Question

(a) Solve this problem for the given parameter values and the given initial condition. (b) Assume the solution \(u(x, t)\) represents the displacement at time \(t\) and position \(x\). Determine the velocity, \(u_{t}(x, t)\). (In Exercises 7-10, assume the series can be differentiated termwise.) $$c=2, \quad l=\pi, \quad u(x, 0)=-\sin x, \quad u_{t}(x, 0)=\sin 2 x+\sin 3 x$$

Short answer

Answer: As time progresses, the velocity of the displacement $u_t(x, t) = 4\pi^2 e^{-4\pi^2t}\sin x$ decreases due to the exponential decay term $e^{-4\pi^2t}$. This means that the overall influence of the initial conditions diminishes over time, as the system relaxes. The $\sin x$ term ensures that there will still be some oscillation in the system, but the amplitude of the oscillation is dampened as time increases.

Step-by-step solution

Find the Fourier coefficients of the initial conditions

To find the Fourier coefficients of the initial conditions, we first need to look at the given conditions: $$u(x, 0) = -\sin x \quad \text{and} \quad u_t(x, 0) = \sin 2x + \sin 3x$$ For the Fourier sine series, we need to calculate each \(B_n\): $$B_n = \frac{2}{l} \int_0^l f(x) \sin \frac{n\pi x}{l} dx$$ We obtain \(B_n\) for both initial conditions \(-\sin{x}\) and \(\sin{2x} + \sin{3x}\): For \(u(x, 0) = -\sin{x}\): Using the above formula, we have: $$B_n = \frac{2}{\pi} \int_0^\pi -\sin x \sin \frac{n\pi x}{\pi} dx$$ If we note that this integral is 0 unless n=1. That is, for this initial condition, only the equal terms will multiply and yield a nonzero result for \(B_1\). Thus, \(B_n = 0\) for \(n\neq1\). For \(u_t(x, 0) = \sin{2x} + \sin{3x}\): Using the the formula, we have: $$C_n = \frac{2}{\pi} \int_0^\pi (\sin{2x} + \sin{3x}) \sin \frac{n\pi x}{\pi} dx$$ Here, only \(C_2\) and \(C_3\) will be nonzero, as only the corresponding terms contribute to the series: $$C_2 = \frac{2}{\pi} \int_0^\pi \sin{2x} \sin {2x} dx = 1$$ $$C_3 = \frac{2}{\pi} \int_0^\pi \sin{3x} \sin {3x} dx = 1$$ For all other \(n\), \(C_n = 0\).

Determine the series solution for \(u(x,t)\) using the found Fourier coefficients

Now that we have the Fourier coefficients \(B_n\) and \(C_n\), we can find the series solution to \(u(x, t)\). Since \(c = 2\) and \(l = \pi\), the general solution for \(u(x,t)\) is given by: $$u(x, t) = \sum_{n = 1}^{\infty} B_ne^{-\frac{n^2\pi^2}{l^2}c^2t}\sin\frac{n\pi x}{l}$$ Now, we can plug in the Fourier coefficients \(B_n\) and \(C_n\) that we found in the previous step. $$u(x, t) = - e^{-4\pi^2t}\sin x$$

Differentiate the series solution with respect to time \(t\) to find the velocity \(u_{t}(x,t)\)

Finally, we need to find the velocity \(u_{t}(x, t)\) by differentiating our solution \(u(x, t)\) with respect to time \(t\). $$u_t(x, t) = \frac{\partial}{\partial t}(- e^{-4\pi^2t}\sin x)$$ Applying the chain rule, we get: $$u_t(x, t) = -(-4\pi^2) e^{-4\pi^2t} \sin x = 4\pi^2 e^{-4\pi^2t}\sin x$$ So, the velocity \(u_{t}(x, t)\) is given by: $$u_t(x, t) = 4\pi^2 e^{-4\pi^2t}\sin x$$

Key concepts

Fourier Series

The Fourier series helps to express complex periodic functions as a sum of simpler sine and cosine functions. This is particularly useful in solving wave equations, where the solution can be cleverly broken down into these trigonometric components. Each term in the series consists of a coefficient and a sine or cosine function with specific frequencies, which correspond to harmonics of the fundamental frequency.

In our problem, the goal is to represent the given initial conditions using the Fourier sine series. This involves calculating Fourier coefficients, termed as \(B_n\) and \(C_n\), from the initial conditions \(u(x, 0)\) and \(u_t(x, 0)\). The formula to find these coefficients is given by:

• \(B_n = \frac{2}{l} \int_0^l f(x) \sin \frac{n\pi x}{l} dx\)

This calculation helps us to recreate the function using a series that can be analyzed and manipulated to understand the physical system being modeled.

Initial Conditions

Initial conditions are crucial for solving partial differential equations (PDEs) because they provide the necessary constraints to find unique solutions. When dealing with PDEs like the wave equation, initial conditions specify the state of the system at the starting time (\(t = 0\)).

In this exercise, the initial condition for displacement is \(u(x, 0) = -\sin x\) and for velocity is \(u_t(x, 0) = \sin 2x + \sin 3x\). The purpose of these conditions is to start the "story" of the wave's behavior at a known point, from which we can apply the governing equations to predict its behavior over time.

Essentially, initial conditions shape how the Fourier series is constructed. They provide the function \(f(x)\) that is then decomposed into the series by calculating the integral to find the coefficients \(B_n\) and \(C_n\). This integration reveals which harmonics are present in the initial disturbance and how they combine to form the initial wave shape.

Velocity Calculation

Velocity in wave motion refers to the rate at which the displacement of the wave changes over time. In mathematical terms, it involves taking the derivative of the wave function with respect to time. This provides insights into how fast each point on the wave is moving as time progresses, which is particularly important when studying wave dynamics in physics.

In our problem, we determine the velocity \(u_t(x, t)\) by differentiating the solution \(u(x, t)\) with respect to time. The originally derived series solution is:

• \(u(x, t) = -e^{-4\pi^2t}\sin x\)

Differentiating this with respect to \(t\) using the chain rule gives:

• \(u_t(x, t) = 4\pi^2 e^{-4\pi^2t}\sin x\)

This formula indicates how the wave's velocity at any point \(x\) is affected by both the initial amplitude \(\sin x\) and the exponential damping factor \(e^{-4\pi^2t}\), demonstrating how energy dissipates over time.

Partial Differential Equation

A Partial Differential Equation (PDE) involves derivatives with respect to more than one variable. Such equations are fundamental in describing physical phenomena like heat conduction, fluid flow, and wave propagation. The wave equation, which describes how waves travel through a medium, is a classic example of a PDE.

The wave equation in one-dimensional form with constant speed \(c\) is presented as:

• \(\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\)

In this exercise, solving the wave equation with given boundary and initial conditions involves two key steps: determining the Fourier coefficients from initial conditions and plugging them into the series solution.

By successfully finding these coefficients and reconstructing \(u(x, t)\), we solve a PDE that models how physical forces shape wave forms over time. Understanding this establishes a link from theoretical mathematics to real-world wave behavior, which has numerous applications in science and engineering.