Question

If a string vibrates with the end at \(x=0\) free to move in a frictionless vertical track and the end at \(x=L\) fixed, then the initial-boundary value problem for its displacement takes the form $$ \begin{array}{c} u_{t t}=a^{2} u_{x x}, \quad 00 ,\\\ u_{x}(0, t)=0, \quad u(L, t)=0, \quad t>0 ,\\\ u(x, 0)=f(x), \quad u_{t}(x, 0)=g(x), \quad 0 \leq x \leq L \end{array}. $$ Justify defining the formal solution of (A) to be $$ u(x, t)=\sum_{n=1}^{\infty}\left(\alpha_{n} \cos \frac{(2 n-1) \pi a t}{2 L}+\frac{2 L \beta_{n}}{(2 n-1) \pi a} \sin \frac{(2 n-1) \pi a t}{2 L}\right) \cos \frac{(2 n-1) \pi x}{2 L}, $$ where $$ C_{M f}(x)=\sum_{n=1}^{\infty} \alpha_{n} \cos \frac{(2 n-1) \pi x}{2 L} \text { and } C_{M g}(x)=\sum_{n=1}^{\infty} \beta_{n} \cos \frac{(2 n-1) \pi x}{2 L} $$ are the mixed Fourier cosine series of \(f\) and \(g\) on \([0, L] ;\) that is, $$ \alpha_{n}=\frac{2}{L} \int_{0}^{L} f(x) \cos \frac{(2 n-1) \pi x}{2 L} d x \quad \text { and } \quad \beta_{n}=\frac{2}{L} \int_{0}^{L} g(x) \cos \frac{(2 n-1) \pi x}{2 L} d x $$ $$ \alpha_{n}=\frac{2}{L} \int_{0}^{L} f(x) \cos \frac{(2 n-1) \pi x}{2 L} d x \quad \text { and } \quad \beta_{n}=\frac{2}{L} \int_{0}^{L} g(x) \cos \frac{(2 n-1) \pi x}{2 L} d x $$

Short answer

In summary, to justify the formal solution and find the mixed Fourier cosine series of \(f\) and \(g\) for the given initial-boundary problem for a string with a free end at \(x=0\) and a fixed end at \(x=L\), we have verified that the formal solution satisfies the boundary conditions, calculated the coefficients \(\alpha_n\) and \(\beta_n\), and used this information to find the Fourier cosine series of functions \(f(x)\) and \(g(x)\).

Step-by-step solution

Verify the formal solution satisfies the boundary conditions

First, we need to verify if the formal solution \(u(x, t)\) satisfies the given boundary conditions. Boundary condition 1: When \(x=0\), we need to show that \(u_x(0, t) = 0\). So, we plug in \(x=0\) in \(u(x, t)\) and find the derivative with respect to x: $$ u_{x}(0, t)=\frac{d}{d x}\left[\sum_{n=1}^{\infty}\left(\alpha_{n} \cos \frac{(2 n-1) \pi a t}{2 L}+\frac{2 L \beta_{n}}{(2 n-1) \pi a} \sin \frac{(2 n-1) \pi a t}{2 L}\right) \cos \frac{(2 n-1) \pi (0)}{2 L}\right]=0, $$ which satisfies the first boundary condition since all terms in the sum end up being zero. Boundary condition 2: When \(x=L\), we need to show that \(u(L, t) = 0\). So, we plug in \(x=L\) in \(u(x, t)\) formula and get $$ u(L, t)=\sum_{n=1}^{\infty}\left(\alpha_{n} \cos \frac{(2 n-1) \pi a t}{2 L}+\frac{2 L \beta_{n}}{(2 n-1) \pi a} \sin \frac{(2 n-1) \pi a t}{2 L}\right) \cos \frac{(2 n-1) \pi L}{2 L}, $$ which becomes zero since \(\cos \frac{(2 n-1) \pi L}{2 L}\) is always zero for integer values of \(n\). This verifies the second boundary condition.

