propagation modes of infrasonic waves in an isothermal atmosphere with constant winds

il.2, 11.3, 11.7 Received 19 November 1965

Propagation Modes of Infrasonic Waves in an Isothermal Atmosphere with Constant Winds

ALLAN D. PIERCE

AVCO RAD, Wilmington, Massachusetts 01887

A dispersion relation for low-frequency disturbances in an isothermal atmosphere with constant horizontal winds is derived. It relates the three components of the wave-propagation vector k to the angular frequency w and depends on the sound speed, the wind velocity v, and the acceleration of gravity g. The relation is obtainable from that for propagation in an isothermal atmosphere without winds if w is replaced by w-k- v. The dispersion relation is studied with reference to the problem of the propagation of waves from a stationary source of limited spatial extent viewed by a stationary observer. The topology of the propagation surface in k space is examined. Propagation modes are defined as corresponding to disjoint portions of the surface in k space for fixed w. The criteria for a given mode existing for given wind speed and frequency are derived and the group velocity of each type of mode is studied. The theory predicts the existence of two wind modes that, at sufficiently high frequencies, have group velocities nearly identical to the wind velocity. Phase velocities of the wind modes also tend to equal the wind velocity. In the latter portion of the paper, the trajectories of fluid particles are studied for each type of wave-propagation mode. It is shown that, in terms of particle motions, the wind modes are physically indistinguishable from the gravity mode predicted for an isothermal atmosphere at rest when viewed by an observer moving with the wind.

INTRODUCTION

NE of the major distinctions between infrasonic wave propagation in the atmosphere and propagation at audible frequencies is that the effects of the Earth's gravitational force and the associated density stratification with height cannot, in general, be ignored for infrasonic waves. Although the effects are usually ignorable for waveperiods less than ! min, they can drastically alter the general characteristics of the propagation when the periods are of the order of 5 min or greater. The effects of gravity on low-frequency waves in an atmosphere initially at rest have been investigated by a number of workers. We cite, for example, the early work by Lamb • on vertical propagation of waves and the more-recent work on long-range propagation of acoustic-gravity waves from nuclear explosions by Harkrider. 2

For propagation involving gravitational effects, the simplest type to study is that in an isothermal atmosphere, since, in this case, the propagation is mathe- matically equivalent to that in a uniform but anisotropic medium. The propagation of low-frequency

x H. Lamb, Hydrodynamics (Dover Publications, Inc., New York, 1945).

•' D. G. Harkrider, J. Geophys. Res. 69, 5295-5321 (1964).

832 volume 39 number 5 part 1 1966

disturbances in an isothermal atmosphere without winds has been discussed in recent years by Hines, a by Moore and Spiegel, 4 and by the present author. 5 The dispersion relation appropriate for this case was found to be

where c is the (constant) speed of sound and o•A and cos are two characteristic frequencies given by the expressions

w.a= ('r/2)g/c, (2)

wB = ('r-- 1)ig/c. (3)

In the above, g is the acceleration of gravity and '• is the ratio of specific heats for air. The dispersion relation, Eq. 1, relates the three components of the wavenumber k to the angular frequency w and serves as a basis for a detailed study of the propagation characteristics in the isothermal atmosphere.

One purpose of this paper is to discuss the modifi- cation of Eq. 1 when constant horizontal winds are present in the atmosphere. As we show, the form of the appropriate dispersion relation is only trivially dif-

a C. O. Hines, Can. J. Phys. 38, 1441-1481 (1960). 4 D. W. Moore and E. A. Spiegel, Astrophys. J. 139, 48-71

(1964). 5 A.D. Pierce, J. Acoust. Soc. Am. 35, 1798-1807 (1963).

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.59.226.54 On: Wed, 10 Dec 2014

11:42:33

INFRASONIC WAVES IN AN ISOTHERMAL ATMOSPHERE

ferent from that above, being given by

k?= (•--co.•)/c •-- (k•4-ky •) (•-co•)/•2 •, (4)

with • representing the Doppler-shifted angular frequency

•=co-k-v, (5)

where v represents the (horizontal) wind-velocity vector. Although, for completeness, we outline the derivation of Eq. 4 from the equations of hydrodynamics in the next section, it should be immediately clear from consideration of the plane-wave case that Eq. 4 represents the extension of Eq. 1 to include winds. A plane wave of frequency co, viewed by a stationary observer in an atmosphere with constant winds, would appear as a plane wave of frequency • in a atmosphere without winds, if the observer were moving with the wind. How- ever, the interpretation of Eq. 4 becomes somewhat more difficult than that of Eq. 1 when one considers Eq. 4 in reference to the problem of the propagation of waves from a stationary source and seeks to find the form of surfaces of constant phase or of surfaces of constant arrival time for wavetrains of limited time dur-

ation and nearly constant frequency. To consider this problem, one must first study the

relation between the direction of energy flow and the direction of the wavenumber vector k. This is done in

Sec. II where we consider the surfaces in k space for constant co described by Eq. 4. The group velocity is in a direction perpendicular to these surfaces.

