Constructive Interference in a Waveguide

A, B, K, internet

2.2 Constructive Interference in a Waveguide

substrate half space and the phase speed along the interfaceV_int reduces to:

V_int = c₁

sin(φ_cr) =c₂. (2.12)

For an incidence angle φ₁< φ_cr, the reflectionR and transmission T coefficients have a value between zero and one. For the critical angle and beyond, no energy leaks outside of the waveguide, T = 0and R = 1, and the amplitude is preserved for all distances A=A₀. Attenuation with distance is expected because of the cylindrical spreading of the waves in the natural waveguide at the desert dunes.

0 20 40 60 80

0 0.2 0.4 0.6 0.8 1

Reflection Transmission Critical Angle

Coefficient

Angle of incidence (degrees)

Figure 2.2: Reflection and transmission coefficient (Lay and Wallace,1995) as a function of angle. Parameters used are ρ₁ =ρ₂ = 1500 kg/m³,c₁ = 200 m/s, c₂ = 350 m/s and H = 2.0 m.

2.2.2 Wave Propagation in a Waveguide Ray tracing

For constructive interference the analysis of the waveguide in terms of ray tracing follows the derivations presented inEwinget al.(1957) andOfficer(1958). Assume that the waveguide depthH is constant across a certain length and that the velocities c₀, c₁, c₂ are constant in each layer. The seismic velocity usually increases with depth in a granular material, but Vriendet al.(2008) showed that this increase is smaller than the large velocity jump across the interface between medium 1 and 2. The propagating waves are traveling in phase within

the waveguide in the case of constructive interference. For a given wave at incident angle φ₁, the extra distance traveled by a waveAB¯ in figure2.1is:

AB¯ = ¯AO+ ¯OB =H

· 1

cos(φ₁) +cos(2φ₁) cos(φ₁)

¸

= 2Hcos(φ₁). (2.13) Officer(1958) defines the condition for constructive interference as the total phase change equal to a factor depending on the mode numbern:

k_nAB¯ −²₁₀−²₁₂= 2nπ. (2.14) The phase change involved with the ray path from A to B depends on the wave number k_n= 2π/λ_n, the wavelengthλ_n=c₁/f_n and the distance traveled by the wave AB.

Substituting the wave number and wavelength and taking the tangent on both sides of equation (2.14):

tan

µ2πHcos(φ₁)f_n c₁

¶

=tan

µ(²₁₀+²₁₂) 2 +nπ

¶

. (2.15)

The phase lag at the surface²₁₀and the bottom²₁₂are derived from the reflection coefficient R (Officer,1958, p. 79) as:

²₁₂

2 =tan⁻¹







 ρ₁

sµ c₁ V_int

¶₂

− µc₁

c₂

¶₂

ρ₂ s

1− µ c₁

V_int

¶₂







. (2.16)

and

²₁₀

2 =tan⁻¹







 ρ₁

sµ c₁ V_sur

¶₂

− µc₁

c₀

¶₂

ρ₀ s

1− µ c₁

V_sur

¶₂







. (2.17)

The current analysis deviates from the treatment by Officer (1958), where ²₁₀=π and

²₁₂= 0, as the waveguide in a sand dune has a mirrored velocity structure for whichc₀ = c₂. In the case of critical refraction, the coupling provides the feedback to the waveguide and the phase speed along the surface interface reduces toV_sur =c₀, producing zero phase lag ²₁₀ = 0at the top surface (Officer,1958, p. 228). Similarly, critical refraction ensures a phase speed of V_int = c₂ along the interface between medium 1 and 2 and zero phase

lag at the bottom surface as well. As a consequence of critical refraction, the phase lag is independent of the density and/or the impedance differences across interfaces. Therefore, equation (2.15) reduces to:

tan

µ2πf_nHcos(φ_cr) c₁

¶

= 0. (2.18)

Solutions are given in terms of the mode number nwithn= 1, 2, 3,...

2πf_nHcos(φ_cr)

c₁ =nπ, (2.19)

and

f_n= n 2

c₁

H s

1− µc₁

c₂

¶₂, (2.20)

The resonant moden= 0is the non-propagating, standing, mode in the waveguide that does not travel in r-direction and has zero phase speed and zero frequency. The n-th overtone exists for frequencies equal or greater than the cutoff frequency as prescribed by equa- tion (2.20).

