Modeling and Simulation of SWARA Path Loss Model for Underwater Acoustic Communication in Multipath Environment

Effective communication in underwater environments is essential for numerous applications, including marine research, oil and gas exploration, and underwater surveillance. However, signal loss during propagation is a significant challenge for underwater communication systems. Accurate mathematical models are necessary to develop and optimize communication systems for underwater environments. This research proposes a mathematical model, SWARA, that incorporates crucial parameters such as transmitter and receiver responses, channel conditions, node placements, and signal frequency to accurately model short-range losses in underwater medium. We conducted tank trials at the UWAA Lab, CARE, IIT Delhi to validate our simulation results, using a chirp signal with a frequency ranging from 25–35 kHz. Our simulation results show that the SWARA model effectively predicts short-range propagation losses in underwater media. Compared to experimental losses, the maximum and minimum simulated losses deviated by only − 0.18–0.10 W/Hz and − 0.36–0.19 W/Hz, respectively, indicating the accuracy of the model. Our tank trials further revealed the significant impact of projector and hydrophone placements on incurred losses, highlighting the importance of careful placement in the design of underwater communication systems. In conclusion, this research proposes the SWARA model for studying short-range losses in underwater tank, which effectively incorporates crucial parameters for accurate modeling. The model has practical applications in the design and optimization of underwater communication systems for short-range applications. This research provides important insights for the development of effective underwater communication systems.


Introduction
When an acoustic signal travels from one point to another, transmission loss (TL) occurs, which reduces the overall strength of the acoustic signal at the receiver end. Path loss models the transmission loss in terms of the function of absorption, spreading, scattering, reverberation, reflection, refraction, and diffraction that occur along the path of the signal in the underwater (UW) channel. A robust propagation model is necessary and should consider the real-time channel conditions, as well as the properties of the transmitter and receiver, to accurately build a propagation loss model. The propagation loss model can be used as the basis for evaluating the performance metrics of deployed nodes.
The accurate prediction of propagation loss is essential to minimize the cost of experimentation required to test a researcher's hardware, prototype, or design, or to test the performance metrics of communication protocols or networks to improve the network's throughput and reliability. To achieve this, a protocol that is aware of the environment must be designed to adjust the parameters of the excitation signal as per the need of the underwater (UW) channel's time and space variations. The parameters mainly include transmission frequency, bandwidth (BW), and power. The choice of transmission frequency depends on the amount of absorption that occurs at the operating frequency (kHz). Transmission BW is selected according to the theoretical upper bound set by Shannon's channel capacity theorem. Transmission power is adjusted to achieve the desired Signal to Noise ratio (SNR) level greater than the noise level to meet the requirements of successful transmission at range 'r' meters. Therefore, four essential steps are needed: (1) Characterization of the underwater acoustic (UWA) channel using a CTD instrument at Tx/Rx nodes, (2) Registration of channel conditions at Tx/Rx nodes, (3) Analysis of propagation loss for acquired channel conditions at Tx/Rx nodes, (4) Selection of the optimum transmission frequency, bandwidth (BW), and power to fulfill successful transmission and reception.
Shallow water exhibits a higher propagation loss due to its varying nature of channel boundaries. Therefore, the occurrence of propagation loss is more significant due to the higher rate of change in the reflection and refraction coefficients of the path reaching towards the receiver. On the other hand, in deep seawater, the propagation loss is moderate due to the slower rate of change in reflection and refraction coefficients. Hence, the nature of the boundary enclosure plays an important role in adding propagation loss at the receiver side.
The modeling approach for predicting propagation loss changes depending on the state of the channel conditions. Nowadays, most authors are trying to develop computationally efficient models that deliver closed approximations of simulated estimates of propagation loss compared to real-time experimental propagation loss for the assumed channel condition. In the last 20 years, the following standard propagation models have been used to study propagation losses: Ray tracing, Normal mode theory, Green's function, Parabolic equation, and Plane wave theory. All five models are derived from the homogeneous Helmholtz equation.
Ray theory provides an insight into how source energy propagates in a given underwater (uw) channel. The analysis of propagation loss is done in two parts: (1) by considering a fan of all rays from the source, where range-dependent propagation loss can be studied for an omni-directional source, and (2) by considering a selected windowed region among all fans of rays (termed as eigen analysis), which aims to provide depthdependent propagation loss [1]. This theory has an advantage in that it delivers a ray trace for both static and dynamic channel conditions by including source directionality. However, for range-dependent analysis, to produce reasonable energy, the spread of source, ray computations need to be performed at all ranges up to the receiver range, which increases its time complexity as the number of channel parameters change over time. Also, it is only applicable for high-frequency approximations, which means that the rate of change of the boundary should be as slow as possible compared to the wavelength of the source's operating frequency (i.e., high frequency) [2].
In the Normal Mode approach, the Helmholtz equation is separated into a one-dimensional depth and range equation. This is achieved by expressing it as a sum of discrete normal modes with one or more branch integrals, considering the vertical stratification and cylindrical symmetry of the underwater channel. The one-dimensional mode provides loss analysis of surface-bottom going rays with higher wave number, whereas the branch integral provides loss analysis of steeper rays with lower wave number [3].
