The two-dimensional wave equation is a well-known partial differential equation (PDE) that describes wave propagation in continuous media. In a homogeneous, bounded domain Ω⊂R2 with absorbing boundaries and obstacles, the classical form appears as: (2)∂2u(x,y,t)∂t2=c2(∂2u(x,y,t)∂x2+∂2u(x,y,t)∂y2)where u(x,y,t) is the wave function representing the amplitude of the wave at position (x,y) and time t, and c denotes the constant wave speed. Typical boundary conditions enforce u=0 on the domain periphery and on any obstacle surfaces, simulating perfect absorption. An initial disturbance at (x0,y0) triggers wave that expand outward until the entire reachable region is covered.

Assuming the environment is bounded within x∈[xa,xb] and y∈[ya,yb], Dirichlet boundary conditions are applied to simulate wave absorption at the boundaries and obstacles: (3)u(xa,y,t)=u(xb,y,t)=0 (4)u(x,ya,t)=u(x,yb,t)=0

At t=0 the robot injects a unit impulse at its start location (x0,y0) (5)u(x,y,t)=U0δ(x−x0)δ(y−y0),u.(x,y,0)

Assume u(x,y,t)=X(x)Y(y)T(t). Substituting into (2) and dividing gives: (6)1c2T′′T=X′′X+Y′′Y=−k2

Introducing another constant −kx2: (7)T′′+c2k2T=0;X′′+kx2X=0;Y′′+ky2Y=0;kx2+ky2=k2

kx=nπxb−xa,ky=nπyb−ya,n,m∈N+

For path planning, an analytical solution of the above PDE provides the temporal evolution of wave within Ω. The final expression for u(x,y,t) can be compactly written in the form: (8)u(x,y,t)=Hcos⁡(ckt)cos⁡(kxx)cos⁡(kyy)where kxand ky Ware integer-based wave numbers determined by the boundary lengths in x and y, and H encapsulates the initial amplitude U0.

Obstacles are inserted as additional boundaries, effectively zeroing out wave amplitude in those regions. Arrival times T(x,y) at which |u(x,y,t)| first surpasses a threshold δ then serve as the basis for constructing a global cost map—each location is labeled by its earliest wavefront encounter.

When implementing numerical methods to solve the wave equation, it is essential to ensure numerical stability. The Courant-Friedrichs-Lewy (CFL) condition provides a criterion for stability in finite difference methods: (9)cΔtΔx≤12where Δt is the time step and Δx is the spatial discretization step. Satisfying this condition prevents non-physical oscillations and ensures accurate numerical results.

The solution in (2) provides a continuous depiction of how the wave expands from (x0,y0). At each location (x,y) the wave amplitude u(x,y,t) eventually surpasses a chosen threshold δ, indicating that point has been “reached” by the wavefront. The arrival time T(x,y) is defined as the earliest t≥0 for |u(x,y,t)|≥δ formally: (10)T(x,y)=min{t≥0|u(x,y,t)|≥δ}

This mapping captures global accessibility: areas near (x0,y0) exhibit small T while cells separated by obstacles or long corridors have significantly larger values.

Fig. 3a illustrates how the total area “covered” by the propagating wave evolves over continuous time according to the analytical two-dimensional wave solution. At t=0, the wave originates at the source location, and initially, only a small region around this source exhibits sufficient amplitude. As time increases, the wave expands outward in a continuous manner, surpassing a chosen amplitude threshold δ across successively larger portions of the domain. It initially rises slowly, reflecting the local nature of early propagation, then accelerates as more corridors become accessible. Eventually, the coverage begins to plateau, indicating that the wave has fully enveloped the free areas of the environment, while obstacles continue to act as absorbing boundaries.

Fig. 3b offers a complementary arrival-time heatmap, again derived from the closed-form wave function. For each point (x,y) within the environment, the earliest time T(x,y) at which the wave amplitude |u(x,y,t)| crosses the threshold δ is recorded. Warmer colors (reds, oranges) correspond to shorter arrival times and thus easier accessibility from the wave source, whereas cooler colors (greens, blues) indicate cells that are reached later due to either greater distances or more complex obstacle placements. The white regions represent obstacles themselves, where wave energy is extinguished by boundary conditions, ensuring that no spurious paths are formed through invalid areas. These smoothly varying contours show natural bending of the wavefront around obstacles, demonstrating how the analytical wave solution inherently encodes corridor effects and global coverage without reliance on incremental or localized expansions.

