Dynamic water body generation method and system based on 3D Gaussian splash

Abstract

The invention relates to a dynamic water body generation method and system based on 3D Gaussian spatter, and belongs to the field of image processing.The method comprises the steps that firstly, a three-dimensional water body model is generated through 3D spatter according to water body pictures of multiple views; 2, determining an internal filling area through an intersection point concept, and carrying out particle filling on the interior of the three-dimensional water body model to obtain a dynamic three-dimensional water body model; and step 3, solving a partial differential equation for describing fluid motion through a smoothed particle hydrodynamic method, physically integrating the three-dimensional water body model, reconstructing a dynamic model and obtaining a water body flow effect. According to the invention, the dynamic three-dimensional water body can be generated according to the two-dimensional water body pictures at different angles; through combination of 3D Gaussian and smoothed particle fluid dynamics, a water body is endowed with a physical attribute, and a water body model is endowed with a physical motion attribute.

Description

Technical Field

[0001] The present invention relates to the field of image processing, and in particular to a method and system for generating a dynamic water body based on 3D Gaussian splashing. Background Art

[0002] Traditional water body generation methods often use different methods to model and draw water bodies. Currently, the Gerstner wave based on linear wave superposition is introduced into the field of water body rendering. Different linear wave functions are superimposed together to form an undulating surface to simulate rough water surfaces such as the ocean, but the dynamic effect lacks details. The existing technology also draws on the ocean empirical spectrum and statistical models, combines a large number of sine waves to generate wave distribution in the frequency domain, and then performs an inverse fast Fourier transform (Fast Fourier Transform, FFT) to transfer the data to the spatial domain to simulate a water surface with a certain visual performance. In order to simulate more detailed effects, the physics-based Lagrangian method studies the changes in the speed, pressure, density and other parameters of a specified particle in the water body over time to study the movement of the entire fluid, but the amount of calculation is too large. The wave particle method uses particles to represent water waves, combining the flexibility of the linear superposition method with the stability and visual details of the FFT-based method, reducing the complexity of dynamic simulation of three-dimensional water waves and allowing water wave reflection. On this basis, some people have proposed the idea of ​​wave packets that can handle water surface reflection behavior and generate attractive interactive water waves.

[0003] However, the above methods all have the problems of slow water body generation speed, inability to generate dynamic water bodies through multi-view two-dimensional images, and lack of dynamic scene reconstruction technology in the field of water body generation. Summary of the invention

[0004] The purpose of the present invention is to overcome the shortcomings of the prior art and provide a method and system for generating dynamic water bodies based on 3D Gaussian splash, thereby solving the shortcomings of the prior art.

[0005] The object of the present invention is achieved by the following technical solution: a method for generating a dynamic water body based on 3D Gaussian splash, the method comprising:

[0006] Step 1: Generate a 3D water model through 3D splashing based on the water body images from multiple views;

[0007] Step 2: determine the internal filling area through the concept of intersection, fill the inside of the three-dimensional water body model with particles, and obtain a three-dimensional water body model for dynamicization;

[0008] Step 3: Solve the partial differential equations describing fluid motion through the smoothed particle hydrodynamics method, physically integrate the three-dimensional water model, reconstruct the dynamic model and obtain the water flow effect.

[0009] The step one comprises:

[0010] Extract image feature points, match feature points at different viewing angles, and estimate the camera’s intrinsic and extrinsic parameters to construct a sparse point cloud of the scene;

[0011] A set of 3D Gaussian distributions are created based on these sparse point clouds. Each Gaussian distribution contains a center position x, a covariance matrix Σ, and a transparency α. The covariance matrix Σ describes the shape and direction of the Gaussian point in three-dimensional space. It is obtained by multiplying a rotation matrix R and a scaling matrix S, expressed as , T represents transpose;

[0012] Given a transformation matrix W and the Jacobian matrix J of the affine transformation, the covariance matrix in the camera coordinates is ;

[0013] Using the GPU-based tiled rasterization renderer, the Gaussian points are quickly sorted, and the color of each pixel is calculated by mixing N ordered points overlapping the pixel. , where m and n are the indices of points that overlap a pixel, For each point's color, is the opacity.

