs = [γp0 /ρ0 (z)]1/2 = [γRT0 (z)/µ]1/2 , and γ being the ratio of specific heats.

8 From the above equations, the acoustic wave equations for the wave variables u1 (t, z), p1 (t, z) and ρ1 (t, z) becomes

2 ∂2 2 ∂ 2 − cs + Ωac (u1 , p1 , ρ1 ) = 0. ∂t2 ∂z 2

(2.4)

where the acoustic cutoff frequency Ωac = cs /2H. Since Ωac = const, we can make space and time Fourier transforms along with derive of the global dispersion relation: (ω 2 − Ω2ac ) = k 2 c2s , where ω is the wave frequency and k = kz is the wave vector. Which implies that the waves are propagating when ω > Ωac and k is real, and they are non-propagating when either ω = Ω0 with k = 0 or ω < Ω0 with k being imaginary; in any other case, the waves are called evanescent waves.

Further work of Lamb’s approach to non-isothermal atmosphere and the resulting new acoustic cutoff frequencies , which has been applied to Earth’s troposphere is described in Chapter 4 and it is also have.The above work is done on the basis of [34].

CHAPTER 3 Cutoff Frequencies for In-Homogeneous Medium 3.1

Introduction The impotance of global cutoff frequencies is that the propagation of different

waves doen’t depend the physical parameters like the speed of sound.The global cutoff frequencies was introduced by H.Lamb(1908). It can be applied various atmospheres like solar atmosphere, stellar atmosphere and any planetary atmospheres like earth itself. In the earth atmosphere, we have gradients of temperature, density and magnetic field which can affect wave propagation. The global cutoff frequencies obtained with the assumption of the isothermal atmosphere is replaced by local cutoff frequencies.Which correctly accounts for the gradients of temperature in the earth atmosphere. A method for determining the cutoff frequency for linear and adiabatic acoustic waves propagating in non-isothermal media without gravity was developed by Musielak et al. (2006). The method has transformations of wave variables which leads to standard wave equation(i.e, Klein-Gordon equations ).Basically, it uses the oscillation theorem to determine the turning point frequencies. Then the physical arguments are used to choose the largest of these turning point frequencies as the acoustic cutoff frequency.The physics arguments could be based on pressure or velocity However the Klein-Gordon equations was first introduced to solar physics by Roberts (1981), which are used to study the propagation of longitudinal (Rae and Roberts 1982), transverse (Musielak and Ulmschneider 2001) and torsional (Routh 9

10 2007) tube waves.Then, the oscillation theorem was first introduced by Schmitz and Fleck (1998) in their studies of the acoustic wave propagation in the solar atmosphere. None of these methods can be taken as a general method to determine cutoff frequencies in inhomogeneous media.So, we use the method Musielak et al. (2006) for the acoustic waves propagating in an non-isothermal medium.

3.2

Wave equation Consider the wave equation which describes the propagation of linear waves in

a in-homogeneous medium along the z-direction[34], we get, ∂ 2ψ ∂ψ ∂ 2ψ 2 − V (z) + P (z) + Q(z)ψ = 0 , 2 2 ∂t ∂z ∂z

(3.1)

where V is the wave velocity, P and Q are given in terms of wave speeds and atmospheric scale heights with their derivatives.

3.3

Transformed wave equation On the transformation of dτ = dz/V the above wave equation becomes

∂2 ∂2 − + ∂t2 ∂τ 2

p V0 + V V

∂ + q ψ(τ, t) = 0 , ∂τ

(3.2)

where V 0 = dV /dτ . To eliminate the first order derivatives w.r.t τ from this wave equation, we use Z 1 ψ(τ, t) = φ(τ, t) exp 2

τ

p V0 + V V

d˜ τ ,

(3.3)

we get the following Klein-Gordon equation

∂2 ∂2 2 − + Ωcr (τ ) φ(τ, t) = 0 , ∂t2 ∂τ 2

(3.4)

11 where Ω2cr (τ )

3 = 4

V0 V

2 −

1 V 00 1 2 2 pV 0 p0 + pV + 2 − +q , 2 V 4 V 2V

(3.5)

where V 00 = d2 V /dτ 2 . Ωcr is the critical frequency (Musielak, Fontenla, & Moore, 1992; Musielak et al. 2006; Routh et al. 2007). By doing the Fourier transform w.r.t time (but not in space because the coefficients are not constant in space) [v(τ, t), b(τ, t)] = [˜ v (τ ), ˜b(τ )]e−iωt , where ω is the wave frequency, we get

∂2 2 2 ˜ )=0. + ω − Ωcr (τ ) φ(τ ∂τ 2

(3.6)

Now, applying the oscillation and turning-point theorems to the following, we state these theorems with only results.

