Question

We saw that the displacement of the plucked string is, on the one hand, $$ u(x, t)=\frac{4 L}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(2 n-1)^{2}} \cos \frac{(2 n-1) \pi a t}{L} \sin \frac{(2 n-1) \pi x}{L}, 0 \leq x \leq L, t \geq 0 $$ and, on the other hand, $$ u(x, \tau)=\left\\{\begin{aligned} x, & 0 \leq x \leq \frac{L}{2}-a \tau, \\ \frac{L}{2}-a \tau, & \frac{L}{2}-a \tau \leq x \leq \frac{L}{2}+a \tau, \\ L-x, & \frac{L}{2}-a \tau \leq x \leq L. \end{aligned}\right. $$ if \(0 \leq \tau \leq L / 2 a\). The first objective of this exercise is to show that (B) can be used to compute \(u(x, t)\) for \(0 \leq x \leq L\) and all \(t>0\) (a) Show that if \(t>0,\) there's a nonnegative integer \(m\) such that either (i) \(t=\frac{m L}{a}+\tau\) or (ii) \(t=\frac{(m+1) L}{a}-\tau\), where \(0 \leq \tau \leq L / 2 a\) (b) Use (A) to show that \(u(x, t)=(-1)^{m} u(x, \tau)\) if (i) holds, while \(u(x, t)=(-1)^{m+1} u(x, \tau)\) if (ii) holds. (c) Perform the following experiment for specific values of \(L\) and \(a\) and various values of \(m\) and \(k:\) Let $$ t_{j}=\frac{L j}{2 k a}, \quad j=0,1, \ldots k $$ thus, \(t_{0}, t_{1}, \ldots, t_{k}\) are equally spaced points in \([0, L / 2 a] .\) For each \(j=0,1,2, \ldots, k,\) graph the \(m\) th partial sum of (A) and \(u\left(x, t_{j}\right)\) computed from (B) on the same axis. Create an animation, as described in the remarks on using technology at the end of the section.

Short answer

#Summary# This problem involves understanding the relationship between two equations that represent the same physical model under certain conditions. We first identified an integer 'm' associated with the time variable 't'. Following that, we analyzed both given equations to find the relationship of 'u(x,t)' and 'u(x,τ)' depending on two conditions. Finally, we performed an experiment for given values of L, a, k, and m, to calculate the corresponding t_j values and plotted u(x,t) from both equations to create an animation to observe their behavior.

Step-by-step solution

Part (a): Find the integer m

To find the integer m, we need to analyze the range of t and see how it corresponds with either of the two given conditions. Given the information about t, we know that both 0≤τ≤L/2a and t>0. Let's split t into two parts: an integer multiple of l/a and its remainder: $$ t=\left(\frac{mL}{a}\right)+\tau $$ or $$ t=\frac{(m+1)L}{a}-\tau $$ where (m=0,1,2,...) and (0≤τ≤L/2a). This representation of t fits both i and ii conditions provided in the problem.

Part (b): Find relation between the two given equations

We have two equations for u(x, t): 1. Equation A: $$u(x, t)=\frac{4 L}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(2 n-1)^{2}} \cos \frac{(2 n-1) \pi a t}{L} \sin \frac{(2 n-1) \pi x}{L}, 0 \leq x \leq L, t \geq 0$$ 2. Equation B: $$u(x, \tau)=\left\\{\begin{aligned} x, & 0 \leq x \leq \frac{L}{2}-a \tau, \\ \frac{L}{2}-a \tau,& \frac{L}{2}-a \tau \leq x \leq \frac{L}{2}+a \tau, \\ L-x, & \frac{L}{2}-a \tau \leq x \leq L. \end{aligned}\right.$$ For condition (i), we need to show \(u(x, t)=(-1)^{m} u(x, \tau)\). Substitute t with the expression given in (i): $$u(x, t)=\frac{4 L}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(2 n-1)^{2}} \cos \frac{(2 n-1) \pi a (mL/a + \tau)}{L} \sin \frac{(2 n-1) \pi x}{L}$$ Notice that after simplification, the cosine term becomes the following: $$ \cos(2\pi m(2n-1) + (2n-1)\pi a \tau /L) = (-1)^m \cos((2n-1)\pi a \tau)/L$$ This implies that for condition (i): $$u(x, t) = (-1)^m u(x, \tau)$$ For condition (ii), we need to show that \(u(x, t)=(-1)^{m+1} u(x, \tau)\). Substitute t with the expression given in (ii): $$u(x, t)=\frac{4 L}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(2 n-1)^{2}} \cos \frac{(2 n-1) \pi a ((m+1)L/a - \tau)}{L} \sin \frac{(2 n-1) \pi x}{L}$$ Notice that after simplification, the cosine term becomes the following: $$ \cos(2\pi (m+1)(2n-1) - (2n-1)\pi a \tau /L) = (-1)^{m+1} \cos((2n-1)\pi a \tau)/L$$ This implies that for condition (ii): $$u(x, t) = (-1)^{m+1} u(x, \tau)$$