The normal modes are computed by selecting preliminary values of horizontal wave number and using the Runge-Kutta numerical approach. This theory is applicable to short-range scenarios and adiabatic approximations (i.e., slowly varying) where the sound velocity profile (SSP) remains stable or varies slightly with respect to range and depth. It takes into account the impacts of changing SSP, boundary environment, and water properties to evaluate the transmission loss at the receiver end [3].
Green's function provides a complete solution to the Helmholtz equation for a channel that consists of mixed liquids and solids by using the Hankel transform. This method can be applied at any range, including the near field. However, there are limitations to this theory, namely, it can only be applied to horizontally stratified mediums, and its computation of propagation loss takes a long time. The parabolic equation approach overcomes the limitations of Green's function by considering the effects of the horizontal variation of SSP with respect to depth and provides a closed approximation to the 3-dimensional modeling of transmission loss. [4].
The plane wave theory is concerned with the analysis of sound waves propagating in the far field direction 'x', where the sound pressure or intensity is calculated at a distance greater than three times the wavelength (i.e. far field distance) from the source. For plane wave propagation in the 'x' direction, the sound pressure and particle velocity in other directions (i.e. y and z) are assumed to be in phase, so that the overall sound pressure or intensity depends on only one spatial variable at a time (i.e. 'x' in this case). As a result, the 3D-Helmholtz equation is simplified to a 1D-Helmholtz equation to represent sound pressure or intensity as a function of the direction of propagation 'x' and time 't'. Thus, the time complexity is reduced compared to conventional ray theory and normal mode theory. This theory is applicable to static tank channels where the properties of the channel are not changing much with respect to time [5].
After reviewing the available literature, we found that empirical models are more easily verified than advanced models. This is because it is impossible to measure all of the channel parameters, such as salinity, temperature, pressure, water density, boundary conditions, wind speed, ocean currents, and noise, using the sensor facilities available at the transmitter and receiver nodes. Additionally, it is difficult to validate the applicability of advanced propagation models to fulfill all of the channel conditions. On the other hand, empirical models are more manageable due to their limited demand for channel parameters, and experiments can be carried out in controlled environments, making it possible to validate their applicability.
Hence, there is a potential for the development of a new empirical model: Shallow Water Acoustics for Random Area (SWARA). SWARA uses (1) physical properties of the channel, (2) source and receiver properties, and (3) noise properties to determine the optimal placement of projectors and hydrophones in a tank by applying the plane wave theory. This approach provides a basic understanding of setting up a preliminary testing environment quickly, without deploying actual nodes, which helps reduce the cost of experimentation in terms of time, human resources, and finances. Such models are necessary at sonar operational sites, where the performance of sonar systems is critically dependent on transmission loss prevention, particularly for shallow water applications.
Section II of this paper introduces a novel mathematical model, called SWARA, for investigating the propagation losses of underwater channels in Indian tanks. The model utilizes the plane wave theory to simulate these losses, considering all possible depth and range placements of projectors and hydrophones to identify the optimal placement that can enhance channel performance.
This paper introduces a new mathematical model, SWARA, in Sect. 2, which is designed to study underwater propagation losses in Indian tank channels. The modeling of uw propagation losses is based on the plane wave theory, and simulations are conducted to determine the optimal placements for projectors and hydrophones at all possible depths and ranges, with the aim of improving channel performance. Section 3 explains the methodology used to calculate path loss, while Sect. 4 presents an analysis of the simulation conducted using the SWARA MATLAB code. The study is applicable to a small rectangular tank measuring 3.85 m in length, 2.4 m in width, and 2 m in depth. In Sect. 5, the verification and validation of the simulated transmission loss are discussed, followed by a comparative analysis and observations of simulated and experimental results of TL in Sect. 6. Finally, Sect. 7 presents the conclusion.

New Mathematical Model for Underwater Propagation Loss using Plane Wave Theory: Introducing SWARA
The shallow water channel is important due to its random temporal and spatial variations. It behaves like a waveguide structure for underwater tank channels, where the acoustic signal radiating from the source spreads out spherically for receivers located at a distance less than the depth of the underwater tank. For receivers located at a distance greater than the depth of the underwater tank, the signal spreads out cylindrically. In static channel conditions, sound rays emitted from the source propagate straight due to no variation in SSP among water layers of the underwater tank. These rays are perpendicular to the wave fronts and propagate in the direction of the wave fronts, as illustrated in Fig. 1.
The propagation of acoustic waves in water is described by the laws of fluid mechanics and the plane wave theory, which is presented below. When a source radiates at a frequency of f 0 kHz, the wave fronts travel in all directions with a particle velocity of U r m/s. U r is the particle velocity in the radial direction from the acoustic center of the source. In the far field (Kr > > 1), the force exerted on a subsequent water particle will cause it to move following the law of conservation of momentum. Between Kr (the wave number for the distance coverage of 'r' m) and & Kr + ΔKr, the sum of forces acting on a water element is equal to the change of momentum, as shown in Fig. 1.
In terms of the acoustic pressure P and the area s of an infinitesimal water element, the rate of change of velocity dU r dt can be expressed as: In the equation above, the vector ⃗ U r represents the particle velocity of water, which is the rate of change of displacement with respect to time. Therefore, the equation can be rewritten as: Note that the expression above is known as the Lagrangian description for the motion of a mass of water element at kr . A more precise balance of momentum can be expressed using the Euler description as follows: In the context of underwater acoustic wave propagation, the total acoustic pressure, denoted by p, is the sum of the static pressure, p sta , and the fluctuating pressure, p flc which is induced by small variations in the acoustic wave.
