Wave Propagation through Random Media

Transcription

1 Chapter 3. Wave Propagation through Random Media 3. Charateristis of Wave Behavior Sound propagation through random media is the entral part of this investigation. This hapter presents a frame of referene from whih to understand the underlying aousti priniples of sound waves as they pertain to the measurements reported herein. Any assembly of partiles whih mutually interat is a medium in whih wave motion may our. A brief disturbane in a small region indues motion in neighboring regions, resulting in some sort of movement whih eventually spreads throughout the medium. This transmission of disturbanes may be alled wave propagation (Baldok, 98). Consider a salar physial quantity u whih is dependent on a single spae oordinate x and on a given time t. If, in some domain of xt-spae, u an be expressed as: where is onstant, u ( x t) = f ( x t), (3..) then u is said to be a wave whih propagates in an x-diretion with veloity (Baldok, 98). Thus, every wave traveling with speed in any x-diretion satisfies the onedimensional lassial wave equation: u x u = t (3..) 9

2 ikewise, a funtion u(r, t) defined on a region of three dimensional spae over some time interval is a plane wave moving in the positive x-diretion by equation (3..). On any plane perpendiular to the x-axis, u is onstant, and the plane moves in a positive diretion with onstant veloity. Hene, we an define this plane as the wave front. A plane wave moving in the diretion of a unit vetor n is defined by: ( r n t) u = f (3..3) From equation (3..3) we an derive the three-dimensional lassial wave equation defined as: u = u t (3..4) satisfied by all plane waves traveling with the same speed (Baldok, 98). As outlined by Tatarskii (96) in his disussion of sound wave sattering in a loally isotropi turbulent flow, the movement of sound waves through a turbulent flow stream is analogous to the phenomenon of eletromagneti wave sattering. The overall veloity of a propagating wave is influened by the relative fluid veloity and surrounding temperatures. Temperature flutuations are important as they ause flutuations in the sound speed. The speed of sound an be expressed as: = γ p ρ = γrt (3..5) for perfet gases, ( R + v ) v γ = (3..6) 3

3 As shown by Eq. (3..5) above, the square of the sound speed is proportional to gamma,γ whih ranges from a maximum of 5 8 for monatomi gases through 7 5 for diatomi gases, R the relative gas onstant, and T the temperature. Tatarskii (96) gives the basi equation for sound propagation in a moving random medium to be written in the form: P = + u i P (3..7) t xi where P denotes the sound wave potential, u i the omponents of the veloity motion of the desribed medium, and the relative sound speed. More speifi to our investigation, Andreeva (3) outlines, at length, the appliability of two well known approximate theories of wave propagation, ray aoustis and the Rytov method. 3. Ray Theory and Statistis Chernov (96) desribed aousti wave propagation in a medium with random inhomogeneities by ray theory and statistis. The theory is appliable provided that a, the sale of the inhomogeneities, is large ompared to the wavelength λ. Thus, this ondition is often satisfied for ultrasoni waves (Chernov, 96). If this ondition is satisfied, the appliation of ray theory is valid in regions of linear dimension, where satisfies the ondition λ << a. However, at larger propagation distanes the theory breaks down, and warrants the use of diffration theory. Therefore assuming the following: 3

4 λ << a (3..) λ << a (3..) and that the transit time of the ray is small ompared to the harateristi sale of hanges of the inhomogeneities in time, the ray equation defined in Eq. (3..3) an be obtained from Fermat s priniple. B A dσ = min (3..3) We an define the refrative index by: n = (3..4) Thus, Eq. (3..3) an be re-written in the form below: B ( x y, z) n, dσ = min (3..5) A Furthermore, we assume that olletively, the ray trajetories are desribed by a family of urves expressed by the equations x = x( u), y = ( u), and ( u) z = that pass through given points A and B (Chernov, 6). The time taken for a ray to travel a given distane hanges with varying onditions in the medium. This results in ray tube deformation, and subsequently intensity flutuations (Chernov, 96). Assuming that deviation of the rays from their 3

5 initial diretion with respet of the x-axis is small, the travel time and mean travel time of a propagating wave over a distane is given by: t = n ( x, y, z) dx (3..6) t = n ( x, y, z) dx = dx (3..7) where the orresponding values of the refrative index n ( x y, z), are taken along the ray. Sine n =, the deviation from the mean time an be expressed by Eq. (3..8) as follows (Chernov, 96). t = t t = ( x, y, z) dx dx = ( x, y, z) n µ dx (3..8) By assuming that the values of x in the orrelation oeffiient N ( x y, z), are of the order a ( x ~ a), hene the mean square transit time flutuations are defined by Eq. (3..9) (Chernov, 96). µ t = N( x,, ) dx (3..9) Thus, we an alulate the phase flutuations S = ω t by Eq. (3..), and subsequently the relative hange in intensity along a path dx by Eq. (3..) below: 33

6 ( x,,) S = µ k N dx (3..) di I = dx µ dx (3..) where k is the wavenumber (Chernov, 96). Subsequently these methods, based in ray theory, are generally used to desribe the propagation of sound in a slowly hanging medium state (Andreeva, 3). The gridgenerated turbulene in our experiment, to a large degree, is omparable to the turbulent atmosphere, featuring flutuations in flow veloity and temperature. Equations (3..) and (3..) were used to determine if the onditions of the ray approximation is satisfied with respet to our investigation. The wavelength was determined by λ = v f, where v is the propagation speed and f the respetive frequeny. Thus, for our khz transduers the wavelength was estimated to be approximately 3mm. The turbulent length sales as outlined in setion. are not appreiably larger than the wavelength, and the left side of equation (3..) as required. As suh, the effet of these flow flutuations on sound propagation annot be treated by the methods of ray aoustis (Andreeva, 3). Rather, a ombination of the statistial representation of isotropi and homogenous turbulene and the Kolmogorov (94) /3 law provide a foundation for the theory of wave propagation in turbulent media. 34

