Question

Explore a function graphically. The wave speed \(c\) of water surface waves depends on the length \(\lambda\) of the waves. The following formula relates \(c\) to \(\lambda\) : $$ c(\lambda)=\sqrt{\frac{g \lambda}{2 \pi}\left(1+s \frac{4 \pi^{2}}{\rho g \lambda^{2}}\right) \tanh \left(\frac{2 \pi h}{\lambda}\right)} $$ where \(g\) is the acceleration of gravity, \(s\) is the air-water surface tension \(\left(7.9 \cdot 10^{-4} \mathrm{~N} / \mathrm{cm}\right), \rho\) is the density of water (can be taken as \(\left.1 \mathrm{~kg} / \mathrm{cm}^{3}\right)\), and \(h\) is the water depth. Let us fix \(h\) at \(50 \mathrm{~m}\). First make a plot of \(c(\lambda)\) for small \(\lambda(1 \mathrm{~mm}\) to \(10 \mathrm{~cm})\). Then make a plot \(c(\lambda)\) for larger \(\lambda\) ) (1 m to \(2 \mathrm{~km}\) ). Name of program file: water_wave_velocity.py.

Short answer

Create a Python script to plot wave speed for small and large wave lengths, utilizing constants and provided formula. Use matplotlib for visualization.

Step-by-step solution

Set Up the Environment

Before solving the exercise, ensure you have a Python environment ready to run the specified code. You will need packages such as matplotlib for plotting, and possibly numpy for handling calculations, which can be installed via pip if not already available.

Understand the Function

The given formula is for calculating wave speed, \( c(\lambda) \), based on wave length, \( \lambda \). Each parameter has a defined constant value or relationship to the others, with notable parameters: gravitational acceleration \( g \), surface tension \( s \), water density \( \rho \), and water depth \( h \).

Define Constants and Variables

In the script water_wave_velocity.py, define the constants like \( g = 9.81 \text{ m/s}^2 \), \( s = 7.9 \times 10^{-4} \text{ N/cm} \), \( \rho = 1 \text{ kg/cm}^3 \) converted appropriately to be consistent with \( g \) in m/s, and \( h = 50 \text{ m} \). Set \( \lambda \) to range according to the two required plotting scales.

Create the Function

Define the function \( c(\lambda) \) as described, utilizing numpy operations for element-wise calculations over the range of \( \lambda \) values. Use \( \tanh \) from numpy for the hyperbolic tangent, ensuring all units are consistent throughout the calculations.

Plot for Small Lambda Values

Using matplotlib, plot \( c(\lambda) \) for \( \lambda \) ranging from 1 mm (0.001 m) to 10 cm (0.1 m). Label your axes, and include a title indicating this is the plot for small \( \lambda \). Use logarithmic scale if necessary for better visualization.

Plot for Large Lambda Values

Similarly, plot \( c(\lambda) \) for \( \lambda \) ranging from 1 m to 2 km (2000 m), again labeling axes and titling the plot for larger \( \lambda \). This ensures the examining of function behavior over large scales.

Save and Test the Script

Save the Python script as water_wave_velocity.py, then run it to check both plots. Verify the output is as expected and adjust scaling or calculations if discrepancies are found in comparison to expected wave velocities.

Key concepts

Function Plotting

In scientific programming with Python, function plotting is a straightforward yet powerful way to visualize mathematical relationships. By plotting the wave speed function, \( c(\lambda) \), you gain insight into how wave speed changes with varying wave length, \( \lambda \).

Utilizing libraries like `matplotlib`, you can create visual representations of functions with ease.

• Start by defining the wave speed function in Python using `numpy` for managing calculations, a key procedure when working with arrays of \( \lambda \) values.
• You can choose to visualize different ranges of \( \lambda \) to see how the graph transitions from small to large wave lengths.
• Functions in Python allow you to dynamically adjust parameters and see immediate visual feedback, helping to reinforce the underlying mathematical principles.

Doing so not only helps in understanding the functions better but also in communicating your findings effectively. Proper labels and titles make graphs more informative, catering to a wider audience.

Numerical Analysis

Numerical analysis is integral in scientific programming, especially when approximating solutions to complex problems. The given formula for wave speed, \( c(\lambda) \), involves variables that can be challenging to compute analytically.

When you utilize numerical methods:

• Libraries like `numpy` enable efficient computation with arrays, allowing you to perform element-wise operations over a range of \( \lambda \) values.
• It is crucial that all units are consistent, particularly when involving constants like gravity \( g \), surface tension \( s \), and density \( \rho \), which can be in differing units.

Numerical analysis helps us approximate the function \( c(\lambda) \) for various conditions, providing practical solutions where analytical methods may be cumbersome or infeasible. This approach is essential in a field where exact solutions are not always easily attainable.

Wave Speed Calculation

Wave speed calculation involves understanding and applying the specific formula provided: \[ c(\lambda)=\sqrt{\frac{g \lambda}{2 \pi}\left(1+s \frac{4 \pi^{2}}{\rho g \lambda^{2}}\right) \tanh \left(\frac{2 \pi h}{\lambda}\right)} \] This equation relates the wave speed \( c \) with the wavelength \( \lambda \), capturing the effects of surface tension and water depth.

Key components that affect wave speed include:

• Gravitational acceleration \( g \) which drives the natural propagation of waves.
• Surface tension \( s \), impacting stability and motion between air and water interfaces.
• Water depth \( h \), influencing how deep-lying wave forces are transmitted.
• Water density \( \rho \) contributes to mass distribution in the medium.

Proper understanding of these parameters helps in engineering applications such as harbor design, ship construction, and understanding wave patterns for environmental assessments.

Python Environment Setup

Setting up your Python environment effectively is a crucial first step in running any scientific programming task. For plotting the wave speed function, make sure your environment is equipped with the necessary packages:

• `matplotlib` is a must for plotting, offering capabilities to create detailed figures and graphs.
• `numpy` will be essential for managing numerical computations, especially when dealing with arrays and trigonometric or hyperbolic operations like `tanh`.
• Ensure your Python environment is consistent in terms of unit measurement since conversions are frequently necessary in scientific contexts.

Use the package manager `pip` to install any missing libraries. By setting up your environment correctly, you pave the way for efficient coding practices and focus more on analyzing results rather than managing technical setbacks.