To classify the various arrivals that may be expected from a stationary source of limited spatial extent, we consider disjoint portions of the surface in k space for fixed w as corresponding to different propagation modes. Within the context of this definition of a wavemode, we find that a new type of wave-propagation mode appears. Modes of this type, which we call wind modes, are possible only when gravity and winds are both si- multaneously present. Although we do not discuss the excitation of these modes, it would appear that any stationary source may excite such modes and that disturbances propagating via these wind-wave modes may be a common phenomena in the atmosphere.

Since the magnitude of the Doppler-shifted angular frequency • in the wind modes is always less than the Brunt frequency we, particle motions in the wind modes when viewed by an observer moving with the wind will always appear as being indistinguishable from those in the gravity mode predicted by Hines 3 for an atmosphere without winds. However, the concept of wind modes, as distinguished from that of a gravity mode, may be useful in the interpretation of waves from stationary sources.

The problem of sound propagation is an isothermal atmosphere with constant winds may appear, at first sight, to have little relevance to the actual problem of sound propagation in the real atmosphere, where tem-

perature and wind velocity vary markedly with height and, to a lesser extent, with horizontal distances. How- ever, the wind modes discussed in this paper tend to travel with a group velocity nearly equal to the wind velocity. This, plus the fact that their wavelengths are comparatively short, would seem to indicate that they tend to follow the wind streamlines and are relatively unaffected by wind shears and vertical temperature gradients.

Furthermore, one promising method of approach to the study of sound propagation in more-realistic atmos- pheres is for one to consider the atmosphere as con- sisting of isothermal layers with constant wind velocity in each layer. This method has been applied in the absence of winds by Press and Harkrider 6 and by Pfeifer and Zarichny 7 and may be extended to include winds. Thus, the problem of propagation in an isothermal atmosphere with constant winds appears fundamental to the study of low-frequency acoustic propagation in the real atmosphere. In a previous paper, the author 8 discussed the propagation of infrasonic waves over large horizontal distances in a temperature- and wind-strati- fied atmosphere. This paper may also be considered as a first step in the study of the residual equations derived in that paper.

The general method of analysis used in this paper was suggested by that which has been successfully used in studying the propagation of electromagnetic waves in ionized gases under the presence of an applied magnetic field? The concepts of wave-propagation surfaces, wavemodes, and group-velocity surfaces that we make use of in Secs. II and III are concepts that have been found useful in the theory of plasma waves. The analogy between the two subjects, acoustic-gravity waves and plasma waves, is immediately clear when one realizes that they are both examples of wave propagation in anisotropic media.

I. DERIVATION OF THE DISPERSION RELATION

The derivation of the dispersion relation Eq. 4 is sufficiently similar to derivations of Eq. 1 already existing in the literature 3.4 that we need only outline the steps involved. The deviations p, p, and u of the pressure, density, and fluid velocity from their ambient values P0, p0, and v are assumed to be of the form

p = p0}P exp[--- i (o•t-- k. x) •,

p=p0•R exp[-i(wt-k. x)•,

u= (U/po D) exp[-- i (o•t- k. x) •,

(6a)

(6b)

(6c)

0 F. Press and D. G. Harkrider, J. Geophys. Res. 67, 3889-3908 (1962).

7 R. L. Pfeifer and J. Zarichny, J. Atmospheric Sci. 19, 256-263 (1962).

8 A.D. Pierce, J. Acoust. Soc. Am. 37, 218-227 (1965). 9 T. H. Stix, The Theory of Plasma Waves (McGraw-Hill Book

Co., Inc., New York, 1962).