3.4

Oscillation and turning point theorems

Oscillation theorem: Consider a O.D.E of the form: d2 y 1 + A(x) y1 = 0 , dx2

(3.7)

Basically these solutions are oscillatory in nature. . Assumption,that there is another equation of the form: d2 y2 + B(x) y2 = 0 , dx2

(3.8)

where, B(x) > A(x) for all x. Then, all of these solutions of Eq. (3.8) would also be oscillatory. The proof of this theorem gives a condition for the existence of oscillatory solutions which is available in the literature (e.g., Kahn 1990). Turning-point theorem: Consider an O.D.E of the form: d2 y 1 + A(x) y1 = 0 , dx2

(3.9)

12 The above equation contains a turning point which separates the oscillatory solutions from and non-oscillatory solutions. Again, if there is an another equation of the form: d2 y2 + B(x) y2 = 0 . dx2

(3.10)

Then, the turning point of this equation can be determined using the condition B(x) = A(x). The proof trivial due to the condition that it requires same form of equations

3.5

Euler’s equation with its turning point frequencies Consider the Euler’s equation (e.g., Murphy 1960) which is of the form CE d2 y + 2y = 0 , 2 dx 4x

(3.11)

where CE is a constant and it also determines the form of solution. When, CE > 1, the equation gives us the oscillatory solutions.similarly, when CE < 1 the equation becomes non-oscillatory. Finally, when CE = 1 we get the turning point, which separates the above two solutions. Using the Fourier transform in time, we get d2 Yi 2 + ω − Ω2i (x) Yi = 0 , 2 dx

(3.12)

where the critical frequencies Ω2i (x), with i = 1 and 2, is different for different wave variables and also different for different models. From Eqs (3.11) and (3.12), and also using the oscillation theorem, we can show that the wave equations given by Eq. (3.12) have oscillatory wave solutions when the condition is [ω 2 − Ω2i (x)] > 1/4x2 is true for all x. We can also show that the turning point theorem can be applied to any wave equation having the condition [ω 2 − Ω2i (x)] = 1/4x2 which is valid for all x

13 3.6

Turning-point frequencies Applying the oscillation and turning-point theorems to the following turning-

point frequencies, Ω2tp,τ (τ ) = Ω2cr (τ ) +

1 , 4τ 2

(3.13)

where Z τ (z) =

z

d˜ z + τC , V (˜ z)

(3.14)

Here, τC is an integration constant and has to be evaluated when models are specified (see Chapters 4, ISA model). From Eq. (3.14), the variable τ (z) has the actual wave travel time tw (z) from the starting point of a model to a given height z.

3.7

The cutoff frequency The turning-point frequencies have two separate solutions as propagating and

non-propagating (evanescent) waves.We know that there is a turning-point frequency for each wave variable, but only one of them can be the cutoff frequency. So,we follow Musielak et al. (2006) and Routh et. al. (2007,2009,2013), and choose the largest turning-point frequency as the cutoff frequency. The choice is physically justified by the fact that in order to have propagating waves at a given height z, the wave frequency ω must always be greater than any turning-point frequency at this height; so, as a result of this choice both the wave variables are always determined by the propagating wave solutions. So, the result of this method can be used to generalize the theoretical problems that were faced by global dispersion method and WKB methods. Finally, in this thesis this turning point result is applied to the troposphere of earth’s atmosphere

14 (see chapter 4).It can also be applied to solar atmosphere and as well as earth’s atmosphere.