[0014] The second step comprises:

[0015] The opacity threshold σ is customized to determine whether the particle exists inside or outside the water body. When the ray enters a high transparency grid from the bottom transparency grid, an intersection point is set;

[0016] After all the intersection points are determined, the internal filling area is obtained, and then the internal particles are filled. The color and opacity of these particles ensure that the color of the particles inside and outside the static water body is consistent;

[0017] The constructed three-dimensional water body model is filled with particles to obtain a static three-dimensional water body model for dynamicization.

[0018] The step three comprises:

[0019] Search the particles globally and calculate the distance between two particles. The density of particle i is obtained by summing: , and the pressure is calculated based on the incompressible condition as , where m j represents the mass of all particles j around particle i, W ij represents the smooth kernel function, k is the fluid elastic modulus, is the reference density;

[0020] The pressure is calculated by taking the gradient of the pressure and finding , applying the Laplace operator to the velocity to calculate the viscosity is , and gravity is , where m i is the mass of particle i, is the pressure gradient, g is the gravitational acceleration;

[0021] Then calculate the acceleration of all particles i through the force, and calculate the velocity and position of the next time step through the acceleration;

[0022] The calculated velocity and position are embedded in the early static three-dimensional water model, and the Gaussian kernel is regarded as a discrete particle cloud to simulate the continuum in space. The particles are subjected to a first-order approximate local affine transformation. For static scenes, The three-dimensional Gaussian distribution of To dynamize the scene, where X p is the location of the Gaussian point, A p is the shape, σ p is the opacity, C p is the color information of the point, X p (t) is the position of the Gaussian point over time, A p (t) is the shape that changes with time;

[0023] The extended Gaussian distribution is , the calculated particle position and velocity are assigned to the Gaussian kernel, so that the particles move along the trajectory of the water body, while ensuring that the spherical harmonic coefficient and opacity of each particle remain unchanged over time, thereby ensuring that the color and transparency of the water body remain consistent during the movement, V p (t) is the velocity of water particles changing with time.

[0024] A dynamic water body generation system based on 3D Gaussian splash, the system comprising a model generation module, a particle filling module and a dynamic water body generation module;

[0025] The model generation module is used to generate a three-dimensional water model through 3D splashing according to the water body pictures of multiple views;

[0026] The particle filling module is used to determine the internal filling area through the concept of intersection, fill the interior of the three-dimensional water model with particles, and obtain a three-dimensional water model for dynamicization;

[0027] The dynamic water body generation module is used to solve the partial differential equations describing the fluid motion through the smooth particle fluid dynamics method, physically integrate the three-dimensional water body model, reconstruct the dynamic model and obtain the water body flow effect.

[0028] The model generation module specifically includes the following contents:

[0029] Extract image feature points, match feature points at different viewing angles, and estimate the camera’s intrinsic and extrinsic parameters to construct a sparse point cloud of the scene;

[0030] A set of 3D Gaussian distributions are created based on these sparse point clouds. Each Gaussian distribution contains a center position x, a covariance matrix Σ, and a transparency α. The covariance matrix Σ describes the shape and direction of the Gaussian point in three-dimensional space. It is obtained by multiplying a rotation matrix R and a scaling matrix S, expressed as , T represents transpose;

[0031] Given a transformation matrix W and the Jacobian matrix J of the affine transformation, the covariance matrix in the camera coordinates is ;

[0032] Using the GPU-based tiled rasterization renderer, the Gaussian points are quickly sorted, and the color of each pixel is calculated by mixing N ordered points overlapping the pixel. , where m and n are the indices of points that overlap a pixel, For each point's color, is the opacity.

[0033] The model generation module specifically includes the following contents:

[0034] The opacity threshold σ is customized to determine whether the particle exists inside or outside the water body. When the ray enters a high transparency grid from the bottom transparency grid, an intersection point is set;

[0035] After all the intersection points are determined, the internal filling area is obtained, and then the internal particles are filled. The color and opacity of these particles ensure that the color of the particles inside and outside the static water body is consistent;

[0036] The constructed three-dimensional water body model is filled with particles to obtain a static three-dimensional water body model for dynamicization.