In the equation, ρ represents the total water density which is the sum of static density ρ sta and fluctuating density ρ flc .
In fluid mechanics, D Dt represents the total derivative. The first term denotes the rate of change with respect to time, while the second term denotes the change with respect to space as sound propagates along 'r' at a velocity of U r . In an underwater channel within a tank, the static pressure p sta and density ρ sta do not vary significantly with time or space, unlike the conditions found in sea channels. As a result, Eq. (4) can be simplified as follows: (1) (Ps) kr − (Ps) kr+kr = Ps dU r dt kr This equation proves that the change in acoustic pressure p flc across a small distance kr causes the water element of unit volume ρ sta to move with acceleration U r t . This equation is generally referred to as the linearized Euler equation, which states that a change in fluctuating pressure causes a water particle to move. By using the principle of conservation of mass (the relation between density and fluid particle velocity) and Eqs. (4) and (6), we obtain the following expression: In the context of water acoustics, the relation between the water fluctuating density ( ρ flc ) and the water particle velocity ( U r ) can be described as higher compression rates in time leading to a steeper negative velocity gradient in space. This implies that a change in acoustic fluctuating pressure results in a change in water fluctuating density and its entropy, as dictated by the laws of thermodynamics.
In the above equation, Blk represents the bulk modulus of water and C w is the speed of sound in the water medium. This equation shows how the fluctuating acoustic pressure is related to the fluctuating water density. It indicates that the speed of propagation is dependent on the medium's characteristics. Therefore, by using Eqs. (6), (7), and (8), we can derive the following relationship: The above equation represents the general one-dimensional wave equation for sound. This equation states that the 3-dimensional wave equation can be simplified to a 1-dimensional wave using plane wave theory. Since the source type is omnidirectional, the use of a spherical coordinate system is more suitable for calculating pressure in the far field. The type of coordinate system used depends on the type of source [6]. In this study, we adopt a spherical coordinate system to conveniently represent the pressure at any point p (x, y, z) at a range of 'r' m from the source in the UW tank. This coordinate system satisfies the governing equation and holds the superposition principle true for each component of the spherical coordinate system (r, Ɵ, φ) [6]. Considering an isotropic medium, the solution to the above wave equation for a point source radiating at a frequency of f 0 at a range of 'r' m is given by the reduced spherical wave equation.
The magnitude of the acoustic pressure of a sound wave propagating in the direction of 'r' with an initial phase 'ϕ' as a function of time 't' is represented by P (r,t) . The static pressure is denoted by p sta , and kr represents the wavelength expressed in terms of λ required to cover a distance of 'r' m.
Acoustic waves are often referred to as longitudinal waves, in which the particle velocity of the medium moves in the direction of the sound wave fronts. These wave fronts initially have a spherical shape (when Kr is around 1) and later become plane in the far field (when Kr > > 1), where the pressure amplitude decreases by a factor of 1/R due to the spherical spread of sound. This implies that the sound pressure in the far field is inversely proportional to the distance from the source [7]. In this study, we consider an omnidirectional point source with an input power of W watts, which radiates sound in all directions with the same intensity at a distance of 'r' m from the center of the source, as shown in Fig. 2.
In the propagation of sound, the physical characteristics and geometry of the channel, which includes the sidewalls and bottom of the uw tank, the type of water and sediment at the bottom during communication, strongly influence the sound's behavior. While normal water has low absorption losses for a moderate range of frequency, it increases as the frequency increases [8]. Hence, absorption losses must be taken into account for higher frequency operations.
The plane wave theory provides a comprehensive mathematical solution for the acoustic wave equation in the far field for all boundary conditions. This theory takes into account the projector and hydrophone properties, as well as the channel conditions, to calculate the transmission loss. The transmission loss represents the difference in the intensity of the acoustic signal from the source to the receiver, as shown in Fig. 2. To achieve effective radiation at the source, Ka should be greater than 1 in the far field region (distance greater than 3λ), and the choice of operating frequency depends on the Ka factor (wavelength in terms of λ to cover the distance of 'a' m). The active average (mean) intensity at a distance of 'r' m from the source is then calculated.
The velocity potential at a distance of 'r' meters can be expressed as follows: To calculate the acoustic pressure at a distance of 'r' m, several parameters must be taken into consideration. These include U 0 , which represents the static pressure divided by the product of water density and the speed of sound in water, as well as the radiation Far field intensity analysis of sound wave using plane wave theory impedance, 1−jKr 1−jKa , and the spreading loss factor, (a/r). These factors collectively contribute to the final value of the acoustic pressure.
The Eq. (11) can be obtained using Eqs. (12) and (13), where ρ w c w represents the characteristic impedance of the channel and −jKa 1−jKa represents the radiation impedance. It should be noted that Eqs. (12) and (13) are valid only when 'r' is greater than 'a'.
Therefore, we obtain.
