Question

If an elastic string is free at one end, the boundary condition to be satisfied there is that \(u_{x}=0 .\) Find the displactement \(u(x, t)\) in an elastic string of length \(L\), fixed at \(x=0\) and freeat \(x=L,\) set th motion with no initial velocity from the initiol position \(u(x, 0)=f(x)\) Where \(f\) is a given function. withno intitial velocity from the initiolposition \(u(x, 0)=f(x),\) Hint: Show that insiamental solutions for this problem, satisfying all conditions except the nonomongent condition, are $$ u_{n}(x, t)=\sin \lambda_{n} x \cos \lambda_{n} a t $$ where \(\lambda_{n}=(2 n-1) \pi / 2 L, n=1,2, \ldots\) Compare this problem with Problem 15 of Section \(10.6 ;\) pay particular attention to the extension of the initial data out of the original interval \([0, L] .\)

Short answer

#Question# Using the step-by-step solution provided, determine the displacement u(x, t) for an elastic string of length L, fixed at one end (x=0) and free at the other end (x=L) given the initial displacement u(x, 0) = f(x) and no initial velocity. #Answer# The displacement u(x, t) can be represented as a Fourier series that satisfies the boundary condition and initial data: $$u(x, t) = \sum_{n = 1}^{\infty} {B_n \sin(\lambda_n x) \cos(\lambda_n a t)}$$ where \(\lambda_n = (2n - 1)\pi / 2L, n = 1, 2, ...\), and the coefficients \(B_n\) can be determined by the initial displacement function f(x): $$f(x) = \sum_{n=1}^{\infty} {B_n \sin(\lambda_n x)}$$

Step-by-step solution

Checking if \(u_n(x, t)\) satisfies the boundary condition

First, we need to test the provided function \(u_n(x, t)\) on the boundary condition \(u_x(L, t) = 0\). Let's find the partial derivative of \(u_n(x, t)\) with respect to x: $$u_{n_x}(x, t) = \frac{\partial}{\partial x}(\sin \lambda_n x \cos \lambda_n a t) = \lambda_n \cos(\lambda_n x) \cos(\lambda_n a t)$$ Now, let's test if \(u_{n_x}(L, t) = 0\): $$u_{n_x}(L, t) = \lambda_n \cos(\lambda_n L) \cos(\lambda_n a t)$$ In order for this to equal 0, we need: $$\cos(\lambda_n L) = 0$$ We know that \(\lambda_n L = (4n - 2) \pi / 2\), so \(\cos (4n - 2) \pi / 2\) must be equal to 0. For all integer values of n, this condition holds true. Therefore, \(u_n(x, t)\) does satisfy the given boundary condition.

Checking the initial position values

We know that the initial position values should satisfy \(u(x, 0) = f(x)\). Let's test if the initial values of \(u_n(x, t)\) given by the provided equation match this: $$u_n(x, 0) = \sin(\lambda_n x) \cos(\lambda_n a (0)) = \sin(\lambda_n x)$$ Notice that the initial data of \(u_n(x, t)\) looks like a Fourier series representation of the function f(x). We know that this series, when properly constructed, will satisfy the boundary conditions and initial displacement of the string. So, now let's write the solution as a Fourier series: $$f(x) = \sum_{n = 1}^{\infty} {B_n \sin(\lambda_n x)}$$ This will give us several solutions as the series converges, so we can consider their combination, which will also satisfy the given boundary condition and initial data.

Comparing with Problem 15 in Section 10.6

The exercise given is similar to Problem 15 of Section 10.6, as both are concerning an elastic string fixed at one end and free at the other end. Both exercises provide a partial differential equation for the displacement of the string. The main difference between the two exercises is the extension of the initial data outside the interval [0, L]. In Problem 15, the initial displacement function f(x) was defined to be \(f(x + 2L) = -f(x)\), while our current exercise provides no guidance on how to extend the initial data outside the interval [0, L], which may affect the general solution of the problem. However, note that it is not necessary to consider the extension of the initial data outside the interval [0, L], as we have already found the displacement for our given initial condition and boundary conditions. The similarity between the two problems helps us in understanding the solution form and method, without needing to perform an explicit comparison with the initial data extension.

Key concepts

Partial Differential Equations

Partial differential equations (PDEs) are a type of mathematical equation involving rates of change with respect to continuous variables. Unlike ordinary differential equations (ODEs), which feature derivatives with respect to a single variable, PDEs involve derivatives with respect to multiple variables. They are crucial in expressing physical phenomena such as heat transfer, fluid dynamics, and in our exercise, the vibrations of an elastic string.

For the elastic string problem, the string's displacement is represented by the function \(u(x, t)\), where \(x\) is the spatial coordinate along the length of the string and \(t\) represents time. The PDE governing the displacement is derived from the physical considerations, such as Newton's laws of motion, and must satisfy certain boundary conditions – for example, in the case of the free end of a string, the boundary condition is that the slope of the string at that point in space is zero, \(u_x(L, t) = 0\). The solution to the PDE must respect all these conditions to accurately describe the behavior of the string.

Fourier Series

A Fourier series is a way to represent a function as the sum of simple sine and cosine waves. By decomposing a periodic function into the sum of simple oscillating functions, Fourier series allows complex functions to be written in terms of an infinite series of sine and cosine functions. This is particularly useful in solving PDEs like the one presented in the elastic string problem.

The Fourier series plays a pivotal role when we try to solve for the initial displacement \(u(x, 0)\). By expressing the function \(f(x)\), which describes the initial shape of the string, as a series of sine functions, each of which satisfies the PDE and boundary conditions, we effectively create a solution that adheres to the initial constraints. In essence, the Fourier series expansion enumerates all the 'modes' in which the string can vibrate independently given the boundary conditions.

Finding the appropriate coefficients for this series is the key to providing a complete solution. These coefficients are typically determined by applying the initial conditions and exploiting orthogonality properties of sine and cosine functions.

Initial Value Problem

An initial value problem (IVP) is a specific type of differential equation problem in which the solution is determined based on the state of the system at a single point in time. Generally, an IVP requires you to solve a differential equation subject to a given set of initial conditions. This is what we see with the elastic string when it is initially set into motion from a specified shape \(u(x, 0) = f(x)\), with no initial velocity.

IVPs are crucial for predicting future behaviors of dynamic systems. Resolving an IVP involves not only finding a function that satisfies the PDE but also ensuring that the function adheres to the initial conditions. The difficulty often lies in knowing the details of \(f(x)\) and finding a solution for \(u(x, t)\) that matches this at \(t=0\). In our problem, this was addressed by constructing a Fourier series that mimics the initial shape of the string. This approach ensures that at \(t=0\), the string takes the shape of \(f(x)\), effectively satisfying the IVP.