Problem 6 - Advanced Engineering Mathematics

Question

The displacement \(u(x, t)\) of a string that is driven by an extemal force is determined from $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}+\sin \psi x \sin \omega t=\frac{\partial^{2} u}{\partial t^{2}}, \quad 00 \\ &u(0, t)=0, \quad u(1, t)=0, t>0 \\ &u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0,0

Step-by-Step Solution

Answer

The solution for \( u(x, t) \) involves a sum of a homogeneous and a particular solution satisfying boundary and initial conditions.

Step 1: Recognize the Equation Type

The given equation \( \frac{\partial^2 u}{\partial x^2} + \sin \psi x \sin \omega t = \frac{\partial^2 u}{\partial t^2} \) is a non-homogeneous wave equation due to the presence of the term \( \sin \psi x \sin \omega t \). This term acts as the forcing function.

Step 2: Find Homogeneous Solution

To find the solution to the homogeneous equation \( \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \), apply separation of variables: assume \( u(x, t) = X(x)T(t) \). This leads to \( \frac{X''(x)}{X(x)} = \frac{T''(t)}{T(t)} = -\lambda \). Solve the resulting ordinary differential equations for \( X(x) \) and \( T(t) \).

Step 3: Apply Boundary Conditions to Homogeneous Solution

The boundary conditions are \( u(0, t) = 0 \) and \( u(1, t) = 0 \). Applying these to \( X(x) \) results in \( X(x) = A\sin(n\pi x) \) where \( n \) is an integer, and \( \lambda = (n\pi)^2 \). Accordingly, \( T(t) = C\cos(n\pi t) + D\sin(n\pi t) \). The homogeneous solution is \( u_h(x, t) = \sum_{n=1}^{\infty} \left[ A_n \sin(n\pi x) (C\cos(n\pi t) + D\sin(n\pi t)) \right] \).

Step 4: Solve for Particular Solution using Method of Undetermined Coefficients

Introduce a particular solution in the form \( u_p(x, t) = a(x) b(t) \), specifically tailored to \( \sin \psi x \sin \omega t \). Substitute into the differential equation and solve for \( a(x) \) and \( b(t) \). As a result, use an educated guess such as \( a(x) = A \sin \psi x \) and \( b(t) = B \cos \omega t + C \sin \omega t \) with coefficients chosen to satisfy the equation.

Step 5: Combine Solutions

The complete solution to the non-homogeneous wave equation is given by summing the homogeneous and particular solutions: \( u(x, t) = u_h(x, t) + u_p(x, t) \). Substitute the coefficients obtained from boundary conditions and initial conditions into the combined solution to finalize the complete expression for \( u(x, t) \).

Step 6: Verify and Simplify Solution

Verify all conditions are satisfied by the combined solution, including boundary conditions \( u(0, t) = 0 \) and \( u(1, t) = 0 \), as well as initial conditions \( u(x, 0) = 0 \) and \( \frac{\partial u}{\partial t}(x,0) = 0 \). Simplify the final expression for clearer insight.

Key Concepts

Method of Separation of Variables

Understanding the method of separation of variables is crucial if you're diving into solving partial differential equations like the wave equation. Here, we begin by assuming a particular form for our solution of the equation, specifically, that it can be expressed as the product of two functions: one that is purely a function of space, and another that is purely a function of time. For instance, for a function \(u(x, t)\), we propose that \(u(x, t) = X(x)T(t)\). This assumption helps separate the original partial differential equation into two ordinary differential equations—one involving only \(X(x)\) and the other only \(T(t)\).This is a powerful tool that turns a complex problem into simpler ones. For the homogeneous wave equation, this method leads us to an eigenvalue problem where we solve the differential equations like \(X''(x) + \, \lambda X(x)=0\) and \(T''(t) + \, \lambda T(t)=0\). By tackling these, we craft solutions which we later superpose, thanks to the linearity of the equation.

Boundary Conditions

Boundary conditions are an essential piece of the puzzle when solving wave equations. They describe constraints that the solution must meet at the boundaries of the spatial domain. For the given problem, the boundary conditions are \(u(0, t) = 0\) and \(u(1, t) = 0\). These conditions tell us that the displacement at both ends of the string is zero for all time \(t\), meaning the string is fixed at these points. This results in the spatial part of the solution, \(X(x)\), taking forms like \(A\sin(n\pi x)\), because sine functions automatically satisfy zero-value conditions at specified endpoints.When you impose these boundary conditions on potential solutions, they significantly restrict the possible forms \(X(x)\) might take, leading us to the quantization of our eigenvalues \(\lambda\), specifically \(\lambda = (n\pi)^2\). Each of these yields a distinct mode of vibration for the solution.

Method of Undetermined Coefficients

The method of undetermined coefficients is used to find a particular solution to the non-homogeneous wave equation. This approach involves making an educated guess about the form of the solution based on the 'forcing' part of the equation—in this case, terms such as \(\sin \psi x \sin \omega t\).The method starts with proposing a specific form for the solution. Here, we conjecture: \(u_p(x, t) = a(x)b(t)\) and choose functions \(a(x)\) and \(b(t)\) that mimic the style of our non-homogeneous term. A sensible guess would be: \(a(x) = A \sin \psi x\) and \(b(t) = B \cos \omega t + C \sin \omega t\).Once we've selected such functions, the next step is finding their unknown coefficients by substituting back into the original differential equation and solving for these coefficients. This process ensures that \(u_p(x, t)\) satisfies the non-homogeneous equation by itself without boundary or initial conditions. Intuitive and straightforward, it complements the homogeneous part to give a complete solution.