This time map provides crucial information about the earliest time the influence from the starting point reaches any given location, which is instrumental in the subsequent path planning stages.

Once T(x,y) established throughout Ω the final path is extracted by traversing from the goal (xg,yg) back to (x0,y0) along cells with strictly decreasing arrival times. A generic neighbor (x′,y′) is deemed valid if: (11)T(x′,y′)<T(x,y)

Ensuring that the path moves “upstream” toward the wave source. However, it is often beneficial to incorporate a cost function that penalizes obstacle proximity, ensuring safer navigation. Let D(x,y) denote the distance from (x,y) its nearest obstacle. A combined cost is then defined as: (12)C(x,y)=γT(x,y)+α[1D(x,y)+ε]where: γ is a weighting factor that adjusts the influence of the arrival time on the cost. α is a weighting factor that adjusts the influence of the obstacle proximity on the cost. ε is a small positive constant to prevent division by zero when D(x,y) approach zero.

During backtracking, each cell’s neighbor set may include 8 adjacent positions. The path candidate (x′,y′) that yields the smallest C(x′,y′) subject to T(x′,y′)<T(x,y) is chosen at each iteration, until eventually (x0,y0) is reached. This process naturally forms a smooth, single-pass route that avoids corners and obstacles, reflecting the wave’s own continuous expansion.

WWS requires a few critical parameters. The wave speed c is chosen as the largest value that still satisfies the Courant–Friedrichs–Lewy stability condition, Its role is to control how quickly the wavefront expands across the grid, ensuring that each cell’s arrival time is distinguishable while adhering to stability constraints [28,29]. The arrival threshold δ defines when a cell is deemed “reached”, balancing early detection against coverage gaps. During backtracking, cost weights αand γ control the trade-off between obstacle avoidance and quick arrival, useful for tuning path safety in narrow passages. Finally, an optional corridor restriction confines wave evaluation to a subregion, reducing computation in large or cluttered maps.

A summary of these main parameters and constraints appears in Table 2. Full derivations are readily found in standard references on wave phenomena, so this section only highlights key points relevant to the wave-based laser simulator proposed in this work.

In this work, the parameters—wave speed c, threshold δ, and cost-function weights α, γ—are maintained as fixed values across all experiments, both indoors and outdoors. This non-adaptive choice was guided by preliminary trials indicating that a single parameter set could provide robust performance in a variety of environments. While this approach does not automatically adjust to factors such as map scale or obstacle density, the global nature of wave expansions in WWS enables it to accommodate moderate variations without specialized tuning. We emphasize that the current study aims to demonstrate the general feasibility and effectiveness of WWS under a single, empirically chosen configuration. Nonetheless, we acknowledge that certain advanced applications—especially those with highly dynamic or drastically varying obstacle layouts—may benefit from online parameter adaptation, a direction we outline in the Conclusion.

Algorithm 1 is the pseudo-code illustrating the key steps of the WWS approach under an analytical wavefront framework. It begins by defining the spatial domain and obstacle/boundary conditions, then computes the arrival-time T(x,y) for each grid cell based on the analytical wave equation solution. Once the global arrival-time map is established, a reverse-search from the goal to the start identifies the collision-free path that minimizes both travel time and obstacle proximity:

Comparison with FMM and Eikonal-Based Methods

While the wave equation underpins our Wave Water Simulator (WWS), many path planning techniques rely on the Eikonal equation, ∥∇T∥=f(x) and its fast numerical solver, the Fast-Marching Method (FMM). In FMM, one typically solves for the arrival-time T(x) of a wave or front propagating at speed f(x) This yields a globally optimal path in a continuous domain by advancing a boundary front outward. However, purely first-order PDE solutions sometimes manifest discretization artifacts or require additional boundary/obstacle labeling to ensure correct collision handling.