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A.D. PIERCE

where P, R, and U are constants. The requirement that Eqs. 6 satisfy the linearized equations of hydrodynamics 8 for an atmosphere with constant sound speed c and constant horizontal wind velocity v leads to a system of five simultaneous linear algebraic equations for P, R, and the three components of U. These are found to have a nontrivial solution only if the relation Eq. 4 holds. In this event, the quantities P, R, and U satisfy certain polarization relations that are similar to those given by Hines • for the isothermal atmosphere without winds. We find

P = c2f• 2h•A,

R= [•22(k•+i.•g/ (2d))-- i (7-- 1)gki-i•'•A,

Ux= •2kxdh•A ,

Uv= •2kydh•A ,

U•= •2 (•22-- ki_i•c•) A ,

where we have abbreviated

(7a)

(7b)

(7c)

(7d)

(7e)

h•= k,q-i(1--'y/2)g/c •, (8a)

ka •= kx•'+kf, (8b)

The quantity f• is given by Eq. 5. The constant A scales the amplitude of the wave and is dependent on the source strength.

The virtue of taking p and p proportional to the square root of the ambient density and of taking u proportional to the inverse of this square root is that the derived dispersion relation Eq. 4 will not contain the factor i. If, k•, kv, and k• are all real, then the energy per unit volume contained in a wave represented by Eqs. 6 will be finite for all x, y, and z. Thus, Eqs. 6 represent a freely propagating plane wave. We are ne- glecting the presence of the ground and, thus, must limit ourselves to waves whose energy density does not diverge at any point in space. With the presence of the ground, the so-called Lamb wave would be possible. s The Lamb wave travels as a surface wave along the ground in the lowest layers of the atmosphere.

The present paper is restricted to plane-wave propagation and ignores the boundary conditions at the earth's surface. By plane waves, we mean waves observed at a sufficiently large distance from a source of limited spatial extent such that wavefronts are approximately planar over distances large as compared with a wavelength.

II. PROPAGATION SURFACE

To facilitate the study of the dispersion relation, Eq. 4, we rewrite it in a nondimensionalized form- i.e.,

n•= (Q•-a•') - (n•2q-n• •) (Q•-aB•.)/Q •, (9)

where n•, nu, n• are the components of the vector

n= kc/co and Q=i-n.M, (10a)

(10b)

M=v/c, (10d)

The vector n has the direction of the phase velocity and a magnitude equal to the speed of sound divided by the phase velocity. It may be considered as the vector index of refraction for acoustic-gravity waves.

We call the surface described by Eq. 9 in the re- fractive-index space (having coordinates n•, nv, n•) the propagation surface. We assume, for definiteness, that • is identically 1.4, corresponding approximately to the appropriate value for air. Then a• will be directly proportional toaB, such that a• 2= (1.225)a• •'. It is clear, therefore, that the form of the surface, Eq. 9, in n space will depend on only the magnitude of M and on the parameter a•. Changing the direction of the wind Mach number M will only serve to rotate the surface about the n• axis. We shall accordingly, for simplicity in what follows, take M to be in the x direction.

The surface described by Eq. 9 is readily seen to be invariant under reflections through either the n•nv plane or the n•-nx plane. It is not invariant under reflections through the nv-n• plane, however, since the quantity Q is not even in n•. The dominant feature exhibiting wind effects is the singular behavior of the surface near n•= 1/M•. In the vicinity of this plane, the quantity n• • will be positive and very large.

If one plots the surface for various choices of a• and Mx, he finds that its form falls into one of four topological categories. For any given pair of values, a• and M•, one of these four topological categories will be realized. In Fig. 1, we divide the plane with axes aB -•

I I I I I i

ANGULAR FREQUENCY IN UNITS OF •B

Fro. 1. Diagram classifying different regions of Mx, co/cos plane according to topological genera of wave propagation surface.



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and Mx into four regions, I-IV, corresponding, respectively, to the four different topological types of the propagation surface. For example, if aB and M x correspond to a point in region I of Fig. 1, then type I topology will be realized. The derivation of the lines separating the four regions of Fig. 1 is given in an':•'un - published report •ø and is omitted here as it is rather lengthy. The line separating regions I and II in Fig. 1 is given parametrically by the equations

(0.225)• a

w/coB (1_ •2)2+0.225, (11a)

[ e) M•= (•2_ 1)2+0.225 _j' (lib)

õ IO I -'- 2o -IO o

-5

-I0 -5 0 I/M x 5 I0 15 n x OR k x c/•

Fro. 2. The wave-propagation surface for M•=0.4, c•B=5.0, illustrating type I topology. The arrows point in the direction of the horizontal component of the group velocity.

as • runs from 0 to 1. The line separating regions II and III is also described by the two same parametric equations---only • must run from (1.225)« to m.

Figure 1 in its application has a certain formal resemblance to the Clemmow-Mullaly-Allis (CMA) diagram ø (which is often used in studying electromagnetic waves in anisotropic plasmas) in that, in each of the various regions in Fig. 1, the topological genera of the surfaces in n space are unchanged. Following the termi- nology used in plasma physics, we refer to each of the four regions in Fig. 1 as bounded volumes. The two dimensional plane having coordinates co/coB and M• are referred to as parameter space.