[0037] The dynamic water body generation module specifically includes the following contents:

[0038] Search the particles globally and calculate the distance between two particles. The density of particle i is obtained by summing: , and the pressure is calculated based on the incompressible condition as , where m j represents the mass of all particles j around particle i, W ij represents the smooth kernel function, k is the fluid elastic modulus, is the reference density;

[0039] The pressure is calculated by taking the gradient of the pressure and finding , applying the Laplace operator to the velocity to calculate the viscosity is , and gravity is , where m i is the mass of particle i, is the pressure gradient, g is the gravitational acceleration;

[0040] Then calculate the acceleration of all particles i through the force, and calculate the velocity and position of the next time step through the acceleration;

[0041] The calculated velocity and position are embedded in the early static three-dimensional water model, and the Gaussian kernel is regarded as a discrete particle cloud to simulate the continuum in space. The particles are subjected to a first-order approximate local affine transformation. For static scenes, The three-dimensional Gaussian distribution of To dynamize the scene, where X p is the location of the Gaussian point, A p is the shape, σ p is the opacity, C p is the color information of the point, X p (t) is the position of the Gaussian point over time, A p (t) is the shape that changes with time;

[0042] The extended Gaussian distribution is , the calculated particle position and velocity are assigned to the Gaussian kernel, so that the particles move along the trajectory of the water body, while ensuring that the spherical harmonic coefficient and opacity of each particle remain unchanged over time, thereby ensuring that the color and transparency of the water body remain consistent during the movement, V p (t) is the velocity of water particles changing with time.

[0043] The present invention has the following advantages: a method and system for generating dynamic water bodies based on 3D Gaussian splashes, which can generate dynamic three-dimensional water bodies according to two-dimensional water body pictures at different angles; by combining 3D Gaussian with Guanghua example fluid dynamics, the water body is given physical properties, and the water body model is given physical motion properties; it is easy to be directly integrated and applied in other three-dimensional engines or simulation platforms, and has good scalability and commercial potential; it can be used as a graphics engine plug-in or a dynamic water body simulation module for game development and digital content creation, and can also be expanded to multiple fields such as fluid mechanics simulation and environmental simulation, and has broad market potential. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 It is a schematic diagram of the process of the present invention;

[0045] Figure 2 Schematic diagram for determining the internal and external boundaries of a water body. DETAILED DESCRIPTION

[0046] In order to make the purpose, technical scheme and advantages of the embodiments of the present application clearer, the technical scheme in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. The components of the embodiments of the present application described and shown in the drawings here can be arranged and designed in various different configurations. Therefore, the detailed description of the embodiments of the present application provided below in conjunction with the drawings is not intended to limit the scope of protection of the application claimed for protection, but merely represents the selected embodiments of the present application. Based on the embodiments of the present application, all other embodiments obtained by those skilled in the art without making creative work belong to the scope of protection of the present application. The present invention is further described below in conjunction with the drawings.

[0047] like Figure 1 As shown, the present invention aims at the problems existing in traditional water body generation: slow water body generation speed, inability to generate dynamic water body through multi-view two-dimensional pictures, and lack of dynamic scene reconstruction technology in the field of water body generation. The present invention combines 3D Gaussian splash with fluid mechanics and proposes a water body generation method integrating physical simulation with dynamic scene reconstruction, which specifically includes the following contents:

[0048] 1. Generate a 3D water model based on multi-view water images through 3D Gaussian splash;

[0049] 3D Gaussian is a way to reconstruct 3D scenes in the form of point clouds. First, the image is calibrated through SfM (Structure from Motion). The calibration process includes extracting image feature points, matching feature points under different viewing angles, and estimating the intrinsic and extrinsic parameters of the camera to construct a sparse point cloud of the scene. Based on these points, a set of 3D Gaussian distributions are created, each of which contains the center position x, the covariance matrix Σ, and the transparency α. The center position is called the mean of the Gaussian distribution:

[0050]

[0051] The covariance matrix Σ can be obtained by multiplying a rotation matrix R and a scaling matrix S, that is:

[0052]

[0053] Among them, R is a 3×3 orthogonal matrix that describes the direction of the Gaussian points, S is a diagonal matrix that describes the scaling of the Gaussian points along the three coordinate axes; the Σ matrix describes the shape and direction of the Gaussian points in three-dimensional space, which is crucial for compactly representing complex geometric structures in the 3D Gaussian point diffusion rendering method. By optimizing the Σ matrix, the 3D Gaussian point diffusion can be made more compact, thereby improving the rendering effect.

