Problem 3 - Advanced Engineering Mathematics

Question

The displacement of a semi-infinite elastic string is det-rmined from $$ \begin{aligned} &a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}, \quad x>0, t>0\\\ &u(0, t)=f(t), \quad \lim _{x \rightarrow \infty} u(x, t)=0, t>0\\\ &u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0, x>0\\\ &\text { Solve for } u(x, t) \end{aligned} $$

Step-by-Step Solution

Answer

The solution is obtained using D'Alembert's formula by applying boundary and initial conditions.

Step 1: Recognize the Type of Problem

This is a wave equation problem with initial and boundary conditions. The equation \( a^{2} \frac{\partial^{2} u}{\partial x^{2}} = \frac{\partial^{2} u}{\partial t^{2}} \) represents a one-dimensional wave on a semi-infinite string.

Step 2: Apply D'Alembert's Solution

The general solution to the wave equation is given by D'Alembert's formula: \( u(x, t) = F(x-at) + G(x+at) \). Here, \( F \) and \( G \) are arbitrary functions determined by the initial and boundary conditions.

Step 3: Use the Boundary Condition at \( x = 0 \)

The boundary condition is \( u(0, t) = f(t) \). Substituting into D'Alembert's solution gives \( F(-at) + G(at) = f(t) \). This relationship will help determine \( F \) and \( G \).

Step 4: Implement the Condition as \( x \to \infty \)

The condition \( \lim_{x \to \infty} u(x, t) = 0 \) implies that \( G \to 0 \) as \( x \to \infty \). So, we can infer that \( G(x+at) \) must vanish as the term \( x+at \to \infty \), suggesting \( G \equiv 0 \).

Step 5: Apply Initial Conditions at \( t = 0 \)

The initial conditions are \( u(x, 0) = 0 \) and \( \left.\frac{\partial u}{\partial t}\right|_{t=0} = 0 \). Since \( G \equiv 0 \), \( u(x, t) = F(x-at) \). Applying \( u(x, 0) = 0 \) gives \( F(x) = 0 \).

Step 6: Define Function \( F \) Consistent with Known Functions

From \( F(-at) = f(t) \), the form of \( F \) consistent with other conditions is that it essentially reflects the initial input \( f(t) \) over time. Given prior conclusions, we directly write \( F \) as the reflection necessary to satisfy the given equation.

Key Concepts

Boundary Conditions

When we talk about boundary conditions in the context of differential equations like the wave equation, we're referring to certain fixed constraints at the boundaries of the spatial domain. These conditions help define how the solution behaves at those boundaries. In this exercise, the boundary conditions are given as:

• For the boundary at the starting point of the string, we have: \( u(0, t) = f(t) \). This tells us that no matter what, at the point where the string begins (\( x = 0 \)), our displacement function \( u \) must equal some function \( f(t) \) of time.
• At the other end, extending to infinity: \( \lim_{x \to \infty} u(x, t) = 0 \). This implies that as we move further along the string (towards infinity), the displacement caused by any waves dies out. Essentially, the effect of the wave becomes negligible at infinity.

Boundary conditions are vital because they help to narrow down the many possible solutions to a differential equation to just those that fit the specific physical situation described. They are crucial for creating a complete and specific model of the wave behavior on the string.

Initial Conditions

Initial conditions specify the state of the system at the beginning of the time under consideration. They are crucial because they define how the system starts out, which influences everything that happens afterward. For wave equations, these conditions often dictate the initial displacement and initial velocity of the wave.

• The initial displacement is given by \( u(x, 0) = 0 \). This means that at time \( t = 0 \), the string starts with no displacement. In simple terms, the string is initially at rest.
• The initial velocity condition, \( \left.\frac{\partial u}{\partial t}\right|_{t=0} = 0 \), tells us that there is no initial velocity, so not only is the string at rest initially, but it is also not beginning to move at all.

These initial conditions help us to determine the specific form of our solution because they offer crucial information about the wave function at its inception. Together with the boundary conditions, they pin down what the exact movement on the string looks like over time.

D'Alembert's Solution

D'Alembert's solution provides a powerful method to solve the one-dimensional wave equation, especially under the constraints of specific boundary and initial conditions. The D'Alembert formula is expressed as:

• \( u(x, t) = F(x-at) + G(x+at) \)

Here,

• \( F \) and \( G \) are functions that need to be determined by inserting the initial and boundary conditions into the formula.
• The terms \( x-at \) and \( x+at \) represent waves traveling in the positive and negative x-direction, respectively.

In the given problem, the boundary condition \( u(0, t) = f(t) \) helps us establish a relationship between \( F \) and \( G \). By also applying the condition at infinity, \( \lim_{x \to \infty} u(x, t) = 0 \), we deduce that \( G \) must be zero to ensure the effect of waves does not persist indefinitely.With \( G \equiv 0 \), this simplifies our solution to \( u(x, t) = F(x-at) \). Applying the initial conditions further refines \( F \), confirming that it must reflect the initial input wave \( f(t) \), ultimately crafting the specific motion of the wave along the string over time.