## Abstract

This study examines the frequency attenuation characteristics of elastic waves in marble, granite, and red sandstone through laboratory tests and numerical simulations based on ABAQUS software on eight rock samples. The correlation between the distance of the decay of the peak frequency and the petrophysical and mechanical parameters is also analyzed. The results show that elastic waves undergo stepwise attenuation of their peak frequency during laboratory attenuation tests. This finding was confirmed by numerical simulations. As elastic waves propagate through a rock medium, the amplitude of the high-frequency peak area near the excitation frequency decreases rapidly, while the proportion of low-frequency signals increases. As the propagation continues, the signal spectrum is dominated by low-frequency components, resulting in a stepwise attenuation of the peak frequency. It has been observed that the step decay distance of the peak frequency of elastic waves varies with the degree of particle bonding, which can be used to characterize the attenuation properties of elastic waves in a medium.

## 1. Introduction

The elastic wave propagation characteristics of a medium can provide insight into its internal properties. In rock mechanics, acoustic emission (AE) monitoring is increasingly prevalent [1]. The AE technique can be used to monitor the elastic waves generated by microcracks when rocks are under loads, making it effective for studying the evolution of damage within the rock [2-4]. Geological exploration utilizes seismic and acoustic waves to extract information about the geological structure and resource reservoirs of a region [5-10]. Techniques such as AE and seismic exploration invert the actual state of the source through the received signal at the sensors. However, elastic wave propagation results in attenuation due to interaction with the medium. Understanding the attenuation characteristics of elastic waves in rock medium is crucial due to potential changes in signal strength, frequency composition, and wave packet shape [11-14].

To estimate the attenuation of seismic waves in different strata, scholars have studied the attenuation coefficients in relation to factors such as stratigraphy, wave speed, frequency, temperature, and pressure [15-20]. In rock masses, the attenuation mechanism of seismic waves mainly includes friction, liquid flow, viscous relaxation, and scattering [21-24].

Laboratory studies have extensively investigated the attenuation of elastic waves in various types of rock under different operating conditions [25]. The effects of fractures, pores, and fluids on the attenuation mechanism can be studied by controlling the loading conditions and the state of the sample [26-28]. The frequency of the waves is a crucial factor that affects their attenuation in the medium. The laboratory study of elastic wave attenuation includes an examination of its frequency dependence and the influence of frequency on the attenuation mechanism. Artificial wave sources, such as ultrasonic waves, are commonly used for this purpose [29-31].

The focus of research on the attenuation of elastic waves in rock medium has always been on exploring the attenuation mechanisms and characteristics of attenuation in different types of rocks [32-34]. However, when analyzing attenuation characteristics, the focus is primarily on time-domain parameters, such as amplitude and energy. A common analysis method involves collecting the elastic wave signal from the excitation wave source at the ends or side wall of small-scale samples, obtaining the waveform amplitude to calculate the attenuation coefficient, Q value, and so forth [33, 35], and then discussing the attenuation characteristics. Elastic waves have several characteristic parameters, including time-domain and frequency-domain parameters. The latter includes peak frequency and center frequency [36-38]. The former can reflect changes in signal propagation, such as amplitude reduction and energy weakening. Due to wave dispersion, different frequency components experience varying degrees of attenuation [32]. As a result, the frequency composition and shape of the wave packet undergo changes during propagation. Furthermore, this study aims to provide reference for understanding the propagation process of elastic waves from the seismic source to monitoring stations in microseismic and seismic monitoring, and to compensate for the attenuation of the received elastic wave signals, thereby gaining a detailed understanding of the characteristics of the seismic source and correcting the frequency dependence of the geological quality factor. At the same time, in local rock mass or laboratory small-scale rock sample AE monitoring, if the attenuation characteristics of elastic waves can be well obtained, the signal received by the sensor can be attenuated and compensated based on the positioning results and the attenuation characteristics of different distances and frequencies with distance in this study, thereby obtaining the frequency characteristics of the seismic source. As the frequency characteristics of the seismic source are related to the scale of the seismic source, this is of great significance in evaluating the internal fracture condition and instability warning of the rock mass.

