# Sound Propagation in the Nocturnal Boundary Layer. Roger Waxler Carrick Talmadge Xiao Di Kenneth Gilbert

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1 Sound Propagation in the Nocturnal Boundary Layer Roger Waxler Carrick Talmadge Xiao Di Kenneth Gilbert

2 The Propagation of Sound Outdoors (over flat ground) The atmosphere is a gas under the influence of gravity thermal stresses the Earth s rotation The ground is a porous elastic solid has large (compared with air) thermal conductivity reflects and attenuates sound properties can differ dramatically from place to place

3 The Diurnal Cycle (fair weather meteorology) The ground is thermally coupled to space heats up during the day cools off at night The air is thermally coupled to the ground Daytime temperature decreases with altitude turbulent 1 km winds slowed by friction Nighttime temperature increases with altitude stable with buoyancy waves stable layer acts as a lubricant for the wind night day neutral Temperature [ C]

4 Refraction of Sound by the Atmosphere The speed of sound in air is c 2 T Temperature gradients sound speed gradients refraction wavefronts propagate towards colder air clear days are upward refracting clear nights are downward refracting night day neutral Sound Speed [m/s] Similarly for wind shear wavefronts propagate towards slower air upwind is upward refracting downwind is downward refracting night day Wind Speed [m/s]

5 Equations of Atmospheric Mechanics The atmosphere is described by fluid mechanics ρ t + (ρv) = ρ ( v t + (v )v) + P = ρgẑ S t + v S = P = ρrt ds R = c p dt R T dp P mass conservation Newton s law heat equation ideal gas law second law The ground is described by poro-elasto-dynamics The two are coupled by interface conditions

6 Meteorology versus Acoustics Variables split into slow (meteorological) and fast (acoustic) terms: v v met v ac ρ P T = ρ met P met T met + ρ ac P ac T ac S S met S ac The meteorological terms split into mean and fluctuating parts: v met v M v turb ρ met P met T met = ρ M P M T M + ρ turb P turb T turb S met S M S turb

7 Local Meteorology For distances km s, times tens of minutes mean quantities depend only on altitude mean vertical wind speed is zero z is altitude and H indicates horizontal v HM (z) ρ M (z) P M (z) T M (z) S M (z) Then dp M dz and v H is arbitrary. example: S M = S = gρ M P M = ρ M RT M S M S = c p ln T M (z) = T () g c p z T M T M () R ln P M P M () Temperature [ C] neutral Wind Speed [m/s]

8 The North American Nocturnal Jets (global generation of local winds) Air flows from warmer regions to cooler regions. During the day flow is impeded by ground friction mediated through the turbulent layer. At night flow is decoupled from the ground. A stiff wind develops above the stable layer. Can be super-geostrophic.

9 Meteorological Measurements meteorological equipment: 1 m tower, tethersonde, sodar

10 Meteorological Data collected in the Delta Locke Station MS 11/9/6 at 18:15 4 T 4 N E Temperature [ C] Wind Speed [m/s] Data is averaged over 15 minute intervals and 15 meter vertical slices

11 The Acoustic Terms Reduces to determining P ac (x H, z, t) from ( 2 H + 2 z 2 + ω 2 c eff (ˆn, z) 2 ) Pac (x H, z, ω) = ˆn is the unit horizontal from source to receiver and c eff (ˆn, z) = 2 T (z) + ˆn v M (z) Downwind Effective Sound Speed 1 c c eff Sound Speed [m/s]

12 Interaction of Sound with the Ground local impedance approximation P ac z z= = iωρ M Z P z= ac Well understood above 2 Hz ground is porous and essentially rigid reaction is essentially local modeled by an impedance condition Normalized Impedance hard backed real imag Frequency [Hz] Local impedances are often used below 2 Hz Deviations from locality have been reported a few Hz

13 Methods of Solution ( 2 H + 2 z 2 + ω 2 c eff (ˆn, z) 2 ) Pac (x H, z, ω) = P ac z z= = iωρ M Z P z= ac geometric: P ac Ae is ; PE: solve 1-way equ. S are wavefront normals ( x ± i 2 z 2 + ω 2 c eff (ˆn, z) 2 ) Pac = FFP: Fourier transform w.r.t. x H modes: Expand w.r.t. eigenfunctions of d 2 dz 2 + ω 2 c eff (ˆn, z) 2

14 Contrasting Day and Night 5 day Sound Speed [m/s] Daytime, 1 Hz source at 1 m Range [km] Transmission Loss [db] Sound Speed [m/s] Nighttime, 1 Hz source at 1 m Range [km] Transmission Loss [db]

15 Modal Expansion for the Acoustic Pressure Eigenvalue problem: of Schrödinger type non-self-adjoint Ducted modes: eigenvectors ψ j (ω, z) eigenvalues k 2 j ( d 2 dz 2 + ω 2 c eff (ˆn, z) 2 k2) ψ = dψ dz z= = iωρ M Z k j = ω c j + iα j ψ() P(x H, z, ω) N(ω) j= ( 2 H + k2) p(k, ω, x H ) = p(k j, ω, x H )ψ j (ω, z) Validity: ducted propagation ranges > 5 m altitudes < 5 m plus source terms.