The variables used in this equation are defined as follows: P r represents the acoustic pressure at a distance of 'r' meters in the far field, U * r is the complex conjugate of the water particle velocity at a distance of 'r' meters from the source, U o is the water particle velocity at a distance of 1 m from the source, and ρ w c w represents the characteristic impedance of the channel. Additionally, I 0 is the average mean intensity at a distance of 1 m from the source, and the 1/2 factor represents the averaging intensity.The effective radiation factor is given by 1/(1 + (1/ka) 2 ).
The size of the radiator (Ka) relative to the wavelength (λ) plays a dominant role in radiation. The radiated sound field is primarily influenced by the distance (Kr) from the radiator relative to the wavelength of interest (λ) [9]. As a result, in order to investigate the propagation of sound waves in a three-dimensional space, one must comprehend the transmission and reflection phenomena that occur at the boundaries (i.e., surface, bottom, and sidewalls) of an underwater (uw) tank.
Impedance mismatch occurs when there is a change in medium, resulting in a discrepancy in the reflection and transmission of pressure or energy bouncing off a flat surface of discontinuity. To illustrate, consider a tub-shaped rectangular underwater tank with dimensions of 4.05 × 2.6 × 2 m for the surface and 3.85 × 2.4 × 2 m for the bottom, as shown in Fig. 3. An Omni source, which radiates at frequency f 0 kHz with an input power of W watts in all directions, causes sound rays to travel from the source to the receiver along several multi-paths.
In this model, we have solely taken into account the eigen paths that are the most significant multi-paths approaching the receiver's vicinity. These paths have been classified into six categories as mentioned below. An eigen ray traveling from the source to the receiver through the surface (water-air) boundary will result in the occurrence of V T , N, NN paths, and traveling via the bottom (water-concrete bottom) boundary will lead to the occurrence of V, N T , N T N T paths, as illustrated in Fig. 4.
In the case of acoustic signals travelling through any of the above-mentioned eigen paths, the velocity of water particles is assumed to be continuous at the surface or bottom boundary, so that the resultant velocity due to the incident and reflected pressure waves is the same as that of the transmitted pressure wave. The equations for pressure and velocity continuity can be expressed as follows:  The variables P inc , P refl , and P trans represent the incident pressure wave, reflected pressure wave, and transmitted pressure wave, respectively. Similarly, the variables U inc , U refl , and U trans represent the incident particle velocity, reflected particle velocity, and transmitted particle velocity, respectively. The minus sign in the equations is due to the 180° phase shift that occurs each time the sound wave bounces off the surface, bottom, or sidewalls of the underwater tank [10]. While acoustic pressure is a scalar quantity that does not depend on the direction of propagation, water particle velocity is a vector quantity and therefore we must consider the direction of propagation. By applying plane wave theory to Eq. (18), we can express the equation for the surface boundary as follows: Similarly, for the bottom boundary: The far field characteristic impedances of water, air, and bottom are denoted by The reflection coefficient for the surface and bottom discontinuity is defined as the ratio of incident pressure wave to reflected pressure wave ( P inc ∕P refl ). Hence, using Eq. (20), the reflection coefficient for the surface discontinuity can be obtained as: Similarly, the reflection coefficient for the bottom discontinuity can be denoted as R wb , and for convenience, we can represent the reflection coefficient for the surface discontinuity R wa as R 10 and for the bottom discontinuity R wb as R 12 .
The reflection coefficients for surface and bottom discontinuities are represented by R 10 and R 12 , respectively, and are defined by oblique wave impedances of water-air and water-bottom.
The SWARA model analyzes the angle-based transmission loss of eigen paths that reach the receiver's vicinity by considering specular reflection. This approach reduces computation time and eliminates the need to calculate transmission loss at every angle, as existing simulators do. Furthermore, to streamline the simulation code's implementation, various assumptions are made regarding the environment (the underwater tank channel) listed below.
1. The water in the tank is homogeneous, isothermal, isotropic, and non-viscous, and there are no layers of different temperatures, salinity, pressures, or sound speeds in the horizontal and vertical water columns. 2. The sound pressure fields are uniform in the far field. 3. Skimming is not handled in the simulation code at this time.
The source and receiver are stationary while sending and receiving the acoustic signal. 5. No other sources of noise or obstacles are present between or around the source and receiver placements. 6. Only the ambient noise due to the platform will be present, and its effects will be considered.
We used Snell's law at a flat interface [11] to investigate sound propagation in the underwater tank. As a result, any incident pressure wave that bounces off the flat interface at an arbitrary incident angle inc experiences linear wave propagation in all three media (i.e. air, water, and concrete bottom) of the underwater tank. Total internal reflection occurs when inc is less than θ sc , greater than θ sc , less than θ bc , or greater than θ bc , where θ sc is the critical angle for the surface interface and θ bc is the critical angle for the bottom interface. Snell's law is not applicable to analyze the transmission of pressure wave when inc is equal to θ sc or θ bc because total internal reflection fails at these angles. Figure 5 illustrates the range of applicability of Snell's law to achieve specular reflection.
In an underwater tank, the interface can be either the water-air interface or the waterconcrete bottom interface. The reflection coefficients R 10 and R 12 for these interfaces are determined using Snell's law with oblique incidence, and their values are presented in Table 1 [12]. When a sound pressure wave encounters any of these interfaces, it is both reflected and transmitted, as illustrated in Fig. 5. The degree of reflection and transmission depends entirely on the characteristic impedance of the respective mediums [13].

