Debris flow seismo-acoustic wave in a finite layer waveguide

. The seismo-acoustic wave detection is a popular method to detect debris flows. It has been successfully used in different early warning systems worldwide. However, more information is embedded in the signals related to flow conditions. This research uses theoretical derivation to connect signals with debris flow propagation speed. We assume debris flow generates seismo-acoustic stress acting on a river bed and the seismo-acoustic waves propagate in a finite layer waveguide below the channel. Then, the propagation of seismo-acoustic stress was solved with the elastic wave equations along the channel. For any fixed point in this waveguide, the frequency of the recording signal can be translated as a function of debris flow propagation speed and seismo-acoustic wave velocity. The result shows that there is a Doppler-like property for a recorded signal where debris flow propagation speed can cause a frequency shift. This result also indicates that different superior frequency bands would be recorded from different media.


Introduction
The acoustic-based detection is a popular way to detect debris flows and has been used in different early warning systems worldwide for decades. From field observation, the high correlation between the energy of ground vibration signal and flow condition has been proved by many studies. Some of the studies also try to calibrate the empirical formula [1][2][3]. However, it only can be used in the location calibrated before and should be modified once the environment changes after a new event.
Some recent studies introduced the bed load seismicity conceptual model [4] to analyze debris flows [5][6]. These approaches are based on the vibration source generated by particle impact and use the Rayleigh-wave propagation function (Green's function) [7] to simulate the receiver's vibration energy. Although these models have been used in the field, the assumptions of the simplified process, the determination of field parameters, the applicability of Green's function, etc., still lack independent verification mechanisms [8].
Because the flow condition still cannot be quantified using a recording signal from a theoretical point of view. To bridge the gap, we aimed to develop a connecting theory between debris flow motion and seismo-acoustic wave propagation.

Ideal waveguide 2.1 Fundamental theory
We consider a debris flow flowing down a channel with a constant wave velocity C [12] and radiating acoustic * Corresponding author: stanscwei@gmail.com waves from its boundaries, such as the free surface or channel bed. Below the channel bed, the underground layer is regarded as a homogeneous and isotropic elastic media. The x -axis coincides with a streamwise direction along the channel bottom. The y -axis is in the transverse direction, and the z -axis is perpendicular to both the x -and y -axis, as shown in Fig. 1(a). An ideal channel shape with radius 1 r is assumed, as shown in Fig. 1 (b).
In this underground layer, the seismo-acoustic wave propagation is governed by the elastic wave equation or Navier's equation [7,[9][10] ( ) where u is displacement vector ( ) ,, where  and  are Lamé's first and second parameter, respectively. The seismo-acoustic source comes from the bottom stress of debris flow a ij zb The stress in the channel layer a ij  can be expressed as [11] When the river channel radius 1 r is much smaller than 2 r , the debris flow along the x -axis can be simplified as one-dimensional, and only the leading order stress a zx zb  = will be survived [12]. Thus Eq. (3) can be simplified as x y z t , black axes describe the moving cylindrical coordinate system in ( , , , ) r    , red slashhatched is debris flow, light gray shadow indicate wave propagation domain.

Governing equations
Since 12 rr , we consider the debris flow as a line source and allow 1 0 r → in the following section. To simplify the problem, we transform the coordinate from the Cartesian system ( , , , ) x y z t to a moving cylindrical system ( ) , , , r    , as the relation below where C is debris flow wave velocity, 0 C is a constant related to wave velocity, sin zr  = . The differential operator of function are listed below With the Helmholtz decomposition theorem, the displacement vector is expended as where  and 11ˆr r e e e rr r r where ψ must satisfy zero divergence condition Due to the nature of line sources, this problem can be reduced to an axisymmetric problem, i.e., Substituting Eq. (11) to Eq. (1), the governing equation can be decoupled as a potential form To simplify the governing equations, we introduce the

Boundary conditions
The corresponding stress boundary condition are listed below.
With the axisymmetric condition, the stress in Eq. (18) and (19)

Separation of variables
We adopted conventional separation of variables to solve the potential  and   . First, we let where ( ) is a positive real number.    (47) The shear waves vanish from this ideal waveguide due to 1 0 rr =→.

Eigenvalue solutions
If we substitute Eq. (42) and (47) The frequency * f at any fixed location x is not only the function of debris flow propagation speed C , but the function of compressional-wave velocity P c . Therefore, it implies the measuring frequency * f is different from the frequency f radiated from debris flow.
In reality, the P c in alluvium medium is around 300~700 m/s, much larger than the debris flow propagation speed C observed from the field (5~20 m/s). By applying the Taylor series expansion to Eq. (52) with The result shows that the measuring frequency of debris flow at a fixed location is proportional to the debris flow propagation speed C in the leading order. If the underground media change in measuring cross-sections, the frequency band might be shifted due to the higher order term of Eq. (53), even if the debris flow passes these cross-sections at the same speed.

Conclusions
We propose an analytical solution for debris flowinduced seismo-acoustic wave propagation on an ideal finite waveguide. Through the finite domain cylindrical seismo-acoustic propagation problem, this research found that the measuring frequencies recorded by a fixed recorder are proportional to the debris flow propagation speed in the leading order. Furthermore, the underground media, such as the soil layer, rock layer, and dam, would affect frequencies with a slight shift in the higher-order term. This finding indicated a Dopplerlike property from the theoretical point of view and showed the potential to evaluate flow conditions based on seismo-acoustic frequencies.