Problem 11 - Advanced Engineering Mathematics

Question

From (8) in the text we have \\[ u(x, t)=\sum_{n=1}^{\infty}\left(A_{n} \cos \frac{n \pi a}{L} t+B_{n} \sin \frac{n \pi a}{L} t\right) \sin \frac{n \pi}{L} x \\] since \(u_{t}(x, 0)=g(x)=0\) we have \(B_{n}=0\) and \\[ \begin{aligned} u(x, t) &=\sum_{n=1}^{\infty} A_{n} \cos \frac{n \pi a}{L} t \sin \frac{n \pi}{L} x \\ &=\sum_{n=1}^{\infty} A_{n} \frac{1}{2}\left[\sin \left(\frac{n \pi}{L} x+\frac{n \pi a}{L} t\right)+\sin \left(\frac{n \pi}{L} x-\frac{n \pi a}{L} t\right)\right] \\ &=\frac{1}{2} \sum_{n=1}^{\infty} A_{n}\left[\sin \frac{n \pi}{L}(x+a t)+\sin \frac{n \pi}{L}(x-a t)\right] \end{aligned} \\] From \\[ u(x, 0)=f(x)=\sum_{n=1}^{\infty} A_{n} \sin \frac{n \pi}{L} x \\] we identify \\[ f(x+a t)=\sum_{n=1}^{\infty} A_{n} \sin \frac{n \pi}{L}(x+a t) \\] and \\[ f(x-a t)=\sum_{n=1}^{\infty} A_{n} \sin \frac{n \pi}{L}(x-a t) \\] so that \\[ u(x, t)=\frac{1}{2}[f(x+a t)+f(x-a t)] \\]

Step-by-Step Solution

Answer

\( u(x, t) = \frac{1}{2}[f(x+at) + f(x-at)] \) represents traveling waves.

Step 1: Initial Condition Setup

We start with the condition \( u_{t}(x, 0) = g(x) = 0 \). This implies that for all terms with sine, the coefficient must be zero, because the derivative with respect to time of a cosine function at \( t=0 \) could be non-zero. Therefore, \( B_n = 0 \), and the expression simplifies to: \[ u(x, t)=\sum_{n=1}^{\infty} A_{n} \cos \frac{n \pi a}{L} t \sin \frac{n \pi}{L} x \] This means our solution only contains cosine terms in time.

Step 2: Rewrite Using Trigonometric Identity

The next step involves rewriting the cosine term using the trigonometric identity for cosine: \( \cos(\theta) = \frac{1}{2}(e^{i\theta} + e^{-i\theta}) \) or its trigonometric counterpart more suited for sines. Applying this: \[ \cos \frac{n \pi a}{L} t = \frac{1}{2}\left( \sin\left(\frac{n \pi}{L} x + \frac{n \pi a}{L} t\right) + \sin\left(\frac{n \pi}{L} x - \frac{n \pi a}{L} t\right) \right) \] Substitute this back into our solution: \[ u(x, t) = \frac{1}{2} \sum_{n=1}^{\infty} A_{n}\left[\sin \frac{n \pi}{L}(x+a t) + \sin \frac{n \pi}{L}(x-a t) \right] \]

Step 3: Using the Initial Condition for Displacement

Given the initial condition \( u(x, 0) = f(x) = \sum_{n=1}^{\infty} A_{n} \sin \frac{n \pi}{L} x \), recognize that the solution matches this at \( t = 0 \). This is used to identify how \( u(x, t) \) transforms with time: if \( f(x) \) is the displacement at time zero, then the expression for displacement evolves with time in terms of traveling waves. Therefore, we identify: \[ f(x+a t) = \sum_{n=1}^{\infty} A_{n} \sin \frac{n \pi}{L} (x+a t) \] and \[ f(x-a t) = \sum_{n=1}^{\infty} A_{n} \sin \frac{n \pi}{L} (x-a t) \]

Step 4: Deriving the Final Expression

Combine these transformed functions to reformulate \( u(x, t) \) with the form: \[ u(x, t) = \frac{1}{2} [f(x+at) + f(x-at)] \] This expression shows that \( u(x, t) \) is the average of the function \( f \) evaluated at two different positions shifted by \( at \) forwards and backwards, representing waves traveling in the positive and negative directions, respectively.

Key Concepts

Wave Equation

The wave equation is one of the foundational partial differential equations that describe how waves, such as sound, light, and water waves, propagate through a medium. It is expressed mathematically by the function \( u(x, t) \), which describes the wave's displacement at position \( x \) and time \( t \).
The standard form of the wave equation is given by:

• \( \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \)

where \( c \) is the speed of wave propagation.
This equation captures the essence of how disturbances in a medium spread out over time. In particular, solutions to this equation can represent a wide variety of waveforms, including simple harmonic waves typically modeled using sine and cosine functions.

• Wave behavior includes reflection, refraction, and superposition.
• The equation can be solved using various mathematical methods, including separation of variables and Fourier series.

Initial Condition

An initial condition in the context of partial differential equations (PDEs) specifies the state of the wave at a starting time, typically \( t=0 \). Initial conditions are critical for uniquely determining the solution of a PDE.
For the wave equation, initial conditions can involve:

• The initial displacement \( u(x, 0) \). This represents the shape of the wave at the starting time.
• The initial velocity \( u_t(x, 0) \). This represents how the wave is changing at the starting time.

In our example, the initial condition \( u_t(x, 0) = g(x) = 0 \) implies that the initial velocity of the wave is zero everywhere. This helps simplify the analysis by eliminating certain terms (like those involving \( B_n \)) from the solution, focusing only on solutions described by cosine functions in time.

Trigonometric Identity

Trigonometric identities offer crucial mathematical tools to simplify and manipulate expressions involving trigonometric functions. These identities help convert complex expressions into simpler forms. In our wave equation problem, we used:

• The identity: \( \cos(\theta) = \frac{1}{2}(\sin(\theta + \phi) + \sin(\theta - \phi)) \)

By rewriting cosine in terms of sine, we leverage the fact that sine functions are well-suited to represent wave motions due to their periodic properties. This step is pivotal in transforming our equation into a form that directly relates to initial waveforms.
Trigonometric identities are especially useful because they:

• Provide alternative perspectives on complex wave forms.
• Simplify integration and differentiation in calculus.

To master wave-related PDEs, becoming comfortable with various trigonometric identities is invaluable.

Fourier Series

The Fourier series allows us to express a function as a sum of sines and cosines. This technique is fundamental to solving partial differential equations like the wave equation, particularly because of its ability to break down complex periodic functions into simpler oscillatory components.
In the context of our problem, the solution \( u(x, t) \) is expressed as a Fourier series with terms \( A_n \sin(\frac{n \pi}{L} x) \) and \( \cos(\frac{n \pi a}{L} t) \).
Fourier series is beneficial because it:

• Transforms complex shapes into sums of simple, understandable components.
• Converges to the function \( u(x, t) \) as more terms are included.

Using Fourier series provides a systematic way to approximate solutions to differential equations, making it a powerful technique in mathematical analysis of wave phenomena.
When dealing with these series, understanding convergence and the role of each term is critical for accurately describing physical systems.