Question

Use the result of Exercise 63 to find a solution of $$ u_{t t}=a^{2} u_{x x}, \quad-\infty< x <\infty $$ that satisfies the given initial conditions. $$ u(x, 0)=x \sin x, \quad u_{t}(x, 0)=\sin x, \quad-\infty

Short answer

Answer: The function u(x, t) that satisfies the given wave equation and initial conditions is: $$ u(x,t) = \frac{1}{2}\left[(x + at) \sin(x + at) + (x - at) \sin(x - at)\right] + \frac{1}{2a}[\cos(x - at) - \cos(x + at)] $$

Step-by-step solution

Apply d'Alembert's formula

d'Alembert's formula for the one-dimensional wave equation is given by: $$ u(x, t) = \frac{1}{2} [ f(x + at) + f(x - at)] + \frac{1}{2a} \int_{x-at}^{x+at} g(s) ds $$ where f(x) represents the initial displacement and g(x) represents the initial velocity. In this problem, the initial displacement is given by \(u(x, 0) = x \sin x\) and the initial velocity is given by \(u_t(x, 0) = \sin x\). Therefore, we have f(x) = x sin x and g(x) = sin x.

Calculate f(x+at) and f(x-at)

To apply d'Alembert's formula, we need to calculate the expressions f(x+at) and f(x-at). Plugging in f(x) = x sin x: $$ f(x + at) = (x + at) \sin (x + at) $$ and $$ f(x - at) = (x - at) \sin (x - at) $$

Integrate g(x) to apply d'Alembert's formula

Now, we need to find the integral of g(s) with respect to s, within the limits x-at and x+at: $$ \int_{x - at}^{x + at} \sin s ds $$ To evaluate this integral, we compute the antiderivative of sin s: $$ \int \sin s ds = -\cos s + C $$ Now, we can evaluate the integral: $$ \int_{x - at}^{x + at} \sin s ds = \left[-\cos(x+at) - (-\cos(x-at))\right] = \cos(x - at) - \cos(x + at) $$

Substitute the results into d'Alembert's formula

Now that we have computed the necessary expressions, we can substitute them into d'Alembert's formula: $$ u(x,t) = \frac{1}{2}\left[(x + at) \sin(x + at) + (x - at) \sin(x - at)\right] + \frac{1}{2a}[\cos(x - at) - \cos(x + at)] $$ This function u(x, t) represents the solution to the given wave equation with the specified initial conditions.

Key concepts

Wave Equation

The wave equation is a fundamental concept in physics and engineering that describes how waves propagate through different media. It's an example of a partial differential equation (PDE) which is used in various fields like acoustics, electromagnetism, and fluid dynamics. The one-dimensional wave equation can be written as
\[ u_{tt} = a^2 u_{xx} \],
where \( u(x, t) \) is the displacement at position \( x \) and time \( t \), \( u_{tt} \) is the second time derivative of \( u \), and \( u_{xx} \) is the second spatial derivative of \( u \). The constant \( a \) represents the speed of wave propagation. Imagine dropping a stone into a still pond and watching the ripples; the wave equation helps predict the pattern of those ripples over time.

Partial Differential Equations

Partial differential equations (PDEs) are equations that involve rates of change with respect to more than one variable. These are vital in mathematically modeling physical systems where multiple factors influence the behavior of the system. PDEs extend beyond the typical one-variable equations (ordinary differential equations, ODEs) by considering functions of several variables and their partial derivatives, which allows them to describe phenomena like heat diffusion, wave propagation, or the growth of biological populations. For example, when you're boiling water, the temperature changes not just over time but also across different locations in the pot; this spatial and temporal change can be depicted by PDEs. They're tougher to solve than ODEs because they require a combination of initial and boundary conditions to come up with a unique solution.

Initial Value Problems

Initial value problems are a specific type of problem in the study of differential equations, whether they are ordinary differential equations or partial differential equations. These problems specify the state of the system at the beginning of the observation,
which is used to determine the unique solution of the equation throughout the time period of interest. The term 'initial value' refers to the given values of the function and its derivatives at the initial time. In the context of the wave equation, initial value problems describe the initial shape and velocity of the wave. Imagine you're watching the first few frames of a movie showing a guitar string being plucked; the initial shape and motion of the string right after it's plucked are the initial values. Accurately capturing these initial conditions is crucial for predicting the subsequent motion of the string and the sound it will produce.