This study investigates elastic wave attenuation in various rock media through laboratory tests and numerical simulations. The peak frequency of elastic waves propagating in marble, granite, and red sandstone at different frequencies in the laboratory decreased stepwise with distance. This finding was further confirmed by numerical simulation. The attenuation distance of the elastic wave peak frequency from the excitation frequency to the low-frequency band varies among different rocks. In numerical simulations, we compared the attenuation distance of the peak frequency in eight types of rocks and the attenuation characteristics reflected by the attenuation coefficient in the time domain. The article discusses the influence of rock physical and mechanical parameters on the attenuation distance of the peak frequency and attenuation coefficient, in combination with the wave impedance and elastic modulus of the rock. The attenuation characteristics of the peak frequencies also reflect the attenuation of the elastic waves, providing a supplement for describing the attenuation characteristics in rock medium.

## 2. Experiment Program

### 2.1. Geologic Background

Seismic waves are mechanical vibration waves that can propagate through rock and other media. The dynamics and motion process of their propagation are closely related to rock properties. Acquiring subsurface structures through seismic waves is an important means of solving the difficulty of matching data at different scales, such as seismic, logging, and ultrasonic core observations. It is one of the most effective tools. Figure 1 shows that seismic waves carry a significant amount of information through the Earth’s medium. Seismic data is used to map the observed waves to the model space, and inversion techniques are employed to obtain information about the underground medium and structure. In modern geotechnical engineering, construction projects face increasingly complex geological conditions and external environments. These include mountain highways, railway road graben slopes, tunnels with intricate geological conditions, bridge foundations [39], urban large-span underground space projects, cross-river and cross-lake highways, underground tunnels [40, 41], and deep underground mining, which is threatened by the “three highs and one perturbation” [42, 43] (high stress, high temperature, karst water pressure, and mining disturbance). It is increasingly important to comprehend the geological conditions and the state of the rock body in the vicinity, and to conduct long-term monitoring of the ground pressure of the rock body during project construction. This will ensure objectivity and precision in the evaluation of the project’s progress. Therefore, seismic wave inversion, analysis of the stress state of the regional rock body, evaluation of crack development and expansion energy, and volume release level are effective measures to ensure the safety and stability of engineering construction under complex rock body conditions. In geological exploration, artificially excited seismic waves can also be used to understand local and regional geological structures, mineral resources, reservoir information, and more.

However, during seismic wave propagation, the microscopic composition of different rock types varies, and seismic wave energy is significantly attenuated due to cracks and pore structures in the rocks. This can alter the frequency composition and shape of signals, leading to a reduction in the resolution of geological structures and imaging accuracy (Figure 2). Therefore, studying the attenuation characteristics of rock materials is significant for predicting the internal structure of rocks. The laboratory investigation of wave propagation attenuation in rock involves controlling experimental conditions such as pressure and loading method, as shown in Figure 3. This approach allows for better understanding of the attenuation mechanism compared to data obtained outside the laboratory [44].

### 2.2. Sample Preparation

The experiment employed long rectangular rods made of three different types of rock: marble, granite, and red sandstone. These larger specimens allowed for a longer propagation path, which was more conducive to studying the attenuation process during propagation compared to the standard cylindrical specimens of 50 mm × 100 mm. The measurements taken for the length, width, height, and mass of the samples are listed in Table 1.

Ultrasonic transducers are positioned at one end of the sample to generate elastic waves of varying frequencies. The response frequencies of the six ultrasonic transducers are 50 kHz, 100 kHz, 250 kHz, 500 kHz, 750 kHz, and 1 MHz, respectively. Figure 4 shows the waveforms of the elastic waves generated at 50 and 100 kHz. The transducers used in the test all have a diameter of 25 mm. The axial lengths of the six transducers are different due to the varying thickness of piezoelectric crystals required to excite ultrasonic waves of different frequencies. However, these length differences do not affect the test. For the elastic wave signal instrument, a 32-channel Vallen AMSY-6 AE instrument was chosen. The signals were detected using a piezoelectric sensor of the VS45-H type, which has a wide frequency response and a resonance frequency of 20–450 kHz. The test conditions were quiet, with the acquisition system threshold set at 35 dB, the preamp gain at 34 dB, and the sampling rate at 10 MHz.