We now describe the surfaces in n space corresponding to three of the topological types. Type IV corresponds to wind Mach numbers greater than 1 and is of neglig- ible interest in the present context.

Figure 2 shows a surface corresponding to M• and co/coB lying in bounded volume I or Fig. 1. The propagation surface is as viewed from above the nx-ny plane. Lines of constant n• are plotted in the Figure. (It should be borne in mind that the propagation surface is even in n•.) Note that the surface is entirely confined between the planes n•= (1-- o•B)/Mx and n•= (1-]-aB)/Mx. The most significant topological feature from the stand- point of distinguishing type I from other types is the "hole" lying between the planes n•= (1--aB)/M• and n•= 1/M•. Also characteristic is the saddle point on the intersection of the propagation surface with the n,-n• plane that lies between the surface n•= (1--aB)/Mx and the hole. The singularity in n• at the plane n•= 1/M• separates the surface into two sheets. Although these sheets are, in a strict mathematical sense, connected at infinity, we shall find it convenient to regard them as virtually disjoint in applying the definition of a wavemode as being wavemotion associated with points on one and only one disjoint sheet. Thus, we say that

•0 A.D. Pierce, "The Propagation of Infrasonic Waves in an Isothermal Atmosphere with Constant Winds," Avco Corp., Wilmington, Mass., RAD-TR-65-21, AFCRL-65-492 (July 1965).

there are two distinct wavemodes in the bounded volume I.

Figure 2 was drawn for aB= 5.0 and M•=0.4, How- ever, the general features of the propagation surface described above are true for any choice of a•B and Mz lying in the bounded volume I. A wind speed of 0.4c is much larger than is generally encountered in the lower atmosphere. Values of Mx between 0.01 and 0.! are more typical. The choice of 0.4 was made to accentuate the effects of winds in the diagram.

In the complete absence of winds, only the mode where n•< 1/M• would be present and its surface would be a hyperbola of revolution about the ns axis. The hole in Fig. 2 would be centered at the origin and the lines of constant n• would be concentric circles. This mode is commonly called the gravity mode. a The effects of winds on the gravity mode's propagation surface are evident in Fig. 2. Note that the hole's center has shifted to the left. A simple analysis shows that, in the limit of small winds, its center is approximately on the n• axis at

n•= --M•(aB2a• 2-- 1)/(a• 2-- 1) 2. (12)

To first order in Mx, its radius will be that in the case of no winds, i.e., [(a• •'-- 1)/(aB 2-- 1)•.

The mode where n•> 1/M• is a mode whose existence depends on the presence of winds. We note that wavelengths in this mode must be less than 2•-M•/w (since k• is greater than w/cM,). These wavelengths will be van- ishingly small in the limit of very small wind speeds. We find it convenient to refer to this mode as the short-

wavelength wind mode. When the bounding surface is approached in Fig. 1

as one passes from bounded volume I to bounded volume II, the left edge of the hole in Fig. 2 will gradually approach the far-left boundary of the propagation surface. Thus, the hole becomes only an inden- tation in the surface. This is the chiefdistinction between

topological types I and II. An example of type II is shown in Fig. 3. This propagation surface (as was the type I surface) is entirely confined between the planes



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A.D. PIERCE

--]-- - 50 -• 15 0 2

• 0 nz=O I I o

-2 0 2 I/M x 4 6 8 n x OR kxC/•

FIG. 3. The wave-propagation surface for Mx=0.4, illustrating type II topology.

nx= (1-aB)/Mxand (ld-aB)/Mx. Here, as before, there are two virtually distinct modes corresponding to nx> 1/M• and n•< 1/M•, respectively. The former we may identify as the short-wavelength wind mode. The latter, although it evolves continuously from the gravity mode as the frequency is increased, is also a mode that would not exist in the absence of winds. (If M x=0, there would be no propagation for o•s<o•<o•A.) We also, accordingly, refer to it as a wind wavemode, calling it the second wind mode when it is necessary to distinguish it from the short-wavelength wind mode.

