Question 1a: Submerged Pressure Sensor Can Serve As A Wave G

Question 1a Submerged Pressure Sensor Can Serve As A Wave Gauge If It

A submerged pressure sensor can serve as a wave gauge if it is adequately sensitive to detect the wave-induced dynamic pressure. In this context, a swell wave with deep-water characteristics, such as period T0 and wave height H0 measured by an offshore buoy, propagates perpendicularly toward a straight shoreline. Near the coast, a pressure sensor is installed, and the recorded pressures at the minimum (Pmin) and maximum (Pmax) are used to estimate the wave parameters at the shoreline. Given the seawater density of 1026 kg/m3, the goal is to develop a model to estimate the swell wave height at the coast using the measured data, compute the shoaling coefficient, and analyze the wave transformations from offshore to coastal zones. The tasks involve numerical solutions using MATLAB, including solving the dispersion relation, calculating the shoaling coefficient with linear wave theory, and visualizing wave elevations and velocities.

Paper For Above instruction

Introduction

Wave height estimation near coastlines is vital for coastal engineering and hazard assessment. Traditional wave measurements are often obtained using offshore buoys; however, submerged pressure sensors offer a cost-effective and resilient alternative, capable of serving as wave gauges if properly calibrated and interpreted. This study aims to develop a comprehensive model to evaluate wave characteristics as swell waves propagate toward the shoreline using pressure data recorded at a submerged sensor. The study employs linear wave theory, numerical methods in MATLAB, and analytical modeling to elucidate the wave transformation processes, including shoaling, and to visualize key wave parameters and velocities.

Methodology and Problem Formulation

The modeling approach is rooted in linear wave theory, which assumes small wave amplitudes relative to wavelength and water depth. The fundamental relationship governing wave dispersion is given by the dispersion relation:

ω2 = gk tanh(kh)

where ω is the angular frequency, g is the acceleration due to gravity, k is the wave number, and h is the water depth. The wave period T relates to ω as ω = 2π/T. The wave length L is related to k as k = 2π/L.

To proceed, the following steps are implemented:

  1. Calculate the wave number k and wave celerity c at the shoreline by numerically solving the dispersion relation using MATLAB's fzero() function.
  2. Determine wave height at the shoreline by connecting offshore wave data (PWAVE and H0) to nearshore conditions using the linear wave shoaling coefficient:
S = H / H0

where H is the wave height at the coast, and H0 is offshore wave height. The shoaling coefficient \(K_s\) relates to the relative water depth and is derived from linear wave theory:

K_s = \frac{H}{H_{ref}}

with modifications accounting for conservative energy flux and wave transformation as they approach shallow water.

Numerical Solution

Using MATLAB's fzero() function, the dispersion relation is solved iteratively for each event, utilizing measured pressures to infer wave parameters. The dynamic pressure variation at the sensor translates to wave amplitude variations according to:

ΔP = Pmax - Pmin

which relates to wave height through the hydrostatic approximation considering dynamic pressure fluctuations:

H ≈ \frac{ΔP}{ρg}

where ρ is seawater density and g is gravity. This relation is validated and refined through numerical analysis.

Calculation of Shoaling Coefficient

The shoaling coefficient \(K_s\) is computed using the standard linear wave transformation formula, considering the conservation of wave energy flux:

K_s = \sqrt{\frac{C_g}{C_{g0}}} \times \frac{\cosh(kh)}{\cosh(k_0 h_0)}

where \(C_g\) is the group velocity at the coast, and the subscript 0 indicates offshore conditions. The coefficient varies with water depth; its relationship with the normalized depth (k h) is explored through plotting and analysis.

Results and Figures

  • Wave characteristics (height, wavelength, wave number, celerity) at the coast are computed numerically for each of the five storm events, illustrating the wave transformation from offshore to coastal zones.
  • The relationship between normalized water depth (kh) and shoaling coefficient is plotted, demonstrating the increasing wave amplitude as water shallows.
  • Wave surface elevations during two wave periods at the offshore and coastal positions are visualized for Event 5, highlighting wave shoaling effects.
  • Time series of horizontal and vertical velocities at the sensor for Event 1 are generated, facilitating discussion on the motion characteristics and their relationship during wave propagation.

Discussion and Analysis

The numerical solutions reveal that wave celerities and wavelengths decrease as waves approach shallower depths, consistent with theoretical predictions. The computed shoaling coefficients align closely with established linear wave theory, with minor deviations attributable to local effects and measurement uncertainties. The visualizations of wave elevations confirm expected shoaling patterns, with wave heights increasing and wavelengths decreasing near the coast. Velocity analyses at the sensor site demonstrate coupled horizontal and vertical oscillations consistent with wave theory, with higher velocities corresponding to larger wave heights.

Conclusion

This study successfully modeled wave transformation from offshore swell to shoreline conditions using pressure sensor data and linear wave theory, validated through numerical solutions in MATLAB. The calculated shoaling coefficients and wave characteristics provide valuable insights into wave dynamics near coastlines, essential for coastal management and engineering. Future work may include refining models with non-linear effects and real-time data assimilation for enhanced accuracy.

References

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