Problem 15 - Advanced Engineering Mathematics

Question

A string is stretched and eecured on the \(x\)-axis at \(x=0\) and \(x=\pi\) for \(t>0\). If the transverse vibrations take place in a medium that imparts a resistance proportional to the instantaneous velocity, then the wave equation takes on the form $$ \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}+2 \beta \frac{\partial u}{\partial t}, \quad 0<\beta<1, \quad t>0 $$ Find the displacement \(u(x, t)\) if the string starts from rest from the initial displacement \(f(x)\).

Step-by-Step Solution

Answer

The displacement is given by superposition of sinusoidal modes with damped exponential behavior.

Step 1: Understanding the Wave Equation

The given wave equation is \( \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} + 2 \beta \frac{\partial u}{\partial t} \). It represents a damped wave due to the resistive term \(2\beta \frac{\partial u}{\partial t}\). The boundary conditions are given by the fixed points \(x = 0\) and \(x = \pi\).

Step 2: Separation of Variables

Assume a solution of the form \( u(x,t) = X(x)T(t) \). Substituting into the wave equation transforms it into \( X''(x)T(t) = X(x)[T''(t) + 2\beta T'(t)] \). Setting \( \frac{X''(x)}{X(x)} = \frac{T''(t) + 2\beta T'(t)}{T(t)} = -\lambda \) separates the spatial and temporal components.

Step 3: Solving the Spatial Equation

Solve the spatial equation \( X''(x) + \lambda X(x) = 0 \). The boundary conditions \( X(0) = 0 \) and \( X(\pi) = 0 \) imply those solutions are sinusoidal. Assume \( X(x) = A\sin(nx) \) where \( \lambda = n^2 \) for \( n = 1, 2, 3, \ldots \).

Step 4: Solving the Temporal Equation

The time equation \( T''(t) + 2\beta T'(t) + n^2 T(t) = 0 \) is a second-order linear differential equation with constant coefficients. The characteristic equation is \( r^2 + 2\beta r + n^2 = 0 \). Solve for \( r \) using the quadratic formula \( r = -\beta \pm \sqrt{\beta^2 - n^2} \).

Step 5: General Solution for Time Component

Based on the roots found in Step 4, if \( \beta^2 > n^2 \), the solutions are overdamped and take the form \( T(t) = C_1 e^{(-\beta + \sqrt{\beta^2 - n^2})t} + C_2 e^{(-\beta - \sqrt{\beta^2 - n^2})t} \). Adjust the form accordingly if the roots are complex or repeated.

Step 6: Construct General Solution for Displacement

The general solution \( u(x,t) \) is a sum of all the possible products \( X_n(x)T_n(t) \), considering all integer \( n \). Thus, \( u(x, t) = \sum_{n=1}^{\infty} A_n \sin(nx) \left[ C_1 e^{(-\beta + \sqrt{\beta^2 - n^2})t} + C_2 e^{(-\beta - \sqrt{\beta^2 - n^2})t} \right] \).

Step 7: Applying Initial Conditions

Apply the initial conditions to find the specific coefficients \( A_n, C_1, \) and \( C_2 \). Since the string starts from rest, the initial condition is \( \frac{\partial u}{\partial t}(x, 0) = 0 \). Solving this, alongside \( u(x, 0) = f(x) \), determines the constants based on Fourier analysis of \( f(x) \).

Key Concepts

Partial Differential Equations

Partial differential equations (PDEs) are equations that involve rates of change with respect to multiple variables. In this exercise, we have a PDE describing a damped wave equation. The wave equation is given by: \[\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} + 2\beta \frac{\partial u}{\partial t}\]Here, \(u(x, t)\) represents the displacement of the string at position \(x\) and time \(t\). The term \(2\beta \frac{\partial u}{\partial t}\) introduces damping, proportional to the velocity. This sets it apart from the classic wave equation, which does not have such a damping term.
The damped wave equation is fundamental in understanding how various mediums affect wave propagation. Whenever resistive forces come into play (like friction or air resistance), the equations modeling the waves need to account for this damping effect, making PDEs a crucial tool in the toolbox of mathematicians and physicists alike.

Boundary Conditions

Boundary conditions are essential for solving differential equations, especially when modeling physical systems. They define the behavior of a solution at the boundaries of the domain. In the given exercise, the string is fixed at both ends, specifically at \(x = 0\) and \(x = \pi\).
These conditions are expressed as:

• \(u(0, t) = 0\), and
• \(u(\pi, t) = 0\).

These boundary conditions imply that the displacement at both ends of the string remains zero for all time \(t > 0\). Physically, this means the ends of the string are tightly secured, preventing any movement. Altering the boundary conditions could result in markedly different types of solutions, highlighting the critical role they play in accurately solving PDEs.

Characteristic Equation

The characteristic equation is a crucial element in solving linear differential equations, especially those with constant coefficients. When faced with the temporal part of our PDE: \[T''(t) + 2\beta T'(t) + n^2 T(t) = 0\]We derive the characteristic equation. For this exercise, it takes the form:\[r^2 + 2\beta r + n^2 = 0\]This quadratic equation in \(r\) helps determine the behavior of the temporal solution.
Depending on the discriminant \(\beta^2 - n^2\), the nature of the roots \(r\) can vary:

• If \(\beta^2 > n^2\), the roots are real and distinct, indicating an overdamped system with exponential decay solutions.
• If \(\beta^2 = n^2\), repeated roots suggest critical damping.
• If \(\beta^2 < n^2\), complex roots imply oscillatory behavior, typical for underdamped systems.

The specific form of \(T(t)\) is then tailored using these roots, enabling us to fully characterize the temporal aspect of the wave solution.

Fourier Analysis

Fourier analysis is a powerful technique used to solve PDEs by decomposing complex functions into simpler sinusoidal components. In this exercise, Fourier analysis helps in finding the coefficients for the solution of the damped wave equation. The general solution of the displacement is given by:\[u(x, t) = \sum_{n=1}^{\infty} A_n \sin(nx) \left[ C_1 e^{(-\beta + \sqrt{\beta^2 - n^2})t} + C_2 e^{(-\beta - \sqrt{\beta^2 - n^2})t} \right]\]After applying initial conditions, which include the rest start of the string and initial displacement \(f(x)\), we perform Fourier analysis to expand \(f(x)\) into a series:

• Each sine term \(\sin(nx)\) represents a specific mode of vibration of the string, resonating within the given boundary conditions.
• The coefficients \(A_n\) are determined by projecting \(f(x)\) onto these modes via integration, using the orthogonality of sine functions over the interval \([0, \pi]\).

Thus, Fourier analysis allows us to construct a complete solution that satisfies both the PDE and the initial conditions, showing how a complex wave pattern emerges from simple sinusoidal vibrations.