[0054] When a new view is obtained, projecting the 3D Gaussian to 2D for rendering requires projection to the image space: given a transformation matrix W, the Jacobian matrix J of the affine transformation, the covariance matrix Σ' in camera coordinates is calculated as follows:

[0055]

[0056] Using a GPU-based tiled rasterizer, the Gaussian points are quickly sorted and the color of each pixel is calculated by blending N ordered points that overlap the pixel:

[0057]

[0058] Where m and n are the indices of the points overlapping the pixel, m refers to the mth point overlapping the pixel, and n is the inner index. For each point's color, is the opacity, which is obtained by multiplying the covariance Σ by the SH color coefficient and opacity of each point.

[0059] By accumulating the opacity and color of each point on the two-dimensional viewing plane, a reasonable color effect as well as the stereoscopic sense and depth of the three-dimensional scene can be obtained on the image plane.

[0060] 2. Fill the interior of the three-dimensional water model with particles;

[0061] Since Gaussian distribution tends to be distributed near the surface of the reconstructed water body, the internal structure of the water body is easily blocked by its surface, resulting in a lack of particles inside. The subsequent water body dynamics will have the problem of sparse water particles, and then the water body dynamic effect will lack realism. In order to fill the particles into the internal hollow area, the present invention discretizes the 3D opacity field of the 3D Gaussian onto a 3D grid.

[0062] In order to determine the boundary between the inside and outside of the water body, the present invention defines the concept of intersection, and determines the candidate grid by projecting rays and checking the intersection. Specifically, a custom opacity threshold σ is used to determine whether a particle exists inside or outside the water body. When the ray is from a low transparency ( ) the mesh enters a high opacity ( ) grid, thereby defining an intersection point where σ a is the current voxel in the 3D opacity field, σb for its neighboring voxels.

[0063] Once all the intersections are determined, the internal filling area is obtained, and then the interior is filled with particles. The color and opacity of these particles are inherited from their adjacent Gaussian kernels, ensuring the consistency of the color of particles inside and outside the static water body. Finally, by filling the reconstructed three-dimensional water model with particles, a static three-dimensional water model for dynamicization is obtained.

[0064] 3. Solve the partial differential equations describing fluid motion by smoothed particle hydrodynamics (SPH) and physically integrate the model;

[0065] The SPH method is a meshless particle adaptive algorithm of pure Lagrangian nature. It regards water bodies as particles, and through the numerical solution of the NS equation, calculates the pressure gradient of the particles and then obtains the movement of the particles. In order to ensure the uniqueness of the solution, initial conditions are required, which include the initial shape of the fluid (the initial position of each particle), the initial velocity of each particle, position constraints (space boundaries) or velocity constraints, etc. Therefore, SPH fluid simulation constitutes a mixed initial-boundary value problem (Mixed Initial-Boundary Value Problem). Several key steps in the SPH solution process need to consider the contribution of neighboring particles within the core radius, which means that for each particle, it is necessary to search for its neighboring particles in the entire problem domain space. For example Figure 2 As shown, firstly, we search for particles globally and calculate the distance between two particles. j Multiply by W ij The weight value of particle i is obtained by summing:

[0066]

[0067] Among them, m j represents the mass of all particles j around particle i, W ij represents the smoothing kernel function.

[0068] According to the fluid elastic modulus k and the custom reference density The calculated pressure is:

[0069]

[0070] The pressure can be calculated by finding the gradient of pressure , m i is the mass of particle i, is the pressure gradient. The viscosity force can then be calculated by applying the Laplace operator to the velocity ,in is the viscosity coefficient, Indicates the velocity diffusion rate. There are also external forces such as gravity , g is the gravitational acceleration, and then for all particles i, the acceleration is calculated by the force, and the velocity and position of the next time step are calculated by the acceleration:

[0071]

[0072]

[0073] in, is the time step, is the current total force, which is the sum of pressure, viscosity and external force, and Ask the position and speed at the current time respectively.