By contrast, WWS leverages the second-order wave equation: (13)∂2u(x,y,t)∂t2=c2(∂2u(x,y,t)∂x2+∂2u(x,y,t)∂y2)

Together with Dirichlet boundary conditions u=0 on obstacles and domain edges. This physically motivated approach encodes obstacles as fully absorbing, preventing amplitude propagation into invalid regions without explicitly labeling them. Furthermore, the oscillatory nature of wave solutions promotes smoother curvature around corners; as the wave amplitude transitions from obstacle surfaces to free space, it avoids the “grid-sharp” expansions that sometimes arise with first-order PDE methods. Table 3 summarizes the conceptual contrasts between WWS and Eikonal-based planners, highlighting how wave-based fluctuations, direct boundary encoding, and corridor constraints distinguish WWS from FMM-like approaches.

Wave Water Simulator (WWS) adopts wave propagation akin to PDE-based planners such as FMM, it does not strictly solve the stationary Eikonal equation ∥∇T∥=f(x). Instead, WWS introduces a second-order wave equation with momentum and transient behaviors. Hence, WWS does not guarantee an exact minimal-distance path in all cases. Nevertheless, if the corridor restriction is sufficiently broad, the wave can explore the entire free space, yielding a trajectory that is empirically close to the global optimum, in narrower corridors the solution remains within 1%–2% of the global optimum in our benchmarks (in Section 3), confirming practical near-optimality.

While the Wave Water Simulator (WWS) also propagates a wavefront, it employs a binary-heap scheduler that always processes the cell with the smallest arrival time. Let Ω contain N free cells. Every cell enters the heap at most once and up to k (≤8) neighbors are relaxed, giving: (14)Time(WWS)=O(Nlog⁡N),Space(WWS)=O(N)

If the optional corridor restriction reduces the working set from N to R cells (R ≪ N in cluttered. maps), the bound becomes O(Rlog⁡R). Back-tracking from goal to start is linear in path length L and does not alter the dominant term.

To demonstrate the core principles of WWS, we first consider a simple 2D domain containing a single rectangular obstacle (Fig. 4). The start point (green) is placed near the lower-left corner, with the goal (red) in the upper-right region. Initially, no precomputed roadmap or heuristic is used—WWS merely propagates a wavefront from the start. As the wave expands, arrival-time contours naturally bend around the obstacle, revealing both safe corridors and areas of higher traversal cost. This obstacle interaction emerges directly from the boundary conditions, which without additional collision checks.

Once the wavefront has covered the navigable domain, a reverse search from the goal follows cells with strictly decreasing arrival times, culminating in a collision-free path that blends efficiency and smoothness. As shown in Fig. 4, the final trajectory deftly avoids “staircase” artifacts and abrupt turns, underscoring how WWS’s global wave equation framework inherently encodes obstacle avoidance and ensures gentle curvature.

To further test WWS beyond a single obstacle, we examined a cluttered 2D domain with multiple static obstacles of various shapes (Fig. 5). These linear barriers and circular impediments form narrow corridors, semi-enclosed passageways, and partial maze structures. Unlike local or heuristic planners susceptible to dead ends in such intricate layouts, WWS starts by globally propagating a wavefront from the start. As the wavefront evolves, arrival-time contours naturally reveal corridors and bottlenecks, offering continuous, domain-wide feedback on accessibility without incremental trial-and-error.

As wave propagation proceeds, WWS constructs a comprehensive arrival-time gradient, “lighting up” optimal routes while circumventing complex obstacles. Once all traversable cells are assigned precise times, the algorithm executes a backward traversal from the goal, following strictly decreasing arrival times. This single global expansion elegantly weaves through tight corridors and avoids local minima, eliminating the need for repeated re-planning or parametric tuning. Together with the simpler example, this scenario demonstrates WWS’s aptitude for both small-scale and maze-like environments, confirming its flexibility, global coverage, and capacity to generate naturally optimized paths.