An Exploratory Study on Methodology for Calculating Path Loss Attenuation in Measured Channel Conditions
To compute the transmission loss for each eigen path, the following steps were taken: 1. To create a scenario where the projector and hydrophone are placed at equidistant depths of 'Z' meters and at a range of 'r' meters. The source and receiver placements were selected according to the Table 3 presented below. The placements were chosen to prevent the occurrence of near field effects. 2. To establish initial boundary conditions, we measured the channel parameters using an instrument. The following parameters were measured: To apply Snell's law and boundary conditions using plane wave theory to calculate respective reflection coefficients, we need to measure channel parameters such as salinity, temperature, pressure, fluid density of water and air with the help of available equipment at the UWAA lab. The measured channel parameters are listed below:  At water to air interface (surface) inc < sc To determine the dependable multipath signals that reach the vicinity of the receiver for a given combination of projector and hydrophone placements. The incident angle incs of the first reliable eigen path of the surface interface was obtained by applying trigonometric relations, as shown below.
The distance between the projector and hydrophone is represented as 'r' in meters, while the distance above the source placement is represented as 'das'.
After obtaining the incident angle incs of the first reliable eigen path of the surface interface using trigonometric relations, we can calculate the incident angle of the first reliable eigen path of the bottom interface, represented by incb . Using this, we can then calculate the incident angles of all the remaining reliable eigen paths that reach towards the receiver proximity using the following equation. Additionally, the variables "dbs" and "das" represent the distance below and above the source placement, respectively.
For the surface interface, the launch angle range begins at θ flas and extends up to 89 degrees, while for the bottom interface, it begins at θ flab and extends up to 89 degrees. Out of all possible combinations of launch angles, we choose only those that satisfy the following equation.
This formula optimizes the selection of reliable eigen paths that reach the receiver by providing the chosen combinations of incident angles using Eqs. (28) and (30) for the given assumed placements of the projector and hydrophone in the UW tank, as illustrated in Fig. 6. 3.1. In order to determine the critical angles for the surface and bottom ( θ sc and θ bc , respectively), we utilized Snell's law and calculated the path length of each eigen path using the Pythagorean theorem.
Here, the critical angle for the surface can be determined using the formula.
The critical angle for the bottom can be determined by applying Snell's law [14], as shown below.

3.2.
To calculate the surface reflection coefficient for V T , N and NN paths, and the bottom reflection coefficient for V, N T and N T N T paths. After identifying the V T , N , and NN paths based on the incident angles obtained from Eqs. (28) to (33), we compute the path lengths using the Pythagorean theorem and trigonometric formulas with the following equations.
We placed the projector and hydrophone at equidistant depths, so for the bottom case, we can obtain path lengths for V, N T and N T N T paths using Eqs. (37), (38), and (39) as follows: The above path lengths can be obtained using Eqs. (37), (38), and (39) for equidistant placements of nodes where das = dar and dbs = dbr, where das is the distance above the source (projector), dbs is the distance below the source, dar is the distance above the receiver (hydrophone), and dbr is the distance below the receiver, as illustrated in Fig. 4. 4. To calculate the path amplitude factor for each eigen ray.
As an eigen ray travels from the source to the receiver, it reflects from one of the boundaries of the underwater (uw) tank, which can be either the surface, bottom or sidewalls. This results in a reduction of energy, determined by the path amplitude factor,paf [15]. paf is calculated by the following equation: The source radius is represented by 'a', while the path length is represented by 'pl', and 'm' and 'n' denote the order of multipath for the respective path that has taken a finite number of bounces from the boundary. The path amplitude factor (paf) is a crucial factor in increasing the transmission loss of waves traveling from the source to the receiver. The greater the paf, the greater the propagation loss.

5.
To calculate pressure at distance 'r' m from rim of source.
As the hydrophone is calibrated based on the output voltage for an incident sound pressure field reaching the receiver hydrophone at time t i , the total pressure field at the receiver hydrophone, which is located at a distance of 'r' meters, will be transformed.
The number of eigen paths that reach the receiver at time t i is represented by nop . The value of nop can vary from 2 to 4 depending on the interference of sound pressure fields that come from the top, bottom or sidewalls of the uw tank. Equation 44 represents the coherent pressure fields. When two or more pressure waves of the same frequency and different phase reach the receiver at time t i , the overall semi-coherent pressure fields will be represented.
The incoherent pressure fields will be represented by Eq. (44) [16]. Consequently, by using Eq. (13), the cumulative pressure P r(r,t i ) can be obtained.
6. To calculate water particle velocity at distance 'r' m from rim of source we use Eq. (12).
7. To calculate acoustic intensity at a distance 'r' m from spherical Omni-source We obtained the intensity at distance 'r' m from an omnidirectional source radiating at a frequency using Eq. (46) and (11).
The intensity at a distance of 1 m from the Omni source can be calculated as follows: To calculate the path transmission loss (in dB) of the respective eigenray.
The aforementioned equation is utilized to compute the propagation loss for every possible combination of Tx and Rx nodes [17]. Table 3 depicts the combinations of projector and hydrophone placements for the given channel conditions.