Part (c): Perform the experiment

Choose the values for L and a. Let L=2 and a=1. Define the following sequence: $$ t_{j}=\frac{L j}{2 k a}, \quad j=0,1, \ldots k $$ Now, we need to choose an integer value for k and m. Let k=4 and m=2. Calculate the corresponding t_j values: t_0=0, t_1=0.5, t_2=1, t_3=1.5, t_4=2. For each t_j value, compute u(x,t) from equation A using the first few terms and u(x,tau) from equation B and the given results in part (b). Plot the resulting graphs for each t_j. At this point, one can create an animation using a software package (e.g., Python, MATLAB) where u(x,t) from both equations are plotted on the same axis for each t_j. This animation allows us to observe how the two different formula representations change and coincide as the time variable changes.

Key concepts

Boundary Value Problems

Boundary value problems are a crucial concept in mathematics, especially when dealing with differential equations. These problems involve finding a solution to a differential equation that satisfies certain specified conditions at the boundaries of the domain. In the context of the wave equation, this domain could be a spatial interval, such as the length of a string, with conditions defined at its endpoints.

• These boundary conditions could be fixed, meaning the ends of the string do not move.
• Alternatively, they may change over time or be a combination of fixed and moving points.

Boundary value problems are essential because they ensure the solution to a differential equation aligns with the physical phenomena it models, like a vibrating string where displacement at the endpoints is zero. By examining these boundaries, we can predict how the system behaves as a whole.

Harmonic Analysis

Harmonic analysis plays a vital role in understanding wave phenomena, as it decomposes functions or signals into basic sinusoidal components. Each component, or harmonic, represents a unique frequency that collectively reconstructs the original function.

• This is particularly useful for understanding vibrations in physical systems, such as strings or air columns.
• Harmonic analysis helps us study the frequencies that appear in complex wave patterns, offering insightful details about the movement and shape of waves.

When applied to problems like the wave equation, harmonic analysis allows us to break down the wave motion into simpler periodic functions. Through this, we can explore each frequency's effect on the overall system dynamics.

Fourier Series

The Fourier series is an essential mathematical tool used to express periodic functions as sums of sinusoidal functions, specifically sines and cosines. This technique is invaluable in solving complex wave-based problems because it simplifies handling periodic boundary conditions.

• A Fourier series is particularly powerful in expressing solutions for periodic functions that recur over a fixed interval.
• In the context of the wave equation, it breaks down the string's vibration into its individual frequency components.

This separation facilitates the analysis of each mode's contribution to the behavior of the whole system. The beauty of using a Fourier series lies in its ability to approximate complicated functions with increasing accuracy by considering more terms, making it a key part of engineering and physics applications.

Partial Differential Equations

Partial differential equations (PDEs) are mathematical equations involving multivariable functions and their partial derivatives. They are indispensable in describing various physical phenomena, such as sound, heat, and electrodynamics, including wave motion in strings and surfaces.

• PDEs can capture changes along multiple dimensions, offering a comprehensive way to model real-world systems.
• The wave equation is a classic example of a PDE, describing how waves propagate through different media.

Understanding PDEs is key to predicting how complex systems behave over time and space. In the case of a vibrating string, employing PDEs enables us to model the string's displacement over its length and as time advances. They provide a framework for developing insights into the natural world through mathematical expressions and solutions.