During the experiment, glue was used to attach iron gaskets to the side wall of the rock rods in order to determine the sensor position. To prevent the sensors from falling off during the experiment, a coupling agent was applied between the sensor, the transducer, and the rock contact surface. Additionally, the sensor was secured with a magnetic clamp. Table 2 and Figure 5 illustrate the positions of the excitation transducers and collection sensors used in the experiment. The sensitivity of each sensor is calibrated through lead breakage tests after bonding. Each sample underwent 10 tests at each frequency, with a single pulse using the excitation system. An acquisition system was used to collect waveform, amplitude, and other elastic wave information.

## 3. Peak Frequency Attenuation Characteristics of Elastic Wave in Laboratory Tests

### 3.1. Attenuation Characteristics of Peak Frequency

Figure 6 shows the peak frequency variation during the propagation process obtained in the experiment. The attenuation of the peak frequency in the rock medium exhibits stepwise attenuation. As the signal propagates, the peak frequency drops steeply to a lower frequency value and continues to propagate after maintaining a distance at the excitation frequency. At excitation frequencies of 50, 100, 250, 500, and 750 kHz, there is a sharp decrease in peak frequency for marble. This phenomenon is not observed at 1 MHz frequency due to the extreme attenuation.

The damping of an elastic wave signal is defined by a faster decrease as the frequency increases. The peak frequency attenuation distance, which is the distance by which the peak frequency decreases from near the excitation frequency value to the lower frequency, varies considerably for various excitation frequencies. The experiments demonstrate that the peak frequency attenuation distance of elastic waves differs among various rocks. Figure 7 illustrates the comparison of peak frequency attenuation of elastic wave signals among different rocks with varying frequencies.

At frequencies ranging from 50 kHz to 1 MHz, the peak frequency attenuation of signals in all three rocks is similar. This is due to the slow attenuation of low-frequency signals and the fast attenuation at 1 MHz. The most significant differences in peak frequency attenuation of elastic waves occur at frequencies of 100 and 250 kHz among the three rocks. Figure 7 b and c shows the peak frequency attenuation distance of marble, granite, and red sandstone, marked by blue, red, and green short lines, respectively. Table 3 lists the attenuation distances for the three rock types.

In Table 3, the peak frequencies of signals with frequencies of 100 and 250 kHz have the longest attenuation distance in granite, followed by marble, and the shortest attenuation distance in red sandstone. The peak frequency of elastic wave exhibits stepwise attenuation during propagation, and the attenuation distance of peak frequency presents a large difference under some excitation frequencies.

### 3.2. Explanation of Peak Frequency Stepwise Attenuation

The characteristics of elastic wave signals are typically divided into time-domain and frequency-domain parameters. Time-domain parameters often exhibit a gradual decrease during propagation. Frequency-domain parameters include peak frequency and center frequency. The center frequency shifts towards the lower frequency band as the signal attenuates, while the peak frequency of elastic waves displays a stepped attenuation characteristic, which distinguishes it from other characteristic parameters.

The peak frequency of the elastic wave corresponds to the frequency with the maximum amplitude in the spectrogram. The attenuation characteristics of this frequency may be related to its composition. For instance, when propagating a 100 kHz elastic wave in marble, Figure 8 displays the signal spectrum collected by sensors at 80, 95, 110, 125, and 140 cm. Figure 7a shows a sharp decrease in the peak frequency of the 100 kHz elastic wave after propagating 110 cm. The spectrum in Figure 8 indicates that the 100 kHz signal in marble has a main peak band near 100 kHz before 80 cm. As it propagates to 90 cm, a peak band near 40–50 kHz begins to appear, and its relative size with the 100 kHz peak band also gradually increases. When propagating to a length of 125 cm, the amplitude of the low-frequency peak band is higher than that of the 100 kHz band, and the signal’s peak frequency decreases and stabilizes around 40 kHz.

The “stepwise attenuation” in peak frequency is caused by the decrease in the amplitude of the high-frequency peak band and the increase in the proportion of signals in the low-frequency band. This occurs because components with higher frequencies decay faster, and after propagating a certain distance, low-frequency components dominate the signal, causing a shift in the dominant frequency.

## 4. Peak Frequency Attenuation Characteristics of Elastic Wave in Numerical Simulation Tests

The laboratory experiment revealed a stepwise attenuation feature in the peak frequency of elastic waves, with varying attenuation distances among different rocks. The most significant difference in peak frequency attenuation distance was observed at excitation frequencies of 100 and 250 kHz. To investigate the generalizability of this experimental phenomenon, we conducted elastic wave attenuation tests on additional rocks using ABAQUS finite element analysis software.