Simulation Analysis of Transmission Loss Using Swara Path Loss Model
We utilized the channel conditions specified in Table 2 to create a SWARA path loss model that simulates the transmission losses for all potential placements of the projector and hydrophone in the UW tank facility at UWAA lab, IIT Delhi, as illustrated in Table 3. The SWARA Matlab code was created using Eqs. (28) to (50), which enabled us to model the effect of source and receiver placements on the occurrence of transmission loss, as outlined in Table 4. The SWARA path loss model is capable of providing the following analyses.
The SWARA path loss model enables us to analyze: (1) the total impact of the surface, bottom, and sidewalls on transmission loss, (2) the incident angle distribution, (3) the distribution of reflection coefficients for the surface and bottom, (4) the distribution of path amplitude factors, (5) the distribution of transmission loss, (6) the delay arrival, (7) the (50) TL Simulated = 10 log 10 I 0(1,t) I r (r,t)  intensity distribution, (8) the power delay profiles of each multipath, and (9) the finite number of available paths for every possible placement of the source and receiver. The aim of this paper is to investigate the impact of variations in the placement of the source and receiver on the overall transmission loss, as presented in Table 4. The minimum transmission loss is attributed to the V T paths, which are derived from the SR category of paths. The term SR refers to surface reflected paths, while BR refers to bottom reflected paths [18].
The minimum transmission loss is caused by the V T paths, which are derived from the SR category of paths and the V paths, which are derived from the LHSDWLR and RHSDWLR categories of paths. These paths are due to the sidewalls of the UW tank. The LHSDWLR and RHSDWLR categories refer to the paths reflected from the lefthand and right-hand sidewalls of the tank, respectively. The collective pressure of all possible paths, including those reflecting from the surface, bottom, and sidewalls, is used to calculate the minimum transmission loss using Eq. (44).
To calculate the maximum transmission loss, the SWARA path loss model will utilize Eqs. (45) through (50) to determine the collective pressure of all feasible paths, including NN and N T N T , reflecting from the surface, bottom, and sidewalls. Depending on the area of interest, we can also consider the effects of N and N T paths reflecting from the surface, bottom, and sidewalls to investigate time-dependent transmission losses, as described previously.
The SWARA path loss model is used to simulate transmission losses under different conditions, as presented in Table 2. The results showed that the simulated transmission loss ranged from 12.48 dB to 88.51 dB, depending on the depth (0.3 m to 1.2 m) and range (2 m to 3.2 m) of the placements of the source and receiver.
Our objective is to validate the SWARA Path Loss Model's simulated transmission losses for various node placements (depths and range) in an underwater tank. To achieve this, we require specific resources including a UW tank, projectors, hydrophones, pre/ Density of water ( ρ w ) 1000 kg/m 3 9 Density of bottom ( ρ b ) 2400 kg/m 3 10 Fluid particle velocity at 1 m from source ( U 0 ) 2.1e -10 m/s 11 Type of projector ITC 1042 Transducer 12 Source radius 17.7 mm 13 Transmit voltage sensitivity 127 dB @ 30 kHz 14 Type of hydrophone Keltron Transducer 15 Receiving sensitivity − 178 dB @ 30 kHz power amplifiers, power supply, data acquisition card, BNC connectors, coaxial cables, PC, and simulation tools. With these resources, we can set up an experiment to analyze the simulated results presented in Table 4. The SWARA Path Loss Model is developed to investigate the propagation losses of acoustic signals in Indian tank channels using the plane wave theory. The transmission loss is analyzed for different placements of projector and hydrophone based on depth and range, to determine the optimal node placement that provides the best channel performance. The simulation is carried out for a small rectangular tank with a length of 3.85 m, width of 2.4 m, and depth of 2 m.

Validation of Simulated Transmission Loss Through Tank Trials at Uwaa Lab Iit Delhi
The resources mentioned above were available at the UWAA lab of CARE, IIT Delhi. As a result, tank trials were conducted from January 7th to January 28th, 2019 to verify and validate the simulated results of the SWARA path loss model. To plan the experiments, we followed the steps outlined below.

To identify the underwater acoustics lab equipped with acoustic communication set up to perform planned experiments
The UWAA lab is equipped with various resources including a UW tank measuring 3.85 m × 2.4 m x 2 m, an ITC 1042 Omni-directional projector, a Keltron Uni-directional hydrophone, a Keltron Pre/Power Amplifier, a 5 V DC Keltron power supply, an NI PCI 6110 DAQ card, an NI 2110 BNC connector, RG-7 (50 Ω) co-axial cables, an Intel Xeon V3 @3.60 GHz × 64 PC with 32 GB RAM, and simulation tools such as Matlab 2017b and LabView 2018. The experimental set up is illustrated in Fig. 7.
To study transmission losses, we generated a chirp signal with a center frequency of 30 kHz and a bandwidth of 10 kHz. We transmitted the signal from the projector (ITC 1042) to the hydrophone (Keltron) at the depths specified in Table 3. These devices were connected to the PC through BNC connectors (NI BNC 2110) using 10 m long coaxial cables with an impedance of 50 Ω.
The signal is acquired at the hydrophone at a sampling rate of 240 kHz (using NI PCI 6110 DAQ), and then processed using LABVIEW and MATLAB simulation tools. Power amplification is performed at the transmitter side and pre-amplification is performed at the receiver side based on the channel conditions (calm, moderate, drastic). 2. To identify depths & ranges at which the experiments are to be performed.