[0074] 4. Physical integration to obtain water flow effect;

[0075] The physical parameters (speed and position) calculated above are embedded in the static three-dimensional water model. The present invention regards the Gaussian kernel as a discrete particle cloud, which is used to discretely simulate the continuum in space and perform a first-order approximate local affine transformation on the particles. For static scenes, A three-dimensional Gaussian distribution, where X p is the location of the Gaussian point, A p is the shape, σ p is the opacity, C p is the color information of the point, and these Gaussian distributions are evolved into dynamic Gaussian distributions through simulation To animate the scene, X p (t) is the position of the Gaussian point over time, A p (t) is the shape that changes with time. On this basis, the present invention extends the Gaussian distribution to , V p (t) is the velocity of water particles that changes with time; the particle position and velocity calculated by SPH are assigned to the Gaussian kernel, so that the particles can move along the trajectory of the water body. At the same time, the spherical harmonic coefficient and opacity of each particle are guaranteed to remain unchanged over time, thereby ensuring that the color and transparency of the water body remain consistent during the movement.

[0076] The above is only a preferred embodiment of the present invention. It should be understood that the present invention is not limited to the form disclosed herein, and should not be regarded as excluding other embodiments, but can be used for various other combinations, modifications and improvements, and can be modified within the scope of the concept described herein through the above teachings or the technology or knowledge of the relevant field. The changes and modifications made by those skilled in the art do not deviate from the spirit and scope of the present invention, and should be within the scope of protection of the claims attached to the present invention.

Claims

1. A method for generating dynamic water bodies based on 3D Gaussian splash, characterized by: The method comprises: Step 1: Generate a 3D water model through 3D splashing based on the water body images from multiple views; Step 2: determine the internal filling area through the concept of intersection, fill the inside of the three-dimensional water body model with particles, and obtain a three-dimensional water body model for dynamicization; Step 3: Solve the partial differential equations describing fluid motion through the smoothed particle hydrodynamics method, physically integrate the three-dimensional water model, reconstruct the dynamic model and obtain the water flow effect.

2. The method for generating a dynamic water body based on 3D Gaussian splash according to claim 1, characterized in that: The step one comprises: Extract image feature points, match feature points at different viewing angles, and estimate the camera’s intrinsic and extrinsic parameters to construct a sparse point cloud of the scene; A set of 3D Gaussian distributions are created based on these sparse point clouds. Each Gaussian distribution contains a center position x, a covariance matrix Σ, and a transparency α. The covariance matrix Σ describes the shape and direction of the Gaussian point in three-dimensional space. It is obtained by multiplying a rotation matrix R and a scaling matrix S, expressed as , T represents transpose; Given a transformation matrix W and the Jacobian matrix J of the affine transformation, the covariance matrix in the camera coordinates is ; Using the GPU-based tiled rasterization renderer, the Gaussian points are quickly sorted, and the color of each pixel is calculated by mixing N ordered points overlapping the pixel. , where m and n are the indices of points that overlap a pixel, For each point's color, is the opacity.

3. The method for generating a dynamic water body based on 3D Gaussian splash according to claim 1, characterized in that: The second step comprises: The opacity threshold σ is customized to determine whether the particle exists inside or outside the water body. When the ray enters a high transparency grid from the bottom transparency grid, an intersection point is set; After all the intersection points are determined, the internal filling area is obtained, and then the internal particles are filled. The color and opacity of these particles ensure that the color of the particles inside and outside the static water body is consistent; The constructed three-dimensional water body model is filled with particles to obtain a static three-dimensional water body model for dynamicization.

4. The method for generating dynamic water body based on 3D Gaussian splash according to claim 1, characterized in that: The step three comprises: Search the particles globally and calculate the distance between two particles. The density of particle i is obtained by summing: , and the pressure is calculated based on the incompressible condition as , where m j represents the mass of all particles j around particle i, W ij represents the smooth kernel function, k is the fluid elastic modulus, is the reference density; The pressure is calculated by taking the gradient of the pressure and finding , applying the Laplace operator to the velocity calculates the viscosity as , and gravity is , where m i is the mass of particle i, is the pressure gradient, g is the gravitational acceleration; Then calculate the acceleration of all particles i through the force, and calculate the velocity and position of the next time step through the acceleration; The calculated velocity and position are embedded in the early static three-dimensional water model, and the Gaussian kernel is regarded as a discrete particle cloud to discretize the continuum in space. The particles are subjected to a first-order approximate local affine transformation. For static scenes, The three-dimensional Gaussian distribution of To dynamize the scene, where X p is the location of the Gaussian point, A p is the shape, σ p is the opacity, C p is the color information of the point, X p (t) is the position of the Gaussian point over time, A p (t) is the shape changing with time; The extended Gaussian distribution is , the calculated particle position and velocity are assigned to the Gaussian kernel, so that the particles move along the trajectory of the water body, while ensuring that the spherical harmonic coefficient and opacity of each particle remain unchanged over time, thereby ensuring that the color and transparency of the water body remain consistent during the movement, V p (t) is the velocity of water particles changing with time.