These two sets of demonstrations—one in a simple environment and another in a considerably more complex map—collectively validate the theoretical and practical foundations of WWS. From a single obstacle to a pseudo-maze, WWS Adapts Gracefully to Complexity: The wave-based approach seamlessly scales from trivial setups to complicated geometries, without losing its capacity to discern global navigational structures. Maintains a Global Perspective: Unlike local-only methods prone to becoming stuck in dead-ends or complex corridors, WWS leverages a global arrival-time map that inherently encodes the relative difficulty of reaching any location in the domain. Produces Naturally Optimal Routes: By following decreasing arrival times from the goal to the start, WWS intrinsically finds paths that minimize travel time and respect obstacle constraints, without manually designed heuristics or extensive parameter tuning.

This section demonstrates the navigation performance of the WWS algorithm in different environments, focusing on its path smoothness, obstacle avoidance capability, and adaptability to maps of varying complexity. Fig. 6 illustrates the optimal paths planned by WWS in several environments with different structures, where the start and end points are marked in green and red, respectively.

WWS exhibits strong adaptability across various environments, ensuring smooth navigation and effective obstacle avoidance. In structured obstacle environments, the algorithm accurately computes optimal paths that allow the robot to traverse narrow passages without abrupt directional changes. In maze-like environments with high obstacle density, WWS effectively avoids local optima and redundant routes while maintaining trajectory continuity. In semi-closed spaces where only specific corridors are available, WWS dynamically adjusts its path to ensure a safe distance from walls and barriers. For randomly distributed obstacles, the algorithm efficiently adapts to varying layouts, utilizing global wave propagation to avoid dead ends. In open environments, WWS generates direct and efficient paths, taking advantage of the available space for rapid goal achievement.

The simulation results confirm that WWS consistently maintains smooth trajectories and robust obstacle avoidance in diverse scenarios. The continuous wave propagation characteristic ensures that paths are both natural and efficient, enabling fluid robotic motion without excessive detours or sudden angular deviations. The algorithm’s ability to adapt to different spatial configurations highlights its potential for real-world navigation applications.

To provide a more comprehensive understanding of how the Wave Water Simulator (WWS) algorithm behaves under different parameter settings. This investigation focuses on the four key parameters, α: obstacle proximity weight in the cost function. γ: arrival-time weight in the cost function. δ: wave detection threshold. c: wave propagation speed.

We selected three representative environments to encompass varying obstacle densities and geometric complexities, environment B, C. For each environment, the parameters are set as: α∈{0.5,1.0,2.0},γ∈{0.5,1.0,2.0},δ∈{0.05,0.1,0.2},c∈{1.5,3,5}.

Table 4 summarizes the aggregated outcomes in Environment B and Environment C for varying α, δ, c and γ. Increasing α beyond 1.0 modestly reduces path length in heavily obstructed scenes (due to stricter obstacle penalties) but can raise computation time by 5%–15%. Higher γ more strongly penalizes large arrival times, causing slightly shorter but more angular routes in some configurations. Lower thresholds δ improve detection of wave arrivals, potentially beneficial for narrow corridors. However, the algorithm sometimes requires additional updates, raising computation time by 10%–20%. Typically decreases arrival-time gradients, slightly reducing path length but having marginal effect on success rate. Overall α = 1, γ = 1, c = 3, δ = 0.1 ensure that WWS remains robust in all tested environments.

WWS was implemented in four distinct outdoor scenarios, designated A, B, C, and D, each characterized by varying degrees of obstacle density, road curvature, and the presence of natural or man-made structures. As shown in Fig. 8, Environment A is characterized by relative openness with sparse obstructions, while scenarios B and C introduce increasingly cluttered layouts, such as denser vegetation or narrow passageways. Environment D combines open regions with corridor-like segments, reflecting typical road networks bordered by trees or low walls. In each case, raw camera images are processed briefly (e.g., perspective alignment, threshold segmentation) to yield binary occupancy maps, where black cells represent obstructed zones and white cells are navigable.