The selection of depths & ranges is decided as per Table 3. 3. To measure the channel conditions, we have used the thermometer and CTD instrument.
The values of reflection coefficients for the SWARA path loss model shown in Table 2 were calculated by applying Snell's law and boundary conditions using these channel parameters. 4. To identify the source & receiver & their operating frequency range to be used for experimentation. We utilized ITC 1042 as the projector to emit acoustic signals from the source to the receiver. For recording the signals sent from the source, we used Keltron 8,240,000,001 as the hydrophone. The operating band of the projector is from 1 to 120 kHz, whereas for the hydrophone, it is 20 kHz to 40 kHz. Therefore, the frequency for the source's excitation signal must be selected within the range of 20 kHz to 40 kHz. We have identified the center frequency for the source's excitation signal as 30 kHz with a bandwidth of 10 kHz (25 kHz to 35 kHz). 5. To identify & design the type of signal to be used for establishing acoustic communication between source & receiver.
We designed a linear chirp signal with a 5 V peak amplitude at a center frequency of 30 kHz and a bandwidth of 10 kHz, as referred from [19]. This excitation signal was sent from the projector to the hydrophone at the depths mentioned in Table 3. A chirp is a sinusoidal signal of frequency f c that varies over time, with the relationship between time and frequency expressed through a polynomial expression that depends on the type of signal (linear, logarithmic, exponential). Chirp signals are utilized in data transmission schemes due to their advantages [20], such as… 5.1. It includes flat amplitude spectrum with independent scalability both in time & frequency domain. 5.2. It includes wide range of frequencies over short interval of time which eliminates influence of low frequency (biological) signals. 5.3. More than 90 percent of energy is present in chirp BW. 5.4. Auto correlation properties of chirps are similar to Impulse response function. 5.5. Shortening of pulse won't affect on general benefits of chirp signals hence pulse compression can be utilized in chirp signals. 5.6. Better Identification of channel impulse response in noisy channel conditions. 5.7. Identification and characterization of transmission parameters like multipath delay spread, coherence time, coherence bandwidth through CIR wiz essential for designing data communication through the available spectrum. 5.8. Chirp spread spectrum is ideal for applications requiring low power usage and needing relatively low data rates (1 mbps or less). The chirp spread spectrum technique is commonly used in sonar applications, as it is resistant to multi-path fading even when the transmitter is operating at very low power. The design parameters for the chirp signal are listed in Table 5. The experiments were conducted in the UW tank facility of UWAA Lab, CARE IIT Delhi, at specified combinations of depths and ranges as shown in Table 3. The tank was filled with pure water up to a height of 1.8 m, and the average water temperature was 20 °C. Power and pre-amplification were not performed at the transmitter and receiver sides, respectively, because the signal-to-noise ratio was favorable for static calm channel conditions (Fig. 9). 7. To analyze TL experimentally performed at decided placements of source & receiver in uw tank. In order to determine the transmission loss at specified depths and ranges, we need to determine the sound wave intensities at 1 m and 'r' m. The sound wave intensity level transmitted by the ITC 1042 at 1 m can be calculated using the equation provided in [21].  The transmit voltage response (TVR) of ITC 1042 at the operating frequency of 30 kHz is shown in Fig. 10, with a value of 127 dB. TVR is a measure of the relative voltage generated at a distance of 1 m by the ITC 1042 projector for a supplied 1 microPascal pressure per 1 V. The sound intensity level at a distance of 1 m from the source is 133.98 dB reference 1 microPascal per 1 V at 1 m. When the receiver is placed at a distance of 'r' meters, the intensity level of the transmitted signal sent from the ITC 1042 source is calculated using an equation given by [21].
The voltage recorded by the Keltron hydrophone is in reference to the received pressure at a distance of 'r' m. The open circuit receiving response (OCRR) of the Keltron hydrophone at an operating frequency of 30 kHz is -178 dB, as shown in Fig. 11. OCRR measures the voltage generated by the hydrophone at a distance of 'r' m for a supplied 1 V per 1 micropascal pressure. The sound intensity level at a distance of 'r' m from the source is 166.73 dB ref 1 V/1μPa @ 'r^' m for one of the time instances of recorded voltage samples of combination index 8. Therefore, the equation for experimental transmission loss is defined using Eqs. (51) and (52) as shown above.   [22] The experiment involved the transmission of a linear chirp signal with a 10 kHz bandwidth centered at 30 kHz from the projector (ITC 1042) to the hydrophone (Keltron 8,240,000,001) at depths specified in Table 3, using Eqs. (51), (52), and (53). The transmission loss at the specified positions of the projector and hydrophone is reported in Table 6 above. The fourth column of Table 6 shows the comparison of the experimental transmission loss with the simulated transmission loss from the SWARA path loss model, thus validating the model.