5. A dynamic water body generation system based on 3D Gaussian splash, characterized by: The system includes a model generation module, a particle filling module and a dynamic water body generation module; The model generation module is used to generate a three-dimensional water model through 3D splashing according to the water body pictures of multiple views; The particle filling module is used to determine the internal filling area through the concept of intersection, fill the interior of the three-dimensional water model with particles, and obtain a three-dimensional water model for dynamicization; The dynamic water body generation module is used to solve the partial differential equations describing the fluid motion through the smooth particle fluid dynamics method, physically integrate the three-dimensional water body model, reconstruct the dynamic model and obtain the water body flow effect.

6. A dynamic water body generation system based on 3D Gaussian splash according to claim 5, characterized in that: The model generation module specifically includes the following contents: Extract image feature points, match feature points at different viewing angles, and estimate the camera’s intrinsic and extrinsic parameters to construct a sparse point cloud of the scene; A set of 3D Gaussian distributions are created based on these sparse point clouds. Each Gaussian distribution contains a center position x, a covariance matrix Σ, and a transparency α. The covariance matrix Σ describes the shape and direction of the Gaussian point in three-dimensional space. It is obtained by multiplying a rotation matrix R and a scaling matrix S, expressed as , T represents transpose; Given a transformation matrix W and the Jacobian matrix J of the affine transformation, the covariance matrix in the camera coordinates is ; Using the GPU-based tiled rasterization renderer, the Gaussian points are quickly sorted, and the color of each pixel is calculated by mixing N ordered points overlapping the pixel. , where m and n are the indices of points that overlap a pixel, For each point's color, is the opacity.

7. The dynamic water body generation system based on 3D Gaussian splash according to claim 5, characterized in that: The model generation module specifically includes the following contents: The opacity threshold σ is customized to determine whether the particle exists inside or outside the water body. When the ray enters a high transparency grid from the bottom transparency grid, an intersection point is set; After all the intersection points are determined, the internal filling area is obtained, and then the internal particles are filled. The color and opacity of these particles ensure that the color of the particles inside and outside the static water body is consistent; The constructed three-dimensional water body model is filled with particles to obtain a static three-dimensional water body model for dynamicization.

8. The dynamic water body generation system based on 3D Gaussian splash according to claim 5, characterized in that: The dynamic water body generation module specifically includes the following contents: Search the particles globally and calculate the distance between two particles. The density of particle i is obtained by summing: , and the pressure is calculated based on the incompressible condition as , where m j represents the mass of all particles j around particle i, W ij represents the smooth kernel function, k is the fluid elastic modulus, is the reference density; The pressure is calculated by taking the gradient of the pressure and finding , applying the Laplace operator to the velocity to calculate the viscosity is , and gravity is , where m i is the mass of particle i, is the pressure gradient, g is the gravitational acceleration; Then calculate the acceleration of all particles i through the force, and calculate the velocity and position of the next time step through the acceleration; The calculated velocity and position are embedded in the early static three-dimensional water model, and the Gaussian kernel is regarded as a discrete particle cloud to discretize the continuum in space. The particles are subjected to a first-order approximate local affine transformation. For static scenes, The three-dimensional Gaussian distribution of To dynamize the scene, where X p is the location of the Gaussian point, A p is the shape, σ p is the opacity, C p is the color information of the point, X p (t) is the position of the Gaussian point over time, A p (t) is the shape changing with time; The extended Gaussian distribution is , the calculated particle position and velocity are assigned to the Gaussian kernel, so that the particles move along the trajectory of the water body, while ensuring that the spherical harmonic coefficient and opacity of each particle remain unchanged over time, thereby ensuring that the color and transparency of the water body remain consistent during the movement, V p (t) is the velocity of water particles changing with time.

Images