The surfaces in n space corresponding to the second wind mode and the short-wavelength wind mode change continuously as one crosses from the bounded volume II to the bounded volume III in parameter space. Hence, one may continue to use these labels for the surfaces where (1-a•)/M•<nx<l/M• and 1/M•<n• < (1-]-as)/M•. However, the bounded volume III is characterized by a third mode whose presence is indicated by a detached spheroidal surface in n space lying to the left of the plane nx= (1-as)/Mx. This mode may be appropriately labeled as the acoustic mode, since it reduces to Hines' acoustic mode a in the limit of zero wind velocities. At the boundary separating

2

-2

I

nz=O ,

0 O.5

0.6

-2 0 2 i/M x 4 n x OR k x c/•

Fro. 4. The wave-propagation surface for Mx=0.4, aB= 10/9, illustrating type III topology.

the bounded volumes II and III in Fig. 1, the acoustic- mode surface is only a point in n space, but it grows in size as the frequency is increased, approaching the surface

nu-t- n z"-t- ( 1 -- M• •') (n •-t- M•( 1 -- M •")-•)" = (1-Mx2) -• (13)

in the limit o•>>o•B. The above formula describes an ellipsoid of revolution about the n• axis, which is elon- gated along the n• direction and centered about the point where n•= M•/(1-M• 2) on the nx axis. In Fig. 4, an example of type iii topology is shown.

III. GROUP-VELOCITY SURFACE

In considering the propagation of plane waves, one must bear in mind that any wave phenomena of a realistic nature must be finite in time duration and must be

generated by a source of limited spatial extent. The concept of a plane wave applies only at relatively large distances from the source where the wavefronts are

nearly planar. To consider the transport of energy, it is convenient to consider the group velocity of the wave governed by Eq. 4. The group velocity will, in the limit of large distances from the source and at time lapses large compared to the time duration of the source, give the direction and speed at which energy is carried by the wave.

The group velocity vg of a given plane wave with wavenumber k and angular frequency o• may be com- puted via the relation

v•= V•o•. (14)

The indicated differentiation may be carried out impli- citly by use of the dispersion relation Eq. 4. Doing this and making use of the notation introduced by Eqs. 9 and 10 we find that,

where

Ug•= M•-l-n•f ,

lggy --- •y f ,

Ugz=ngQ'.f/(Q2-a•2),

(15a)

(15b)

(lSc)

/=Q(Q•.-as•.)/[Q4-as•.(n•2-+-nu)-I. (16)

We have, for convenience, introduced [Ig=¾g/C, which represents the group velocity in units of the sound speed.

It should also be pointed out that it follows from Eq. 14 that the group velocity points in a direction normal to the propagation surface in n space. To de- termine the sense of the group-velocity direction (i.e., inward or outward), we write the denominator in Eq. 16 as

Q2(Q2-aB'.)-•[(Q'.-a•'.)'.-l-aB2(a.4'.-aB'.-l-nz'.)], (17)

using Eq. 9. It follows that the quantity f is positive or negative if Q is positive or negative, repectively.



11:42:33


Hence, ugy has the same or opposite sign as ny, de- pending on whether n• is less or greater than 1/M•, respectively. Also, ug• has the same sign as nz if 1/M•<n•< (i+aB)/M• or n•< (1--aB)/M•. It has the opposite sign to nz if (1--a•)/M•<n•<l/M•. These observations enabled us to sketch in the arrows in Figs. 2-4, showing the directions of the projection of the group velocity on the horizontal plane.

It should be noted that u• and k are not necessarily parallel. This is, of course, only an additional aspect of the fact that the medium is anisotropic. Since the expressions Eq. 15 also depend on the frequency, the propagation will also be dispersive.

The plot of the surface (or surfaces) in ugh, space, as n ranges over the propagation surface, has direct physical interpretation as it is geometrically similar to the surface at which energy arrives at a given time from a point source. The shape of these surfaces will allow one to assess just how rapidly the energy spreads in various directions. Cross sections in the ugy= 0 plane of three such surfaces are shown in Figs. 5-7.

Figure 5 corresponds to the case aB= 5.0 and Mx= 0.4. The identification of the curves as corresponding to the gravity mode or the short-wavelength wind mode was

0.4

02

UgzO

-O2

-O4

i GRAVITY •' i MODE, nx<O I

_ , /-MODE,n x >0

-04 -02 0 02 04 06 08 IO 12 Ugx

Fro. 5. Cross section for ugu=0 of the group-velocity surface when Mx=0.4 and aB=5.0. The coordinates ugx and ugz are the x and z components of the group velocity in units of the sound speed c. The dashed line corresponds to ugx = Mx.

0.4

0.2

Ug z 0

-0.2

-0.4 o

WIND MODE WIND MODE !_

0.2 0.4 0.6 0.8 1.0 1.2 1.4

Ugx

Fro. 6. Cross section for ugv=0 of the group-velocity surface when M•=0.4 and a•=5/3.