CHAPTER 4 Propagation of Acoustic Waves in troposphere of Earth’s Atmosphere 4.1

Acoustic Wave Equations Consider the 1-D atmospheric model whose gradients of density, temperature

and pressure occur along the z-axis. Propagation of linear and adiabatic acoustic waves is described in terms of basic 1-D hydrodynamic equations given by [34] ∂ρ ∂(ρ0 u) + = 0, ∂t ∂z ∂u ∂p ρ0 + + ρg = 0 , ∂t ∂z ∂p dp0 dρ0 ∂ρ 2 +u − cs +u = 0, ∂t dz ∂t dz

(4.1) (4.2) (4.3)

where u, p and ρ are the perturbed velocity, pressure and density respectively. Also, ~g is gravity, cs is the speed of sound, ρ0 and p0 are the background gas density and pressure respectively. The background medium is assumed to be in hydrostatic equilibrium, which implies that dp0 /dz = −ρ0 g. Also, the speed of sound is given as cs = [γp0 /ρ0 ]1/2 = [γRT0 /µ]1/2 , where γ is the ratio of specific heats, R is the universal gas constant, µ is the mean molecular weight and T0 is the background temperature. For stratified and non-isothermal medium, T0 = T0 (z), cs = cs (z), with both density Hρ and pressure scale heights Hp are also being functions of z. Now qi , where i = 1, 2 and 3, with q1 = u, q2 = p and q3 = ρ, on combining the linearized and 1-D hydrodynamic equations we get the following wave equations 2 ∂2 c2s (z) ∂ ˆ −1 ∂ 2 ˆ − cs (z) 2 + Li qi = 0 , Li ∂t2 ∂z Hi (z) ∂z 15

(4.4)

16 The above equation describes the propagation of linear and adiabatic acoustic waves in a non-isothermal atmosphere. ˆ 1 = ˆ1, Here L ˆ 2 = ˆ1 − g L

∂ ∂t

−2

∂ ∂z

and

2 ˆ3 = ∂ . L ∂z 2

(4.5)

Now, H1 (z) = Hp (z) and H2 (z) = H3 (z) = −Hρ (z) with Hp (z) =

1 dρ0 (z) 1 dp0 (z) and Hρ (z) = , p0 (z) dz ρ0 (z) dz

W.K.T Hρ (z) 6= Hp (z) and

1 Hρ

=

1 Hp

+

(4.6)

Hp 0 . Hp

. On transformation of qi = Lˆi q1i , Eq. (4.4) becomes

4.2

∂2 c2s (z) ∂ ∂2 2 − cs (z) 2 + q1i = 0 , ∂t2 ∂z Hi (z) ∂z

(4.7)

Local acoustic cutoff frequency Consider the transformation dτ =

dz , cs (z)

(4.8)

Using the wave equations for u1 and p1 ; Now rho1 and p1 have the same behavior.So, on introducing u2 (τ, t) and p2 (τ, t), we get ∂ 2 u2 ∂ 2 u2 − + ∂t2 ∂τ 2

∂ 2 p2 ∂ 2 p2 − + ∂t2 ∂τ 2

c0s cs + cs Hp

c0s cs − cs Hρ

∂u2 = 0, ∂τ

(4.9)

∂p2 = 0. ∂τ

(4.10)

and

Here, we can apply the above wave equations in their standard (or Klein-Gordon) forms. On removal of first order terms from the above equation and applying the

17 transformation to it, we get the following equations : i i h h Rτ 0 Rτ 0 u2 = u˜(t, τ ) exp 1/2 τ0 (cs /cs + cs /Hp ) and p2 = p˜(t, τ ) exp 1/2 τ0 (cs /cs − cs /Hρ ) . Which implies

∂2 ∂2 2 − + Ωcr,u,p (τ ) u,˜p(τ, t) = 0 , ∂t2 ∂τ 2

(4.11)

So,the critical frequency of velocity and pressure Ωcr,u and Ωcr,p (Musielak, Fontenla, & Moore, 1992; Musielak et al. 2006; Routh et al. 2007) are Ω2cr,u (τ )

3 = 4

0 2 2 cs 1 c00s 1 cs Hp0 1 cs − + + , cs 2 cs 4 Hp 2 Hp2

(4.12)

Ω2cr,p (τ )

3 = 4

0 2 2 cs 1 c00s 1 cs 1 cs Hρ0 , − + − cs 2 cs 4 Hρ 2 Hρ2

(4.13)

and

here c00s = d2 cs /dτ 2 . Now the critical frequencies in terms of z is given by 0 Ω2cr,u (z) = (ωac + ωas )2 + 2ωac ωas − cs ωas ,

(4.14)