Continuous guided wave

An alternative approach to derive this formula is to analyze the waveguide in the contin- uous sense following Sleep and Fujita (1997). For wave propagation at long ranges and at moderate to low frequencies, the normal-mode solution combines interference effects from all ray paths. The trial functionφof a planar geometry with a standing wave in z-direction and a propagating wave in r-direction with rigid boundaries is:

φ=cos(k_zz)exp(i[k_rr−ωt]). (2.21) The wave propagates within the waveguide in horizontal direction with a constant phase velocity V₁ = ω₁/k₁, with k₁ = p

k²_r+k²_z and ω₁ = 2πf₁. At the upper boundary z = 0, the boundary condition of zero displacement ∂φ/∂z = 0 is satisfied. At the bottom the boundary condition at z=H is satisfied ifk_zH=nπ, with integer n. The trial function φ

including all modesn is:

φ= X∞

−∞

φ_ncos µnπz

z₀

¶

exp(i[k_rr−ωt]). (2.22)

The wave equation is:

∂²φ

∂t² =c²₁

·∂²φ

∂r² + ∂²φ

∂z²

¸

, (2.23)

with the p-wave velocityc₁=p λ₁/ρ₁.

Substituting the trial function from equation (2.22( into the wave equation (2.23) gives:

−ω_n² =c²₁£

−k²_r−k²_z¤

=c²₁

·

−k²_r−

³nπ H

´₂¸

. (2.24)

The incident angle φ₁ is orientated as:

tan(φ₁) = µ k_r

nπ/H

¶

, (2.25)

such that:

ω_n=c₁

³nπ H

´ ptan(φ₁)²+ 1 =c₁

³nπ H

´s 1

cos(φ₁)² =c₁

³nπ H

´ 1

cos(φ₁). (2.26) Restructuring this equation in terms of the frequency and substituting incidence at the critical angle φ₁ =φ_cr gives:

f_n= ω_n 2π = n

2

c₁ H

r 1−

³c1

c2

´₂, (2.27)

which is the same resonance relation as equation (2.20) found via ray tracing.

2.2.3 Phase Velocity

The booming waves travel at a phase speed V situated between the seismic speed of the dry layer of sand c₁ and the seismic speed of the denser, deeper layer of sand c₂. As the subsurface structure of the dune changes in the uphill or downhill direction (illustrated in figure 2.1), the phase velocity also changes independent of the frequency of the source. At a given moment in time a seismic sensor measures the global booming frequency and the

local phase velocity. The phase velocity adapts as the waves move into a different velocity or layering structure. A sensor on the desert floor, located roughly 500 meters from the booming dune slope, measured the same booming frequency (82 Hz) as the local recording, as shown in figure 2.3, but a much higher phase velocity (∼500 m/s).

0.2 0.4 0.6 0.8

0 100 200 300 400 500

Time (s)

Frequency (Hz)

120 110 100 90 80 70 60 50 40

0 8

Voltage (10-

3

V) Power 0.001 0

Frequency (Hz)

200 100 300

0

Time (sec) 0.5 1 0

500

400

_-60

-40

-80 -100 -120

-8

Figure 2.3: Seismic response of a booming emission at a distance of approximately 0.5 km.

The local phase velocity depends on the local depth of the layeringH_A, the local velocities c_1Aandc_2Aand the global booming frequencyf =f_cutoff obtained from equation (2.27). For known dimensions, velocities and booming frequency, the local phase velocityV_Ais obtained by solving the transcendental equation:

tan



f_cutoff2πH_A c₁

s 1−

µc_1A V_A

¶₂

= ρ₁ ρ₂

r³c1A

VA

´₂

−

³c1A

c2A

´₂

r 1−

³c1A

VA

´₂ . (2.28)

In the example of figure 2.1, the depth of the layering increases uphill from the source creating a longer wavelength λ_a > λ_b and physically signifies the approach of the dune crest (Vriend et al., 2010a). As the subsurface velocity commonly is smaller close to the crest (Vriendet al.,2007), the phase velocity decreases significantly uphill from the source V_a < V_b. The example also shows a thinning in downhill direction, where the subsurface velocity increases, occurring in grainfall areas (Vriend et al.,2010a). The local wavelength decreasesλ_c< λ_b and the phase speed increasesV_c> V_b. An alternative waveguide structure

of thickening in downhill direction occurs at the transition of grainfall and grainflow regions.

The increase in subsurface velocity and the increase in depth are two competing factors that have an opposite effect on the phase speed. Usually the increase in downhill velocity dominates the increase in depth such that the effective phase speed increases.