Question

Find the general solution of the wave equation $$ \frac{\partial^{2} u}{\partial t^{2}}-a^{2} \frac{\partial^{2} u}{\partial x^{2}}=0, \quad 00, $$ such that \(u(0, t)=0, u_{x}(\pi, t)=0\).

Short answer

The general solution of the wave equation satisfying the given boundary conditions is as follows: $$ u(x,t) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{(2n-1)x}{2}\right)\left[C_n\cos(a\lambda_n t) + D_n \sin(a\lambda_n t)\right] $$ Here, \(\lambda_n = \frac{(2n-1)}{2}\) and \(B_n, C_n, D_n\) are constants to be determined by initial conditions or other constraints.

Step-by-step solution

Let's assume that the solution has the form \(u(x,t) = X(x)T(t)\). Substitute this form into the wave equation: $$ X(x) \frac{d^2 T(t)}{dt^2} - a^2 T(t) \frac{d^2 X(x)}{dx^2} = 0. $$ Now, divide both sides by \(X(x)T(t)\) to separate the variables: $$ \frac{1}{a^2}\frac{\frac{d^2 T(t)}{dt^2}}{T(t)} = \frac{\frac{d^2 X(x)}{dx^2}}{X(x)} $$ #Step 2: Solve the ODEs in x and t#

Since the left side of the equation depends only on \(t\) and the right side only on \(x\), they must both be equal to a separation constant, which we will call \(-\lambda^2\). This gives us two ODEs: 1. Time equation: \(\frac{d^2T(t)}{dt^2} + a^2\lambda^2 T(t) = 0\) 2. Position equation: \(\frac{d^2X(x)}{dx^2} + \lambda^2 X(x) = 0\) #Step 3: Solve the position equation#

The position equation is a second-order linear ODE with constant coefficients, which has a well-known solution: $$ X(x) = A \cos(\lambda x) + B \sin(\lambda x) $$ Apply the first boundary condition: \(u(0,t) = X(0)T(t) = 0 \Longrightarrow X(0) = 0\) $$ A \cos(0) + B \sin(0) = 0 \Longrightarrow A=0 $$ Thus, the position function becomes: $$ X(x) = B \sin(\lambda x) $$ #Step 4: Apply the second boundary condition#

The second boundary condition is \(u_x(\pi,t) = 0\). Compute the derivative of \(X(x)\) with respect to \(x\): $$ X'(x) = B\lambda \cos(\lambda x) $$ Now evaluate \(X'(\pi)\): $$ B\lambda \cos(\lambda \pi) = 0 $$ For non-trivial solutions, \(B\neq 0\), so we need \(\cos(\lambda \pi) = 0\). This occurs for \(\lambda_n = \frac{(2n-1)}{2}\) where \(n\) is a positive integer. Hence, we have: $$ X_n(x) = B_n \sin\left(\frac{(2n-1)x}{2}\right) $$ #Step 5: Solve the time equation#

The time equation is also a second-order linear ODE with constant coefficients: $$ \frac{d^2T(t)}{dt^2} + a^2\lambda_n^2 T(t) = 0 $$ The solution of this ODE is given by: $$ T_n(t) = C_n\cos(a\lambda_n t) + D_n \sin(a\lambda_n t) $$ #Step 6: Find the general solution and apply boundary conditions#

The general solution is the product of \(X_n(x)\) and \(T_n(t)\): $$ u_n(x,t) = \left[B_n \sin\left(\frac{(2n-1)x}{2}\right)\right]\left[C_n\cos(a\lambda_n t) + D_n \sin(a\lambda_n t)\right] $$ Using the first boundary condition \(u(0,t)=0\), we get: $$ u_n(0,t) = \left[B_n \sin(0)\right]\left[C_n\cos(a\lambda_n t) + D_n \sin(a\lambda_n t)\right]=0 $$ This holds for every \(t\), so the overall solution is a sum of these \(u_n(x,t)\): $$ u(x,t) = \sum_{n=1}^{\infty} u_n(x,t) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{(2n-1)x}{2}\right)\left[C_n\cos(a\lambda_n t) + D_n \sin(a\lambda_n t)\right] $$ Here, \(u(x,t)\) is the general solution of the wave equation satisfying the given boundary conditions.

Key concepts

Partial Differential Equations

Partial differential equations (PDEs) form a cornerstone of mathematical physics and applied mathematics, characterizing various phenomena where change takes place across multiple variables. They can describe how physical quantities such as heat or sound waves propagate through space over time. The wave equation presented in our exercise is a classic example of a second-order PDE, where the notation \frac{\(\partial^{2} u\)}{\partial t^{2}} - a^{2} \(\frac{\partial^{2} u}{\partial x^{2}}\) indicates that the change in the function \(u(x,t)\) depends on both space (x) and time (t).

Students will recognize PDEs such as the wave equation by the presence of multiple derivatives with respect to more than one variable. Solving these equations often involves finding a function that satisfies all conditions laid out by the problem, which may include initial or boundary values. The intricacy of PDEs necessitates specialized methods such as separation of variables, a technique we utilize to resolve the wave equation in the provided solution.

Boundary Conditions

Boundary conditions are essential constraints for solving differential equations, particularly critical when dealing with PDEs. These conditions serve as additional information that the solution must satisfy on the edge of the domain over which the equation is defined. In the context of our wave equation, the domain is given by \(00\), and we are provided with two boundary conditions: \(u(0, t)=0\) and \(u_x(\pi, t)=0\).

These conditions guide us to the particular solution that not only fulfills the wave equation but also observes the physical constraints of the problem. For instance, \(u(0, t)=0\) could represent a fixed end of a string at position \(x=0\), while \(u_x(\pi, t)=0\) implies that there is no slope (or no transverse velocity) at \(x=\pi\), perhaps indicating a free end. Implementing the boundary conditions as demonstrated in the solution steps is critical for finding the correct expression for \(X(x)\)

Separation of Variables

Separation of variables is a widely-used method for solving PDEs, including wave equations. The method relies on the possibility to decompose a complex, multivariable problem into simpler, single-variable problems. In the case of the wave equation, we assume a solution of the form \(u(x,t) = X(x)T(t)\), where \(X\) depends only on space and \(T\) only on time.

Through this process, as seen in the solution steps, we separate the wave equation into two ordinary differential equations (ODEs) for \(X\) and \(T\) respectively. By solving each ODE using standard techniques and applying boundary conditions, we construct the full solution as a product of these individual functions. This approach simplifies the original problem markedly, enabling us to systematically address the wave equation and signal qualified solutions that align with both the mathematical model and the physical interpretation of the scenario.