### 4.1. Construction of Numerical Models

The geometric model used in the numerical simulation has dimensions of 2000 mm in length, 40 mm in width, and 40 mm in height, which correspond to the laboratory test samples. A sinusoidal function load with periods of 10^{–5} s and 4 × 10^{–6} s is applied at the center of one end of the model to simulate elastic waves of 100 and 250 kHz, respectively. The load value is 5 MPa. The mesh size is determined to be 5 mm in accordance with the Nyquist Sampling theorem.

Regarding the material properties of the model, we selected rock materials used in laboratory experiments and five rock samples [14, 24, 29, 43]. Table 4 shows the corresponding physical and mechanical parameters of the materials. To ensure the reliability of the numerical simulation, we verified it by comparing it with the laboratory attenuation test. We then explored the generalizability of the conclusions.

Based on the given model, numerical calculations can be used to obtain stress, strain, displacement, and other state parameters of each element node in the model. These parameters represent the mechanical state of particles in the elastic wave-disturbed rock medium. In one-dimensional problems, the axial stress component can reflect the main internal particle disturbance as the elastic wave propagates along the long axis of the rock rod. Figure 9 displays the stress history of granite at the center of the excitation end face at 100 and 250 kHz. The stress analysis is based on the stress history of various positions near the central axis of the rock rod. The maximum stress point in the waveform diagram corresponds to the stress amplitude of the elastic wave at this position. The spectrum information corresponding to the waveform can be obtained through Fast Fourier Transform (FFT). The frequency at which the amplitude is highest on the amplitude–frequency diagram is known as the peak frequency.

### 4.2. Simulation Results

Figure 10 shows the peak frequencies of elastic waves propagating at 100 and 250 kHz in marble, granite, and red sandstone, as determined by numerical calculations. The results are consistent with laboratory tests, as shown in Figure 7. The peak frequency attenuation also exhibits a stepwise attenuation in the numerical simulation results. Table 5 shows the attenuation distances of the peak frequencies at 100 and 250 kHz for the three types of rocks, as obtained by the two experimental methods.

The numerical simulation and laboratory experiments revealed significant differences in the attenuation distance of the peak frequency in various rocks. However, it is important to note that the peak frequency attenuation distance obtained from numerical simulation and laboratory experiments cannot be directly compared due to the different excitation signal intensities used. However, the relative magnitudes of the three rock attenuation distances are consistent in both test methods, which verifies the reliability of applying numerical simulation to explore the attenuation characteristics of the peak frequency of elastic waves in rock materials. Based on this, the attenuation of elastic waves with peak frequencies of 100 and 250 kHz in five other types of rocks was obtained, as shown in Figure 11 and Table 6.

The peak frequencies of elastic waves in five types of rock exhibit a step-like damping during propagation, which is a common feature across rock types. The attenuation distance of the peak frequency varies significantly among different rocks, with limestone exhibiting the longest distance and gray sandstone exhibiting the shortest. The relative magnitude of the attenuation distance among different rocks is similar at 100z and 250 kHz. The type of rock affects the attenuation distance of the peak frequency.

### 4.3. Relationship Between Peak Frequency Attenuation and Rock Physical and Mechanical Parameters

The peak frequency of the elastic wave experiences different attenuation distances in various rocks during numerical simulations and laboratory experiments. This reflects the varying attenuation of the elastic wave in different rocks. The response characteristics of rocks to load are often related to their physical and mechanical properties. The attenuation characteristics of elastic waves in a rock medium reflect the response state of the rock when disturbed by dynamic loads. Commonly used parameters to describe the attenuation of elastic waves in the medium include the attenuation coefficient (α) and Q value [12, 27].

Based on the stress waveforms obtained from numerical simulations at varying distances from the rock rod, the stress amplitudes can be determined. The amplitude of the elastic wave is related to the distance it travels, as shown in equation (1) [13]:

where *A*_{0} is the amplitude at the source, *α* is the attenuation factor, *x* is the distance of signal propagation, and *A* is the amplitude of the wave propagating from the source to *x*.