Comparing Experimental and Simulated Transmission Loss: Observations and Analysis
The overall transmission loss is dependent on surface and bottom reflection, path amplitude factor, water particle velocity, radiation, and characteristic impedance as demonstrated in Eqs. (47) to (50). The occurrence of transmission loss is reliant on the instantaneous distribution of these factors. Table 7 displays the range of incident angles of the eigen rays' bounces from the surface, bottom, and sidewalls that reach the receiver. The directivity pattern of the projector is based on its transducer equivalent beam pattern [23]. The SWARA path loss model also incorporates the projector's beam pattern and directivity to choose the orientation of the projector and hydrophone to ensure the reception of the acoustic signal from the source to the destination during deployment.
In accordance with Table 7, the directivity pattern of the projector should encompass incident angles ranging from 17 • to 42 • for surface bouncing, 28 • to 56 • for bottom bouncing, and 26 • to 50 • for sidewalls to ensure signal reception at the receiver for all 17 combinations. This is because the ITC 1042 projector is of the spherical beam type, which includes all the beams mentioned above in its directivity pattern [22].
The reflection coefficient at the surface is a crucial factor in determining the propagation loss. When a pressure wave hits the surface, bottom, or sidewalls, it undergoes a phase shift of 180 • , indicating a negative sign. Simulation results indicate that for the bottom boundary, R 10 ranges from 0.99 to 1, meaning that most of the incident acoustic pressure is reflected from the surface of the underwater tank. For the surface boundary, R 10 ranges from 0.953 to 1, indicating that the reflected pressure is slightly less than that of the bottom. The bottom reflection coefficient, R 12 , is 1 for the surface, bottom, and sidewalls, except for a few N and N T paths.
The spreading loss depends on distance travelled by respective eigen path. From simulations it is observed that path amplitude factor increases with increase in path lengths. Minimum spreading loss is occurred at V T and V paths. Maximum spreading loss is occurred at NN and N T N T paths.
The range of path amplitude factor for sidewalls, surface, and bottom boundary is 1.01e-5 to 9.11e-9, 1.1e-5 to 9.48e-8, and 1.84e-5 to 9.64 e-7, respectively. The range of path amplitude factor is lower for the bottom compared to surface and sidewalls, as illustrated in Figs. 12, 13, and 14. Simulation results indicate that sidewalls contribute more to the overall transmission loss than the surface and bottom of the tank, with the bottom contributing the least.
The simulation of transmission loss caused by sidewalls is performed by placing projectors and hydrophones at the center of the width of the underwater tank to prevent near field effects. The optimal placements will be selected by considering the minimum deviation in both maximum and minimum simulated and experimental TL, as presented in Table 8. Combination numbers 3, 5, 8, 10, 12, 13, 14, and 15 exhibit a perfect agreement between the theory and experiments. Therefore, the SWARA path loss model can be considered effective in providing range and depth-dependent TL analysis. Deviation of maximum TL simulated value Y times from experimental value Y

Conclusions
The results of the experiment indicate that the absorption from the sidewalls and bottom, as well as the high characteristic impedance of water and interference among sound pressure fields reflecting from the boundaries of the underwater tank, considerably change the propagation loss of acoustic signals. Smaller tanks with smaller dimensions will have a greater number of multipaths. Therefore, to accurately calculate the propagation loss at the receiver, it is important to consider the specific treatments of the pressure field at the location. The simulation of propagation loss is carried out using plane wave theory, which calculates the difference between effective plane wave intensities at a distance of 'r' meters and 1 m from the acoustic source. The sound intensity level at both distances is calculated using the mean squared pressure fields recorded by the projector. The difference between the calculated intensities is referred to as experimental transmission loss (TL).
Among the 17 combinations analyzed, 8 show a close match between theoretical and experimental results. Combination number 10, in particular, exhibits perfect agreement between theory and experiment. For combinations 3,5,8,10,12,13,14, and 15, the deviation of simulated transmission loss ranges from -3% to 8% of the experimental value.
In contrast, the simulated values of TL for the remaining 9 combinations (no. 2,6,7,9,11,16, and 17) are within -0.18 to 0.19 times the experimental transmission loss, indicating reasonable agreement between theory and experiment. However, the simulated values for combinations no. 1 and 4 fall outside the acceptable range of reasonable limits.
Based on the literature, the deviation between simulated and experimental TL can be attributed to several factors, including: 1. Assuming a constant theoretical source level at 1 m from the acoustic source, while in practice the source level depends on the accurately measured reverberation time 'T60' of the UW tank. 2. Assuming a uniform distribution of the sound field in the far-field, whereas it depends on the placement of the hydrophone in the reverberant field, i.e., whether it is near the surface, at the middle of the depth, or near the bottom. 3. The total mean squared sound pressure fields at the hydrophone are a combination of reflected sound fields from the boundaries of the UW tank. It is practically impossible to segregate the respective sound field at the point of measurement using only one hydrophone. Thus, an array of hydrophones is required to segregate each multipath arrival and its direction at the measurement point. 4. Uncertainties in the measuring equipment, including the known projector and hydrophones. Table 6 and 8 demonstrate a close agreement between theory and experiment, indicating that the SWARA mathematical model's interpretation of transmission loss for the specified placements of the projector and hydrophone is valid and considered physically accurate.
Therefore, the proposed SWARA path loss model can be considered as a forward propagation model that allows for a detailed simulation of transmission loss for all possible placements of projectors and hydrophones. This mathematical model provides a specific analysis of multipaths based on the path and time, which is quite useful for analyzing the operational site before deploying the acoustic set-up.