7 3.3 Kolmogorov s /3 aw and Resulting Travel Time Equations Kolmogorov s /3 law states that the veloity flutuations at two different points in the flow stream are proportional to the distane between these points raised to the /3 power (Frish, 995). Sound wave propagation between two transduers over a distane, in a diretion t and opposite diretion t, an be desribed by a derivation of the flowmeter equation (Andreeva, 3). Aordingly, by Andreeva (3) we an express the aousti travel times of these waves as: t t = = dy u t + dy + u t u dy u dy (3.3.) (3.3.) where t denotes the travel time in the undisturbed media, and the sound speed. The veloities u and u are defined by: u = U sin β + u (3.3.3) u = U sin β + u (3.3.4) where U is the mean flow veloity, and u u are the respetive flutuations along the sound path. Negleting terms of order U /, U /, the time differene is expressed by Eq. (3.3.5) as follows: 35

8 t = t + t ( ) = t u u dy udy (3.3.5) o The turbulent veloity flutuations in Eq. (3.3.5) whih influene the aousti travel time an be desribed by a random funtion of time and position. Thus, Eq. (3.3.5) is rewritten as: t = 4 ( y ) u( y ) dy dy u (3.3.6) Invoking the Kolmogorov /3 law, we aount for the orrelation of flutuations at different points in the flow, and so (Andreeva, 3): 3 [ u ( y ) u( y )] = C R (3.3.7) where R is defined by the distane between the points y and y, and C a onstant harateristi of turbulene having dimensions of m 3 s. Under the isotropi and homogenous turbulent assumption, the left hand side of Eq. (3.3.7) is expressed by Eq. (3.3.8). 3 [ ] [ u( y ) u( y )] = ( u( y )) u( y ) u( y ) C R = (3.3.8) Thus, from transduer geometry we an define: ( y ) + y R = δ (3.3.9) ( y ) y R = (3.3.) 36

9 By way of Eq. (3.3.9) and Eq. (3.3.) and use of the /3 law, we rewrite the integrand of Eq. (3.3.6) as follows: /3 { ( ) } /3 ( ) ( ) { } ( ) ( ) /3 /3 u y u y = C R R = C y y y y + δ (3.3.).5 Substituting Eq. (3.3.) into Eq. (3.3.6) we define the expression: 5/3 δ t = C onst (3.3.) where the onstant is determined experimentally by Obukhov (95) to be equal to 3 (Andreeva, 3). 3.4 Fermat s Priniple Fermat s priniple states that the path of a ray of light between two points is the path that minimizes the travel time (for whih it is an extremum). For example, let the ray traveling from a soure in medium v loated at (, a) to a reeiver in medium v at (b, -), pass through the interfae between the two media at some point (x, ). To ompletely determine the ray path, we must find x suh that the total travel-time T(x) for the path defined by x is an extremum. 37

10 Figure 3- - Ray geometry defining minimum travel path The time neessary for the ray to travel from (, a) to (b, -) via (x, ) is defined by: [ + ] ( ) a + x ( b x) T ( x) = + (3.4.) v v Thus, to define x suh that T(x) is an extremum, we set the first derivative to zero: dt ( x) dx = = = (3.4.) v x ( ) a + x v ( b x) b x sin i sin i [ ] v v + Hene, we arrive at Snell's aw: sin i sin i v v = (3.4.3) 38

11 Thus, so that the refrated ray path is atually a minimum time path we note that: d T ( x) dx = v a + [ + ] ( ) a + x v ( b x) > (3.4.4) This relationship is also true for sound waves propagating between two different medium. With regard to our experiment, the priniple states that the travel path between the ultrasoni transmitter and reeiver always is an extremum. Hene, the length of the travel path, measured along a ray, is always shorter then any other path (Andreeva, 3). 3.5 Caustis The basi equations and methods of ray theory whih desribe wave motion throughout random media may beome invalid as rays ome within lose proximity to eah other to form austis. The singularities of geometrial optis are austis, and their systematization by atastrophe theory is perhaps the most signifiant appliation of that theory to date (Nye, 999). The diffration patterns assoiated with the austis are dominated by wave disloations, whih are line singularities of the phase, analogous to rystal disloations. A polarized wave field possesses an even finer struture of singularities (Nye, 999). Aording to ighthill (978), a austi is defined as a boundary between a region with a ompliated wave pattern, due to interferene between two groups of waves. In terms of ray theory, by Spetzler and Snieder (), the onept of austis is understood as the fous point in spae through whih rays go; whereas the 39

12 onsequene of their prodution in a wavefield is that the amplitude is infinitely high at the fous point beause the geometrial spreading fator is zero at the austi point. The austi phenomenon has been the fous of several investigations, suh as those onduted by White et al. (988), where he used limit theorems for stohasti differential equations on the equation of dynami ray traing to predit when austis start to develop in Gaussian random media (Spetzler, ; Snieder, ). In addition, Kravtsov (988) gave a thorough desription of austis, and Brown & Tappert (986) used Chapman's method to write expliitly the variation of -D and 3-D wavefields in the viinity of fous points (Spetzler, ; Snieder, ). ighthill (978) uses the Airy integral to address the ontraditions of a straightforward ray aoustis theory and aount for the loal singularities in the presene of the austis. However, for our purposes we will narrow the sope of our interest to the behavior of propagating waves before they strongly interat to form austis, but, interat strongly enough for their influene to be seen in the non-linearity of the amplitude variane. 4