Therefore, the attenuation coefficient of the elastic wave is determined by the decrease in amplitude during its propagation, indicating the attenuation of the elastic wave in the medium. Table 7 compares the attenuation coefficient of elastic waves in various media and the distance over which the peak frequency attenuates. The attenuation coefficient and distance are inversely related for the eight types of rocks in the numerical simulation. Rocks with smaller attenuation coefficients have larger peak frequency attenuation distances and elastic waves propagate with less attenuation. Therefore, the intensity of attenuation can be determined by comparing the attenuation distance of the elastic wave in the medium.

The attenuation of elastic waves is affected by the tightness of particle bonding, the gap between particles, and the integrity of particles in the rock medium. These microstructural properties are reflected in the physical and mechanical parameters of the rock, such as density and wave velocity. The product of these parameters is the wave impedance, which reflects the overall nature of the rock. The rock’s elastic modulus provides a comprehensive reflection of its internal properties. Figures 12 and 13 demonstrate that the attenuation coefficient and attenuation distance vary with P-wave impedance and Young’s modulus, respectively.

Figure 12 illustrates that the attenuation coefficient decreases as the wave impedance increases at 100 and 250 kHz, while the peak frequency attenuation distance shows an increasing trend. In Figure 13, the attenuation distance has a significant increasing trend with the increase of elastic modulus, and the attenuation coefficient gradually decreases. This correlation is particularly significant at 100 kHz. The attenuation coefficient of stress amplitude in different types of rocks decreases as wave impedance increases. This is consistent with the fact that the attenuation distance of peak frequency is larger and attenuation in propagation is slower with higher wave impedance. Therefore, the attenuation characteristics reflect the comprehensive properties of rocks to some extent.

The simulation of elastic wave attenuation tests in rocks indicates that the attenuation distance of the peak frequency of elastic waves varies among different rocks and reflects their attenuation characteristics. Rocks with larger attenuation distances have smaller attenuation coefficients. The parameters for attenuation in both the frequency and time domains reflect the characteristics of elastic waves in the medium. Analyzing the variation of the peak frequency’s attenuation can provide a supplementary basis for describing the attenuation characteristics in rock mediums.

## 5. Conclusion

This study investigated the propagation of elastic waves in a rock medium and identified certain parameters that decrease with increasing propagation distance. Specifically, the peak frequency of elastic waves was observed to attenuate stepwise in laboratory experiments, with variations in the attenuation distance depending on the rock type and frequency. This attenuation behavior is due to the rapid decrease and attenuation of the high-frequency component near the excitation frequency, accompanied by increasing dominance of the low-frequency range. As a result, the peak frequency transitions from the excitation frequency to the low-frequency band, exhibiting a stepwise attenuation pattern primarily caused by the accumulation of the low-frequency signals.

The numerical simulations confirmed the presence of the stepwise attenuation phenomenon in a wide range of rock types, indicating its universal nature during wave propagation. The attenuation distance and coefficient of the elastic wave’s peak frequency serve as indicators of signal attenuation characteristics in the medium. It is worth noting that rocks with larger attenuation distances also exhibit smaller elastic wave attenuation coefficients. These two attenuation parameters are closely related to the physical and mechanical properties of the rock.

An increase in the wave impedance and elastic modulus of the rock corresponds to an increase in the peak frequency attenuation distance and a decrease in the attenuation coefficient. The impedance of the wave, which is the product of the density and the wave velocity, reflects the overall characteristics of the rock, representing the tightness, particle gaps, and integrity of its internal structure. On the other hand, the elastic modulus reflects the rock’s resistance to deformation, providing insights into its internal structure and composition properties.

Furthermore, the peak frequency attenuation distance in the medium follows the same variation trend as the wave impedance and elastic modulus of the rock. Consequently, the use of peak frequency attenuation distance as a measure enables a reliable characterization of elastic wave attenuation characteristics in rock. The distance at which the peak frequency is attenuated corresponds to the attenuation parameters in the time domain and rock mechanics parameters. This demonstrates the significance of frequency domain parameters in describing the attenuation characteristics of elastic waves in the medium.

## Data Availability

Data will be made available on request.

## Conflicts of Interest

The authors declare no conflicts of interest.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 42172316) and the Natural Science Foundation of Hunan Province (Grant No. 2021JJ30810).