Question

Compute velocity and acceleration. In a laboratory experiment waves are generated through the impact of a model slide into a wave tank. (The intention of the experiment is to model a future tsunami event in a fjord, generated by loose rocks that fall into the fjord.) At a certain location, the elevation of the surface, denoted by \(\eta\), is measured at discrete points in time using an ultra-sound wave gauge. The result is a time series of vertical positions of the water surface elevations in meter: \(\eta\left(t_{0}\right), \eta\left(t_{1}\right), \eta\left(t_{2}\right), \ldots, \eta\left(t_{n}\right)\). There are 300 observations per second, meaning that the time difference between to neighboring measurement values \(\eta\left(t_{i}\right)\) and \(\eta\left(t_{i+1}\right)\) is \(h=1 / 300\) second. Write a Python program that accomplishes the following tasks: 1\. Read \(h\) from the command line. 2\. Read the \(\eta\) values in the file src/random/gauge. dat into an array eta. 3\. Plot eta versus the time values. 4\. Compute the velocity \(v\) of the surface by the formula $$ v_{i} \approx \frac{\eta_{i+1}-\eta_{i-1}}{2 h}, \quad i=1, \ldots, n-1 $$ Plot \(v\) versus time values in a separate plot. 5\. Compute the acceleration \(a\) of the surface by the formula $$ a_{i} \approx \frac{\eta_{i+1}-2 \eta_{i}+\eta_{i-1}}{h^{2}}, \quad i=1, \ldots, n-1 $$ Plot \(a\) versus the time values in a separate plot.

Short answer

Plot the elevation, velocity, and acceleration against time using the provided formulas.

Step-by-step solution

Read Inputs

First, read the time difference per sample, \(h\), from the command line. Since there are 300 observations per second, you'll typically have \(h = \frac{1}{300}\). Then, load the wave elevation data from the file `src/random/gauge.dat` into an array called `eta`. Use `numpy` for numerical operations and `matplotlib` for plotting.

Plot Elevation Data

Create an array `time` which represents the time values corresponding to each \(\eta\). This is accomplished by multiplying the index of each \(\eta\) by \(h\). Use `matplotlib` to plot the elevation `eta` against `time`. The plot will provide a visual representation of the water surface elevation over time.

Compute Velocity

Define an array `v` to hold the velocity values. For each \(i\) from 1 to \(n-1\), compute the velocity using the formula: \[v_{i} \approx \frac{\eta_{i+1} - \eta_{i-1}}{2h}\] This formula captures the rate of change of \(\eta\) per unit time, approximating the derivative. Exclude the boundary points \(t_0\) and \(t_n\) since they require data outside the available range. Plot `v` versus `time`.

Compute Acceleration

Define an array `a` to hold the acceleration values. For each \(i\) from 1 to \(n-1\), compute the acceleration using the formula: \[a_{i} \approx \frac{\eta_{i+1} - 2\eta_{i} + \eta_{i-1}}{h^2}\] This formula computes the second derivative of \(\eta\), representing the rate of change of velocity per unit time. Like before, exclude \(t_0\) and \(t_n\) due to boundary limitations. Plot `a` versus `time`.

Key concepts

Python Programming

Understanding Python programming is crucial for tackling the given exercise efficiently. Python is a high-level programming language known for its readability and simplicity, making it ideal for scientific computations. The task involves reading data and performing numerical differentiation to compute wave velocities and accelerations.

Firstly, you need to read the time step, \( h \), from the command line. This will dictate how you handle successive data points for computation. Then, using Python libraries like `numpy` and `matplotlib`, you can import the data from a file and store it in an array named `eta`.

Once you have the data, the next step is to visualize it, which Python excels at with libraries like `matplotlib`. With simple commands, you can plot the elevation against time, which helps in visually analyzing the data before performing further computations.

Python’s rich ecosystem of libraries makes it a powerful tool for scientific computing tasks like this, enabling clear and efficient code to handle complex numerical calculations.

Wave Analysis

Wave analysis in this context involves understanding wave motion through data collected by ultra-sound wave gauges. The primary focus here is on measuring the displacement of the water surface over time.

In the exercise, you are given elevation data points \( \eta(t_0), \eta(t_1), \ldots, \eta(t_n) \), which represent the height of wave surfaces at different time intervals. These measurements form a time series that helps in analyzing how waves propagate through the medium—in this case, a wave tank.

The task is to compute two main parameters of wave motion—velocity and acceleration. Velocity gives an idea of how fast the wave surface is changing its elevation, while acceleration indicates how the velocity of the surface changes over time. This analysis aids in understanding the dynamic behavior of waves, useful for modeling natural phenomena like tsunamis.

Scientific Computing

Scientific computing provides the framework for applying computational techniques to solve scientific problems, such as the numerical differentiation present in this exercise.

The main computation tasks involve estimating the derivatives of the wave height data. Numerical differentiation allows you to calculate the velocity and acceleration without needing an explicit mathematical function, which is common when dealing with discrete data samples.

For velocity, the formula used is:
\[ v_{i} \approx \frac{\eta_{i+1} - \eta_{i-1}}{2h} \]
This approximates the rate of change of elevation data to derive the velocity of the wave particles.

Acceleration is computed using the second derivative approximation:
\[ a_{i} \approx \frac{\eta_{i+1} - 2\eta_{i} + \eta_{i-1}}{h^2} \]
This captures how the velocity of the particles is changing, which gives insights into the force acting on them. By leveraging scientific computing tools, these calculations become more precise and efficient.

Data Visualization

Data visualization is an important step that aids in interpreting the computational results of any scientific experiment. In this exercise, you visualize both the original elevation data and the computed velocity and acceleration.

Using `matplotlib`, visualizing the dataset begins with plotting the wave elevation \( \eta \) against time, which gives an overall impression of the wave profile over time. It shows peaks and valleys, helping observe natural wave behavior directly.

After calculating velocity, a separate plot demonstrates how quickly the water surface is rising or falling over time. This is crucial for understanding the immediate responses to disturbances.

Similarly, plotting acceleration lays out the rate at which the wave speed itself changes. The plots provide visual insights that numbers alone cannot. This two-dimensional representation helps in identifying trends, cycles, and anomalies in the wave behaviors effectively, making data easy to comprehend and analyze.