0 . Ω2cr,p (z) = (ωac + ωas )2 − 2ωac ωas + cs ωas

(4.15)

and

where ωac =

γg 2cs

=

cs 2H

is the original Lamb acoustic cutoff frequency and ωas =

1 dcs . 2 dz

Remember the isothermal atmosphere of both the critical frequencies can be reduce to Lamb’s cutoff ωa c. Using the Fourier transform in time and applying the oscillation to turningpoint theorems (Musielak et al. 2006; Routh et al. 2007, 2010, 2013) we get the following turning-point frequencies Ω2tp,u,p (τ ) = Ω2cr,u,p (τ ) +

1 , 4τ 2

(4.16)

18 The turning-point frequencies have two separate solutions one is propagating and the other is non-propagating (evanescent) waves. However, there is a turning-point frequency for each wave variable, only one of them can be the cutoff frequency. So,we follow Musielak et al. (2006) and Routh et. al. (2007, 2010), and identify the largest turning-point frequency as the cutoff frequency. (i.e, to check whether Ω2cr,p (z) > Ω2cr,u (z) or vice versa ) By using similar conversion for the turning-point frequencies Ω2tp,u (τ ) and Ω2tp,p (τ ) we get, Ω2tp,u,p (z)

=

Ω2cr,u,p (z)

1 + 4

Z

z

d˜ z + τC cs (˜ z)

−2 ,

(4.17)

and the cutoff frequency given by Ωcut, (z) = max[Ωtp,u (z), Ωtp,p (z)] .

(4.18)

So the condition for propagation of waves is ω > Ωcut and similarly for non-propagating waves is ω ≤ Ωcut .(Refer Hammer et al. 2010), [34]. Here the cutoff frequency is a local quantity, which describes the relation between height with respect to the frequency of acoustic waves.

4.3

Cutoff Frequency for Troposphere To derive the annlytical cutoff frequency, consider the variation of temperature

T w.r.t height [ISA Paper] and is given by the formula T = Tc − Ch

(4.19)

6.5 where Tc is background temperature =288.15k, C= 1000 in terms of meters for height.

Consider, speed of sound equation and apply the above equation in it. So, we get Cs = [γRTc − Ch/µ]1/2

(4.20)

19 Then, Cs = Cso [1 − ah]1/2

(4.21)

Where, Cso = Cs = [γRTc /µ]1/2 and a= TCc Now, ωac =

Cso [1 − ah]1/2 M g 2K(Tc − Ch)

(4.22)

Figure 4.1. Lamb cutoff frequency. Again, ωas = −

acso 4[1 − ah]1/2

(4.23)

0 ωas =−

a2 cso 4[1 − ah]3/2

(4.24)

On differentiation,

Substituting the above expressions in eqations(15) and (16) and we get Ω2cr,p (z) > Ω2cr,u (z). So the turning point frequency of equation (16) takes the form,

20

Ω2tp,u,p (z) =

2 2 2 Cso [1 − ah]M 2 g 2 a2 Cso a2 Cso + + 2 4K 2 (Tc2 + C 2 h2 − 2Tc ch) 8(1 − ah) 16(1 − ah)

(4.25)

The square root of the above expression leads to the cut off frequency of troposphere.

Figure 4.2. A plot of cutoff freequecy versus height of the troposphere.

21 4.4

Conclusion The plot of cutoff frequency (per sec) versus height (Km) describes the variation

of cutoff frequency in non-isothermal medium for earth’s troposphere.For troposhere the value zero corresponds to the sea level and 1200 Km is the upper limit. This graph was obtained on the basis of analytical solution of the turning point frequency method (refer equation (6)).During integration transformations we have choosen the integration constant to be zero ( i.e, the starting point of the wave propogation).Finally, we can conclude that the cutoff frequency is not a global quantity, but it is a local cutoff. Therefore, for a acoustic-gravity wave to propagate in earth’s troposphere, the frequency of the wave at a height has to be higher than the cutoff frequency at that height.Here, the cutoff frequency almost linearly increases with respect to height in earth’s troposphere. so this imposes a propagation restriction on the waves. The waves which have frequency higher than the local cutoff value they will propagate higher in the atmosphere; while other waves which have less frequency than cutoff value will be reflected back or be evanescent.We compared our result with the Lamb’s cutoff calculated locally at each height and found our cutoff frequency is greater than Lamb’s cutoff which is expected because our results are derived taking non-isothermality of the atmosphere in to account.