0.4

0.2

Ug z o

-0.2

-0.4

SECOND --/'•"" WIND MODE

-0.4 -0 2 0 0.2 0.4 0.6 0.8 1.0 Ugx

Fro. 7. Cross section for u•u=O of the group-velocity surface when M•=0.4 and aB= 10/9.

made using the fact that n•< 1/M• corresponds to the gravity mode, while nx> 1/M• corresponds to the wind mode. The Figure is characteristic of type I propagation, using the classification system described in Fig. 1. Figures 6 and 7 are characteristic of higher frequencies. Figure 6 is for the same parameters, aB= 1.667 and M•=0.4, as was Fig. 3, and corresponds to type II propagation. Figure 7 is characteristic of type III propagation, as was Fig. 4.

As a further example of a group-velocity surface, a contour plot of the gravity mode for a•=5.0 and M•=0.4 is given in Fig. 8. This should be compared with Fig. 5, which shows a cross section of the same surface. Note that the surface has a hole in the center

and is topologically equivalent to a toroid. We consider the group-velocity surfaces for the wind

modes in more detail in the next section.

05

0 0.5 1.0 Ugx

Fro. 8. Group-velocity surface of the gravity mode when a•=5.0 and M•=0.4. Lines of constant ug• are shown for ugh>0 only, since the surface is even in ug,.

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A.D. PIERCE

IV. WIND WAVEMODES

The wind wavemodes are undoubtedly the most novel development of the theory, since they would not be predicted in a formulation that neglected either winds or gravitational effects. To isolate the salient features of these modes, we examine them in the limiting case of high frequencies and low winds, aB2<<l and Since Q2<aB •' for the wind wavemodes, the quantity nx must be of the order of 1/M• and will, therefore, be a very large number. It follows that, for the terms ap- pearing in Eq. 9,

(n•q-nu •) (a•-Q•)/Q2>>a•t•-Q •, (17)

unless Q• is virtually identical to a• •. It is, therefore, apparent that the dominant features of the wind modes may be studied by approximating the dispersion relation, Eq. 9 as

n•2= (n•q-nu •) (a•-Q•)/Q •. (18)

The group velocity (in units of the sound speed) com- puted from this approximate dispersion relation is then

ug•=M•q-n•a•-•Q(aB•-Q•)/(n•q-nu•), (19a)

ugu=nua•-2Q(a•-Q•)/ (n•q-nu•), (19b)

ugh= -- n•a•-•Q3/(n•q-nu2). (19c)

It follows from these equations that the group-velocity surface for the wind wavemodes is described by the equation

{ V•/I-V•+Mx(ug•-M•)-]}•=aB•ug?/V •, (20) where

V•= (u•- M•)•+u•+u• •. (21)

Note that the right-hand side of Eq. 20 is much less than 1 since we are assuming au•<<l. It follows that Eq. 20 will not be satisfied in the high-frequency limit unless

I M•(u•- M•)I>> V•. (22)

Thus, the V • term can be neglected in the denominator of the left-hand side of Eq. 17. With this neglect, the equation may be cast in the form

(•+•+•)•= •, (23)

where we have abbreviated

•= (ugh- M•)/(MxaB), (24a)

•= u•/ (M•aa), (24b)

•'= u•/ (M•aa). (24c)

The surface in •, •/, •' space described by Eq. 23 is easily visualized by plotting lines of constant •/in the •, •' plane. This we have done in Fig. 9. The surface is even in each of the coordinates •, •/, •. Thus. only lines of positive n are plotted. As is evident by examination of Eq. 23, the surface cannot intercept the planes •=0

04

0.2 i=0.15 0.19

• 0

-0.2

-04- -0.6 -0.4 -0.2 0 0.2 0.4 0 6

Fro. 9. The group-velocity surface for the wind modes in the limit of o•>>o•B and Mx<<l. The coordinates t/, n, and •' are proportional to ugx--M•, ugu, and ugz, respectively.

or •'=0, except at the origin. Thus, the surface will consist of four separate surfaces that touch at the origin.

The resemblance of Fig. 9 to the group-velocity surfaces in Figs. 5-7 is particularly evident for the short wavelength wind mode. The second wind mode ap- proaches the limiting form at a slower rate with increasing o•.

