Problem 26 - Advanced Engineering Mathematics

Question

Starting at \(t=0\), a concentrated load of magnitude \(F_{0}\) moves with a constant velocity \(v_{0}\) along a semi-infinite string. In this case the wave equation becomes 21 $$ a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}+F_{0} \delta\left(t-\frac{x}{v_{0}}\right) $$ where \(\delta\left(t-x / v_{0}\right)\) is the Dirac delta function. Solve this PDE subject to $$ \begin{aligned} &u(0, t)=0, \quad \lim _{x \rightarrow \infty} u(x, t)=0, t>0 \\ &u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0, x>0 \end{aligned} $$ (a) when \(v_{0} \neq a\), and (b) when \(v_{0}=a\)

Step-by-Step Solution

Answer

For (a) when \(v_0 \neq a\), use D'Alembert's solution. For (b) when \(v_0 = a\), account for resonance with the wave speed.

Step 1: Identify the wave equation

The given wave equation is \[ a^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} + F_{0} \delta\left(t - \frac{x}{v_0}\right). \]The terms represent the elastic wave propagation with a moving load, incorporating the Dirac delta function to model the concentrated load at specific time-space locations.

Step 2: Determine boundary and initial conditions

The boundary conditions are \( u(0, t) = 0 \) and \( \lim_{x \to \infty} u(x, t) = 0 \). The initial conditions are \( u(x, 0) = 0 \) and \( \frac{\partial u}{\partial t} \big|_{t=0} = 0 \). These conditions ensure the wave starts from rest and returns to zero at infinity, while being fixed at the origin for all times.

Step 3: Solve part (a) using D'Alembert's solution for \(v_0 \neq a\)

For \( v_0 eq a \), the equation can be solved using D'Alembert's principle. The solution is of the form:\[ u(x, t) = \frac{F_0}{2a} \left( H(t - \frac{x}{a}) - H(t - \frac{x}{v_0}) \right). \]Here, \( H(t) \) is the Heaviside function, ensuring that the effect of the force is wave-like along the string as \( t \geq \frac{x}{a} \).

Step 4: Verify solution for part (a)

To ensure correctness, substitute back the derived solution into the original wave equation and verify it satisfies both the equation and the initial/boundary conditions. This involves checking the transformations of the partial derivatives and affirming the initial rest state is satisfied.

Step 5: Solve part (b) when \(v_0 = a\)

For \( v_0 = a \), the load moves at the wave speed, which causes a singularity. The solution can be expressed using the influence of the delta function:\[ u(x, t) = \frac{F_0}{2a} H(t - \frac{x}{a}) \left( t - \frac{x}{a} \right). \]This solution accounts for both the singularity and the initial rest conditions, resonating at the wave speed.

Step 6: Verify solution for part (b)

Check that the solution for \( v_0 = a \) holds by plugging it back into the equation, ensuring it maintains the initial conditions and the effect of a resonant force throughout its course along the string.

Key Concepts

Partial Differential Equation

A partial differential equation (PDE) involves partial derivatives of unknown functions with respect to more than one independent variable. In the context of the wave equation, a PDE is used to describe the propagation of waves along a medium.
The general form of the wave equation is \[ a^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} + F_{0} \delta\left(t - \frac{x}{v_0}\right). \]This equation combines second derivatives in both space, \( x \), and time, \( t \), creating a link between how a wave evolves over time and its spatial nature.
The term \( \delta\left(t-x / v_{0}\right) \) in the equation represents a disturbance applied at a specific time and location, modeling a load moving along the string. Understanding PDEs is crucial, as they model many physical processes, from heat distribution to quantum mechanics. By analyzing these equations, you can predict how systems change and behave over time and space.

Initial Conditions

Initial conditions specify the state of a wave at the very beginning of observation, which is vital to solve a PDE like the wave equation. In the exercise, we have:

• \( u(x, 0) = 0 \): This condition states that initially, the wave is at rest and does not have any displacement anywhere along the string.
• \( \frac{\partial u}{\partial t} \big|_{t=0} = 0 \): This tells us that initially, there is no velocity or initial wave motion.

Initial conditions ensure that we have a complete picture of the system from the start, allowing us to predict future behavior. They form a critical part of the overall solution process, as they are necessary to integrate partial differential equations effectively.

Dirac Delta Function

The Dirac delta function, \( \delta(t-a) \), is a special function often used in physics and engineering to represent an idealized point load or impulse occurring at a specific point in time or space. In the wave equation given in the exercise:
\[ F_{0} \delta\left(t - \frac{x}{v_0}\right) \]the Dirac delta function represents a concentrated force of magnitude \( F_0 \) moving along the string.

• It has the property that it is zero everywhere except at \( t = \frac{x}{v_0} \).
• Its integral over the entire space is equal to one, effectively capturing an impulse or load at that specific point.

In solving the wave equation, the Dirac delta function helps model the effects of real-world concentrated loads in a simplified mathematical form, providing solutions that reflect the intensity and precise location of applied forces.

Boundary Conditions

Boundary conditions in PDE problems describe the behavior of the wave at the edges or limits of the domain under investigation. For the given exercise, the boundary conditions are:

• \( u(0, t) = 0 \): It implies that the displacement is zero at the origin, meaning the string is fixed at the starting point and cannot move.


• \( \lim_{x \to \infty} u(x, t) = 0 \): As \( x \) approaches infinity, the solution returns to zero, showing that disturbances caused by the wave dissipate and eventually vanish at a distance.

Boundary conditions are essential for ensuring the solution is physically realistic, as they dictate how the wave interacts with its environment. Proper understanding and implementation of boundary conditions guarantee that the solution is consistent and complete, fulfilling both the physical context and mathematical requirements of the problem.