PROPAGATION OF ACOUSTIC WAVES IN EARTH’S ATMOSPHERE

by MADHU KASHYAP

Presented to the Faculty of the CPGS of The Jain University in Partial Fulfillment of the Requirements for the Degree in

Master of Science

Jain University June 2014

23 4.5

References

1 H. Lamb, Proc. Lond. Math. Soc., 7, 122 (1909). 2 H. Lamb, Proc. R. Soc. London, A, 34, 551 (1910). 3 H. Lamb, Hydrodynamics (Dover, New York, 1932). 4 G.B. Whitman, Linear and Nonlinear Waves (Wiley, New York, 1974). 5 P.M. Morse, and H. Feshbach, Methods of Theoret. Physics (Mc Graw Hill, New York, 1953). 6 P.M. Morse, and K.U. Ingard, Theoretical Acoustics (Princeton Uni. Press, Princeton, 1986). 7 E.M. Salomons, Computational Atmospheric Acoustics (Kluwer Acad. Publ., Dordrecht, 2002). 8 L.M.B.C. Campos, Rev. Mod. Phys., 58, 117 (1986). 9 D.W. Moore, and E.A. Spiegel, Astrophys. J, 139, 48 (1964). 10 D. Summers, Quart. J. Mech. Appl. Math., 29, 117 (1976). 11 J.H. Thomas, Ann. Rev. Fluid Mech., 15, 321 (1983). 12 L.M.B.C. Campos, Rev. Mod. Phys., 59, 363 (1987). 13 P.Bala Subrahmanyam, R.I. Sujith, and T.C. Lieuwen, J. Vibration and Acoustics, 125, 133 (2003). 14 T.D. Rossing, and N.H. Fletcher, Principles of Vibrations and Sound (SpringerVerlag, New York, 2004). 15 D.R. Raichel, The Science and Applications of Acoustics (Springer Science+Business Media, New York, 2006). 16 F. Schmitz, and B. Fleck, Astron. Astrophys., 337, 487 (1998). 17 T.M. Brown, B.M. Mihalas, and J. Rhodes, J. in Physics of the Sun, Ed. Peter A. Sturrock, Vol. 1, 177 (1987).

24 18 C.J. Hansen, D.E. Winget, and S.D. Kawaler, 1985, Astrophys. J. 297, 544 (1985); 19 Z.E. Musielak, D.E. Winget, and M. Montgomery, Astrophys. J. 630, 506 (2005). 20 C.J. Hansen, D.E. Winget, and S.D. Kawaler, 1985, Astrophys. J. 297, 544 (1985); 21 T.M. Brown, and R.L. Gilliland, Ann. Rev. Astron. Astrophys. 32, 37 (1994); 22 Z.E. Musielak, D.E. Winget, and M. Montgomery, Astrophys. J. 630, 506 (2005). 23 N. Suda, K. Nawa, and Y. Fukao, Science 279, 2089 (1998); J. Rhie, and B. Romanowicz, Nature 431, 552 (2004). 24 N. Kobayashi, and N. Nishida, Nature 395, 357 (1998). 25 D. Deming et al., Astrophys. J. 343, 456 (1989); 26 U. Lee, Astrophys. J. 405, 359 (1993). 27 Hammer, R., Musielak, Z. E., & Routh, S. 2010, AN, 331, 593 28 Musielak, Z.E., Musielak, D.E., and Mobashi, H. 2006, Phys. Rev. E, 73, 0366121 29 Musielak, Z.E., Routh, S., & Hammer, R. 2007, ApJ, 660 30 Routh, S., Musielak, Z.E., Hammer, R., Sol. Phys. 246, 133 (2007) 31 Routh, S., Musielak, Z.E., Hammer, R., ApJ, 709, 1297-1305(2010) 32 Routh, S., Musielak, Z.E., Hammer, R., ApJ, 763, 44 (2013) 33 Vernazza, J. E., Avrett, E. H., and Loeser, R. 1981, ApJS, 45, 635 34 Routh,University of Texas Arlington ,propagation and generation of waves in solar atmosphere(2009) 35 DOUGLAS O. REVELLE, Annals of the New York Academy of Sciences Volume 822, Near-Earth Objects: The United Nations Conference pages 284302, May 1997 36 Tom Benson, NASA Earth Atmosphere model