Question

Carry out the following steps. Let \(L=10\) and \(a=1\) in parts (b) through (d). (a) Find the displacement \(u(x, t)\) for the given initial position \(f(x) .\) (b) Plot \(u(x, t)\) versus \(x\) for \(0 \leq x \leq 10\) and for several values of \(t\) between \(t=0\) and \(t=20\). (c) Plot \(u(x, t)\) versus \(t\) for \(0 \leq t \leq 20\) and for several values of \(x .\) (d) Construct an animation of the solution in time for at least one period. (e) Describe the motion of the string in a few sentences. \(f(x)=\left\\{\begin{array}{ll}{1,} & {L / 2-12)} \\ {0,} & {\text { otherwise }}\end{array}\right.\)

Short answer

Based on the solution and the plots generated, describe the motion of the string: 1. Initially, the string moves following the shape of the initial position function, f(x). This shape resembles a bump centered at the middle of the string. Since it is a wave equation, the bump oscillates up and down over time. 2. The displacement of the string is generally greater around the central point, where the bump was at the initial moment (x=L/2). This is because the energy initially concentrates in the center before being redistributed along the string. 3. As time progresses, the motion of the string may become more organized or chaotic depending on the Fourier coefficients and the type of wave. For this specific problem, the motion may become more organized into a set of standing waves. 4. If any pattern emerges, it might be related to the spatial and temporal frequencies of the Fourier coefficients. The string may exhibit periodic behavior with certain periodicity in both space and time domains. This could lead to repeating patterns of motion along the string and similar displacement profiles at different times.

Step-by-step solution

Find the displacement u(x, t) for the initial position f(x)

To find the displacement u(x, t), we first need to find the Fourier coefficients, \(A_n\) and \(B_n\), for the given initial position function f(x): \(A_n = \frac{1}{L} \int_0^L f(x) \cos \left( n \pi x / L \right) dx\) \(B_n = \frac{1}{L} \int_0^L f(x) \sin \left( n \pi x / L \right) dx\) Since the given function f(x) is not an even function nor an odd function, you will need to compute both the \(A_n\) and \(B_n\) integrals. However, notice that f(x) is nonzero only on the interval \((L/2-1, L/2+1)\). Hence, the integrals will be computed on this interval. \(A_n = \frac{1}{10} \int_{L/2-1}^{L/2+1} \cos \left( n \pi x / 10 \right)dx\) \(B_n = \frac{1}{10} \int_{L/2-1}^{L/2+1} \sin \left( n \pi x / 10 \right) dx\) Compute the integrals above to get the coefficients \(A_n\) and \(B_n\). With these coefficients, you can write the solution to the wave equation in terms of the Fourier series: \(u(x, t) = \sum_{n=1}^{\infty} \left[ A_n \cos \left( \frac{n \pi x}{L} \right) - B_n \sin \left( \frac{n \pi at}{L} \right) \sin \left( \frac{n \pi x}{L} \right)\right].\)

Plot u(x, t) vs. x for several values of t

Using the displacement function u(x, t) obtained in step 1, plot it versus x for \(0\leq x \leq 10\) and several values of \(t\) between \(t=0\) and \(t=20\). You can use a software like MATLAB, Python, or Mathematica to generate the plots.

Plot u(x, t) vs. t for several values of x

Using the displacement function u(x, t) obtained in step 1, plot it versus t for \(0\leq t \leq 20\) and several fixed values of x. You can use the same software as aforementioned to generate these plots.

Construct an animation of the solution in time

Create an animation of the solution u(x, t) in time by plotting u(x, t) for at least one period. This will give visualization of the motion of the string over time. The animation can be generated using software like MATLAB, Python, or Mathematica.

Describe the motion of the string

By analyzing the plots or the animation created, describe the motion of the string. Some aspects to consider while describing the motion are: 1. How does the string move initially? Does it oscillate? 2. Is the displacement greater at particular points on the string? 3. How does the motion change over time? 4. Do any patterns emerge in the motion of the string?

Key concepts

Fourier Coefficients

Understanding the concept of Fourier coefficients is essential in solving boundary value problems involving vibrational modes, heat transfer, and wave propagation. These coefficients represent the amplitudes of individual sine and cosine terms in the Fourier series, which is a way to express a function as an infinite sum of sines and cosines.

In the context of the given problem, the function representing the initial displacement of a string, denoted as f(x), is broken down into its Fourier components. To calculate the coefficients A_n and B_n, one must perform integrals of the product of f(x) and cosine or sine functions over the relevant domain. For cases where the function, f(x), has specific values within a certain interval, the calculation will focus on that interval for improved efficiency and accuracy.

These coefficients are fundamental as they directly influence the shape and behavior of the vibrational modes represented by the series. They essentially capture the weight of each harmonic component in the overall motion of the string.

Wave Equation Solution

The wave equation is a second-order linear partial differential equation that describes the propagation of various types of waves, such as sound waves, light waves, and water waves, as well as the vibration of strings and membranes. Solving the wave equation typically involves finding a function u(x, t) that represents the displacement of points on the medium over time.

For a string fixed at both ends, the solution can be expressed using the Fourier series, which combines the relevant sine and cosine terms along with the calculated Fourier coefficients. This approach allows us to model the initial conditions of the problem and predict the subsequent motion of the string. A key advantage of using the Fourier series is its ability to handle complex initial conditions and produce an exact solution that can be analyzed for any point on the string and at any moment in time.

Partial Differential Equations

Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. The wave equation is one of the most studied PDEs, especially within the field of engineering, physics, and applied mathematics because it accurately models the behavior of waves.

The complexity of PDEs often necessitates the use of approximation methods for solutions. In our problem, the Fourier series provides a method for expressing the solution to a PDE in terms of an infinite sum of sine and cosine functions. This technique is particularly useful for solving boundary value problems, such as the vibration of a string with fixed ends, because the series inherently satisfies the physical constraints imposed upon the system at the boundaries.

MATLAB or Python Plotting

The visual representation of data is a powerful tool for understanding complex equations and their solutions. To graph the wave equation solution, software like MATLAB or Python can be utilized.

By plotting the function u(x, t) against x or t with fixed parameters, one can observe the characteristics of the wave over time or at specific points in space. In computational environments like MATLAB or Python, users can employ built-in functions and libraries, such as NumPy and Matplotlib in Python, to create plots that are both informative and visually appealing. These plots provide students with an intuitive grasp of phenomena such as wave propagation and string vibration, which might be challenging to conceptualize through equations alone.

String Vibration Animation

Animation is a powerful tool that provides dynamic visualizations that static plots cannot offer. By constructing an animation of the string vibration, we bring the solution of the wave equation to life.

The process involves calculating the displacement u(x, t) over a range of time and then rendering a sequence of frames to visualize how the shape of the string changes with time. This can be accomplished using programming libraries available in MATLAB or Python, which have capabilities for creating animations. Through animation, students can observe the periodic motion and the formation of standing waves, thereby gaining a deeper understanding of the physical process behind the mathematical solution.