Equations 24 and Fig. 9 show that the group velocity of the wind modes in the limiting case under consideration is virtually identical to the wind velocity. The deviation Ivg-v I, where vg is the group velocity and v is the wind velocity, is of the order of

cM • = (q•- 1)lgv•/ (25)

This number is a characteristic velocity for the wind modes. It gives the order of magnitude of the speed with which an explosive disturbance of nearly constant frequency •0 and with a spectrum of wavenumbers will diverge from a point moving with the wind velocity. This point will appear effectively as the source of the disturbance although the actual source (according to our hypothesis) is stationary.

V. PARTICLE ORBITS

Some insight into the nature of the various propagation modes may be obtained by considering the motion of a fluid particle during the passage of a plane wave. If v-4-u is the total fluid velocity (and v is the ambient wind velocity) at point x and time t, then the motion of a fluid particle is described by the equation

dx/dt=v+u. (26)

If the perturbed velocity u is now given by the real part of Eq. 6c, then the motion in a plane wave is described for small amplitudes by

x= x0+ vt+ x', (27)



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where x' is given by the real part of

(i/•) (IJ/po•) exp[- i(•2t- k. x0)•. (28)

The vector x0 represents the approximate position of the particle at time t-0. The remaining symbols are defined in Sec. I of this paper.

The first two terms in Eq. 27 represent unperturbed motion with the wind, while the third term represents the disturbance and describes the perturbed displacement of the fluid particle. The components of x ' are each periodic in time with an angular frequency of

Since the polarization relations, Eqs. 7c and 7d, re- quire that x'/y•=kx/k•, the vector x' will at all times lie in a plane that contains the vector k and the z' axis. The tip of the vector x' will describe an ellipse in this plane as t varies. The parametric equations for this ellipse are readily seen from Eqs. 7 to be

x'= (kx/k•)Ca sin(9t--•0--t•),

y'= (k•/ka)Ca sin(f•t--/•0--3t),

z'=C• sin (gt-- •0),

(28a)

(28b)

(28c)

where Ca, C•, 150, and 3t are real quantities that do not depend on t. The parameter 3t is the phase of h• as defined by Eq. 8a. The ratio Ca/C•, which we denote by a, is given by

a=Ca/C•= kac2[k•2 - (1--«'y)•'g2/c4]¾ (92-- ka22. (29)

If the quantity 3t were 0 or ,r, the path described by Eqs. 28 would be a straight line and one could speak of a linearly polarized wave. The line of polarization would make an angle tan-•a with the z' axis.

In the more general case, one may, with some effort, derive expressions characterizing the shape and orien- tation of the ellipse described by Eqs. 28. The ratio of the semiminor-axis length to the semimajor-axis length is

o o

where r = ( 1 -- T) ¾ ( 1 q- T) •, (30)

T = [ 1 -- 4a" (sin2u) / (1 + a•)"• L (31)

The angle qb that the semimajor axis makes with the horizontal component of k is obtainable from the equations

sin2•=2a(cost•)/•(l+a•)T•, (32a)

cos2qb= (a 2-- 1)/•(i+a•)T•. (32b)

With the above definition of qb, the angle that the semimajor axis makes with the -{-z' axis will be

An additional characterization of the particle motion is that of the sense in which the vector x' revolves as t varies. Consider first the case where ft> 0. Then, from Eqs. 28, one sees that x' will be positive when z' is both

zero and increasing with time if q•< 0, where

q•= (kx/ka)a sint•. (33)

If q•> 0, then when z' is zero and increasing with time, x' will be negative. It is clear that the particle motion when viewed from a point on the --y' axis at a large distance from the origin will appear to be clockwise if qx>0, and counterclockwise if q•< 0. If, on the other hand, ft< 0, then the motion will appear to be counterclockwise if qx> 0 and clockwise if q•< 0. Consideration of the form of q• as given by Eq. 33 allows one to con- dense these observations to the following statement. If one views the particle motion such that the direction of ka/ft points to one's right and the z' direction is upward, then the motion will be clockwise if a sinu>0 and counterclockwise if a sin3t < 0.

We may note that sin3t is always positive and that cos3t has the same sign as k,. Also, the quantity ft 2-- kH2c • (and, therefore, the quantity a) is positive for the acoustic mode and is negative for the gravity mode and for the two wind wavemodes. The proof of this latter statement follows from two facts. First, one may always find a point on the propagation surfaces of the acoustic mode for which •2--ka2c2>0, and, similarly, may always find a point on the propagation surfaces for any of the other modes for which this quantity is negative. Second, one may show that the line FF--ka"c" does not intersect the line k?=0 in the k•, ku, plane, where k, 2 is given by Eq. 4.