16 Finding the Modes (the atomic analogy) Effective Potentials night day down no Effective Potentials potential: ω2 c 2 eff bound states: ducted continuum states: upward propagating altitude, meters Modes at 4 Hz sound speed (m/sec) a non-self-adjoint tunneling problem

17 Modal Dispersion for a Simple Duct (temperature inversion, no wind) Dispersion for the surface and every 2 n th refracted mode. Sound Speed [m/s] Frequency [Hz] refracted modes surface mode Sound Speed [m/s] surface 336 mode Frequency [Hz] refracted modes Attenuation [1/m] e-4 1e-5 1e Frequency [Hz] surface mode refracted modes Mode Mag Squared [1/m] e Frequency [Hz] surface mode refracted modes

18 Modal Dispersion for a Complex Duct (temperature inversion and down wind) Dispersion for the surface and every 2 n th refracted mode. Sound Speed [m/s] surface 34 mode Frequency [Hz] refracted modes Sound Speed [m/s] surface 34 mode Frequency [Hz] refracted modes Attenuation [1/m] e-4 1e-5 1e Frequency [Hz] surface mode refracted modes Mode Magnitude [1/m] Frequency [Hz] surface mode refracted modes

19 Tloss [db re 1 m] Tloss [db re 1 m] Propagation from a Point Source P e i π 4 N(ω) j= e ik j r 8πkj r ψ j(ω, z S )ψ j (ω, z) Ground to Ground Transmission Losses at 3 km Total Modal Frequency [Hz] Frequency [Hz] Tloss [db re 1 m] Tloss [db re 1 m] surface mode refracted modes Frequency [Hz] surface mode refracted modes Frequency [Hz]

20 Surface Mode vs Refracted Modes At long ranges the surface mode - is very sensitive to the ground surface - is attenuated primarily by the ground - has no low frequency cutoff - carries energy up to about 1 Hz - propagates at roughly the ground sound speed - propagates horizontally At long ranges the refracted modes - are not very sensitive to the ground surface - are attenuated primarily by the atmosphere - have cutoff frequencies - carry energy up to about 1 Hz - propagate with speeds greater than the ground sound speed - propagate at shallow angles to the horizontal

21 The Narrow Band Near-ground Structure Near-Ground Mode Mags at 1 Hz, downwind surface mode Effective Sound Speeds 1 c c eff Sound Speed [m/s] The Quiet Height The surface mode has no nodes. The lowest node of the refracted modes are at the same altitude. P eik r i π 4 8πk r ψ (ω, z S )ψ (ω, z) N(ω) e ik j r i π 4 + 8πkj r ψ j(ω, z S )ψ j (ω, z) j=1 As the surface mode attenuates a quiet height develops.

22 The Quiet Height Experiment source 1.8 km mics 4 m altitude (m) db 15 Hz db rel mic 1 altitude (m) db 2 Hz db rel mic 1 SPL (db re 2 µpa) 6 rms frequency (Hz) altitude (m) db 25 Hz db rel mic 1 altitude (m) db 3 Hz db rel mic 1 Altitude (m) Sound Pressure Levels re. 1 st Mic 125 Hz 15 Hz 175 Hz 2 Hz 225 Hz 25 Hz 275 Hz 3 Hz 325 Hz 35 Hz 1 db db rel mic 1

23 The Broad Band Pulse Tail Point source with spectrum Q(ω) at 3 km range P(r, z, t) = N(ω) Q(ω) j= e ik j (ω)r iωt i π 4 8πkj (ω)r ψ j (ω, z S )ψ j (ω, z) dω 3 1 mode 1 3 Pa modes 6 ms Sound Speed [m/s] 4 modes 16 modes.5 Pa.5 Pa 1 ms 32 modes 1 ms

24 The Geometric Acoustics of the Distinct Early Arrivals No Wind Down Wind Range [km] Range [km] Range ground reflections 6 km 5 km 4 km 3 km 2 km Travel Time re Range/c(z I ) [sec] Range ground reflections 5.5 km 5 km 4.5 km 4 km 3.5 km 3 km 2.5 km Travel Time re Range/c(z I ) [sec]

25 Recent Pulse Propagation Experiments 4 Downwind Effective Sound Speed mics c source c eff km Sound Speed [m/s] met 1 m mic altitude [m] run 1, pulse 58.8 sec A visible surface mode tail propagates horizontally amplitude decreases with altitude Front arrives at a shallow angle h dl d 2θ dl b θ

26 Signal Variability (short term stability long term variability) pulse number run 1, lowest mic.1 sec run number st Pulses, lowest mic.1 sec

27 Yet to be Done (we ve got ourselves a probe) Study data inversions for source location for elevated wind jets for ground properties Investigate the convergence zone theoretically experimentally Study the variation in pulse duration study arrival time variability identify the source experiment planned for the spring Study upwind propagation Study poor weather propagation propagation in fog under cloud cover during rain

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