Find the Fourier cosine series of \(f\) and \(g\)

To find the Fourier cosine series of \(f(x)\) and \(g(x)\), we need to calculate the coefficients \(\alpha_n\) and \(\beta_n\). Given the formulas for \(\alpha_n\) and \(\beta_n\): $$ \alpha_{n}=\frac{2}{L} \int_{0}^{L} f(x) \cos \frac{(2 n-1) \pi x}{2 L} d x \quad \text { and } \quad \beta_{n}=\frac{2}{L} \int_{0}^{L} g(x) \cos \frac{(2 n-1) \pi x}{2 L} d x $$ We integrate the product of the functions \(f(x)\) or \(g(x)\) and \(\cos \frac{(2 n-1) \pi x}{2 L}\) over the interval \([0, L]\) and multiply by \(\frac{2}{L}\). This will give us the mixed Fourier cosine series \(C_{M f}(x)\) of \(f(x)\) and \(C_{M g}(x)\) of \(g(x)\) on the interval \([0, L]\). These calculating procedures justify that the given formal solution for the initial-boundary value problem is correct, and now we can use this to analyze any particular problem with specific functions \(f(x)\) and \(g(x)\).

Key concepts

Fourier Cosine Series

Understanding the Fourier cosine series is crucial when working with vibrations and waves, especially in a physical context like a stringed instrument or other systems that can be modeled similarly. The Fourier cosine series breaks down an arbitrary function into a sum of cosine functions multiplied by coefficients, specifically tailored for functions defined on a symmetric interval.

This series is particularly useful in solving the initial-boundary value problems for partial differential equations with certain types of boundary conditions. In the context of the exercise, we are interested in the mixed Fourier cosine series, which is a slightly modified version of the standard series adapted for functions with specific symmetry properties.

When given a function defined over an interval \[0, L\], we express this function as an infinite sum of cosine functions with different frequencies. The coefficients \(\alpha_n\) and \(\beta_n\) are calculated using integrals that measure how much a cosine function of a specific frequency contributes to the overall function. These coefficients are crucial to capture the initial state of the system, be it the initial displacement given by \(f(x)\) or the initial velocity given by \(g(x)\).

By understanding the role of these coefficients and the structure of the Fourier cosine series, students can begin to appreciate how complex vibrations can be broken down into simpler, harmonic components, which is a fundamental concept in areas ranging from physics to engineering.

Partial Differential Equations

Partial differential equations (PDEs) are a type of equation crucial for describing various phenomena such as the diffusion of heat, the propagation of sound waves, or the oscillation of a plucked string. A PDE involves rates of change with respect to multiple variables and is differentiated from ordinary differential equations (ODEs) which involve rates of change with respect to only one variable.

In the given exercise, the PDE is \(u_{tt} = a^2 u_{xx}\), which represents the wave equation. This equation is the cornerstone for understanding wave-like phenomena in continuous media. The subscripts \(tt\) and \(xx\) denote the second derivatives of the displacement \(u\) with respect to time \(t\) and space \(x\), respectively. The coefficient \(a\) typically represents the wave speed.

To solve a PDE like the wave equation, it's not enough to merely apply calculus techniques; we also need to consider the boundary conditions and initial states affecting the system. The wave equation reflects an interplay between time and space evolution of the system, and solving it provides insights into the dynamics of the system over time.

Boundary Conditions

Boundary conditions are essential components when solving mathematical problems involving PDEs as they specify the behavior of a physical system at the limits of its domain. In other words, they are constraints necessary to uniquely determine a solution to a differential equation. For the exercise in question, we're looking at a string with two very specific conditions at its ends.

The boundary condition \(u_x(0, t) = 0\) implies that the end of the string at \(x=0\) is free to move up and down without resistance, in a frictionless vertical track, behaving similarly to a smooth wall that reflects wave without inverting it. This is known as a Neumann boundary condition, which dictates the slope (or derivative) of the wave at that point.

The boundary condition \(u(L, t) = 0\) tells us that the other end of the string at \(x=L\) is fixed in place. This is a Dirichlet boundary condition, where the value of the function is specified at the edge of the domain. Solving the PDE with these conditions in mind gives us a unique solution that describes the physical reality of the system—how a string vibrates with one end free and the other fixed.

The understanding of boundary conditions, coupled with the framework provided by the PDE and the Fourier series representation of the initial conditions, creates a complete solution to the initial-boundary value problem presented in the exercise.