It is clear from Eq. 32 that 0<qb<,r/2 if acos3t>0 and that ,r/2<•<,r if acos3t<0. This, in conjunction with the statements made in the preceding paragraphs, implies that 0<qb<•r/2 or ,r/2<qb<,r if k,>0 or k•<0, respectively, for the acoustic mode. The converse is true for the gravity and wind modes.

It also follows from our previous statements concern- ing the sense of the particle motion that the motion will be clockwise for the acoustic mode and counterclockwise for the other modes when viewed such that the

direction of k•i/9 points to one's right and the z' axis is upward.

The conclusions made in the preceding paragraphs are summarized in Fig. 10.

The fact that x', y', and z' vary sinusoidally with time with a period of 2,r/ft rather than 2,r/c0 is of par- ticular relevance in the interpretation of the wind modes. If f•2 is less than c0a 2, then the particles behave as though they were moving in a gravity mode with the absence of winds. Even for c02>>c0a 2, it is possible to have ft2<c0a 2 for some range of k and yet satisfy the dispersion relation, Eq. 4. An observer moving with the wind would be unable to distinguish a wind mode from a gravity mode.

The minimum period of particles moving in the wind modes is 2,r/c0•. This period, with values of g and c characteristic of the lower atmosphere, is typically of the order of 5 min. However, the waveperiods as seen by an observer at rest may be much less than this.



11:42:33

A. D. PIERCE

Fro. 10. Sketches of fluid trajectories relative to a point moving with the wind. The arrows indicate the sense in which the perturbed displacement moves around the ellipse. The coordinate •' is the component of the perturbed horizontal displacement in the direction of kale2, where ka is the horizontal component of the wavenumber vector and •2 is the Doppler-shifted angular frequency. (a) Acoustic mode for ks>0. (b) Acoustic mode for ks <0. (c) Gravity and wind modes for ks>0. (d) Gravity and wind modes for ks <0.

VI. CONCLUDING REMARKS

In this paper, we have classified the various waves that would propagate out from a stationary source of limited spatial extent in an atmosphere with constant winds in terms of propagation modes defined from a consideration of the topology of the surfaces in k space for fixed co. It must be borne in mind that this defi-

nition of the propagation modes is in reference to this given experimental situation. Indeed, as we have shown, the wind modes are physically indistinguishable from the gravity modes in terms of particle motions when viewed by an observer moving with the wind. The characteristic properties of the wind modes that justify their being given a separate label from the gravity mode is their tendency to propagate only downwind and their existence at frequencies higher than the Brunt frequency when viewed by an observer at rest. Propagation in the wind modes is characterized, particularly at the higher frequencies (i.e., co>coB, by phase and group velocities

nearly identical to the wind velocity. The former would imply that wind-mode wavelengths (especially at low wind speeds) would be much shorter than acoustic-mode wavelengths for the same frequency.

Although the model adopted in this paper (of an isothermal atmosphere with constant winds) is admit- tedly idealized, it would appear that this prediction of the wind modes should also be relevant for the actual

atmosphere. In other words, it should be possible to excite wavelike disturbances that travel downwind with

the wind velocity and that have wavelengths of the order of the wind speed divided by the frequency.

The questions of whether or not such disturbances are excited by natural causes, and whether they are present frequently in the atmosphere, deserve careful consideration. Cook and Young n report that pressure fluctuations recorded onmicrobarovariographs are some- times caused by "the turbulent passage of the wind over obstructions on the landscape, such as buildings, trees, and hills," and that "the noise pressure is conveered through the atmosphere at approximately the mean wind speed." It is possible that the wind modes may afford a mechanism by which these pressure fluctuations are transported. There are, of course, competing mechanisms such as waves guided along a discontinuity of wind speed or density. Turbulence, in which the non- linear terms of the equations of hydrodynamics play an important r61e, could also be a mechanism. Just which of these mechanisms is responsible for the pressure fluctuations is a topic that deserves further study and experimentation. It may be that wind modes are not responsible for these fluctuations but are responsible for fluctuations high in the atmosphere that are not as easily observed.

ACKNOWLEDGMENTS

The research reported in this paper was sponsored by the U.S. Air Force Cambridge Research Labora- tories, Office of Aerospace Research. The author should like to thank Susan Darcy and Mark Bernstein, for their very able assistance with the numerical compu- tations. He would also like to thank Prof. CO. Hines, for an enlightening discussion regarding the physical nature of the wind wavemodes and for a critical review

of this paper.

n R. K. Cook and J. M. Young, Sound-Its Uses and Control 1, No. 3, 25-33 (1962).

840 volume 39 number 5 part I 1966


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propagation modes of infrasonic waves in an isothermal atmosphere with constant winds

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