To plan a route from the start to the goal, WWS computes a wave propagation heatmap indicating how quickly each free cell is reached by the wave. The color-banded arrival-time displays reveal that warm regions (reds, oranges) denote rapid wave arrival, whereas cooler hues (greens, blues) appear in more distant or indirectly accessible areas, with obstacles remaining non-traversable (shaded or black). This heatmap visually highlights how the wave solution systematically accounts for all potential pathways around obstacles, bending around occlusions in a smooth gradient. For instance, in Environment B, the wavefront contours demonstrate gentle detours around clusters of obstacles, whereas in a tight corridor setup (e.g., C or D), the contour lines compress, revealing complex geometry. These arrival-time gradients guide the final path extraction, ensuring every feasible corridor is explored in a unified framework.

The resulting WWS paths across Environments A–D have been shown to exhibit smooth, direct navigation, even in cluttered terrain such as Environment C, where WWS avoids abrupt turns by leveraging the continuous time gradient, resulting in a path that gracefully curves around obstacles instead of making sharp angular deviations. Likewise, in corridor-like areas (Environment D), WWS blends open-area traversal with narrow corridor following, producing a coherent trajectory well suited for real vehicle motion. In comparison to earlier laser-simulator (LS) approaches, which can yield longer or more conservative routes, WWS’s wave-based formulation naturally balances obstacle clearance with minimal travel distance, achieving shorter, more efficient paths in these outdoor tests while maintaining robust collision avoidance. Compared to earlier laser-simulator approaches (LS, GLS), which can produce longer or more conservative paths due to partial expansions, WWS’s wave-based methodology systematically balances obstacle avoidance and travel distance. Although WWS’s coverage of the entire start–goal corridor can lead to higher computational complexity than LS or GLS, it remains more efficient than A*’s exhaustive global search in many cluttered or large-scale outdoor scenes.

As shown in Fig. 9, The experiments were conducted in three separate indoor laboratory settings, denoted Environments A, B, and C. Each environment featured a flat floor but varied in terms of the arrangement of obstacles. Most notably, a centrally placed tire and wooden barriers were present in Environments A and B, forcing the route to deviate. The first image in each series shows the original indoor scene, revealing how obstacles restrict the corridor or block direct travel. After this, a minimal image-processing step was applied to the scene, converting it into a binary occupancy map. This map is represented in the second image of the series, where black cells represent obstacles, and white cells denote free space.

Following the formation of the occupancy map, the WWS algorithm calculates an arrival-time heatmap (third image), illustrating the propagation of a wave from the robot’s initial position (or goal) and its subsequent trajectory around obstacles. Each colored band in the heatmap denotes the time (or cost) required to reach that specific region. In instances where obstacles impede direct lines of travel, the wavefront curves around the obstructions, resulting in elevated arrival-time zones behind the occluding objects. This process ensures that WWS exhaustively accounts for every navigable corridor, even narrow passages. In the fourth image, the final WWS path is obtained by tracing a sequence of cells with strictly decreasing arrival times, producing a smooth, arc-like trajectory around the central tire or wooden barriers without abrupt angular turns.

In practice, WWS’s continuous wave-propagation paradigm fosters both comprehensive route exploration and high path smoothness, even in tight corridors. By expanding over the entire corridor region connecting start and goal, WWS can thoroughly map each free cell’s accessibility. This global perspective is akin to A* in that it scans a significant portion of the grid; however, WWS restricts its search to a corridor around the start–goal line, preventing a full-map scan. Consequently, WWS tends to exhibit greater computational cost than purely local methods (LS, GLS), which only sample or locally expand around the current robot location. Meanwhile, A* often has the highest overall complexity due to an exhaustive search across the entire occupancy map.

Overall, the indoor results confirm that WWS delivers high-quality navigation—a global coverage ensures no corridor is missed, while the wave-based approach gives smoother turns. Although WWS’s corridor-wide scanning increases computational burden compared to LS and GLS, it typically remains more efficient than A* in cluttered or large testbeds. Meanwhile, methods like LS, GLS, and A* are prone to “Z”-shaped or jagged trajectories, especially in obstacle-rich domains, unless extensive smoothing is applied post hoc. Hence, these findings underscore that WWS, despite its somewhat larger wave expansion, achieves shorter and smoother paths with practical computational requirements—an appealing combination for real-world robotic navigation in laboratory-scale indoor environments.