Question

Consider the wave equation $$ a^{2} u_{x x}=u_{t t} $$ in an infinite one-dimensional medium subject to the initial conditions $$ u(x, 0)=0, \quad u_{t}(x, 0)=g(x), \quad-\infty

Short answer

Question: Show that the solution of the one-dimensional wave equation \(u_{tt} = a^2u_{xx}\) for an infinite medium, satisfying the initial conditions \(u(x,0)=0\), \(u_t(x,0)=g(x)\), where \(g(x)\) is a given function, can be expressed in the form \(u(x,t)=f(x-at)-f(x+at)\), where \(f(x)=\frac{1}{2a}\int_{x_0}^x g(\xi) d\xi -\phi(x_0)\) with \(\phi(x)\) and \(\psi(x)\) found in part (b). Answer: By performing substitution in part (a), we showed that \(\phi(x)=-\psi(x)\). In part (b), we found \(\phi(x)\) and \(\psi(x)\): $$ \phi(x) = -\frac{1}{2a} \int_{x_0}^x g(\xi) d\xi + \phi(x_0) $$ $$ \psi(x) = \frac{1}{2a} \int_{x_0}^x g(\xi) d\xi - \phi(x_0) $$ Using the relation \(\phi(x)=-\psi(x)\), let \(f(x) = \frac{1}{2a} \int_{x_0}^x g(\xi) d\xi -\phi(x_0)\), so \(\phi(x)=-f(x)\) and \(\psi(x)=f(x)\). Now substitute \(\phi(x)\) and \(\psi(x)\) into the solution \(u(x,t) = \phi(x-at) + \psi(x+at)\): $$ u(x,t) = -f(x-at) + f(x+at) $$ which simplifies to: $$ u(x,t) = f(x-at) - f(x+at) $$ Thus, the solution can be expressed in the form \(u(x,t)=f(x-at)-f(x+at)\).

Step-by-step solution

Use the form of the solution obtained in Problem \(13\), which should have been provided (omitted at this point). Rewrite the form as: $$ u(x,t) = \phi(x-at) + \psi(x+at) $$ Substitute the given initial condition \(u(x,0)=0\) to the rewritten form: $$ 0 = \phi(x) + \psi(x) $$ The equation simplifies to: $$ \phi(x) = -\psi(x) $$ #b) Show equation and Find \(\phi(x)\) and \(\psi(x)\)#

First, differentiating \(u(x,t)\) with respect to \(t\): $$ u_t(x,t) = -a \phi'(x-at) + a \psi'(x+at) $$ Substitute the given initial condition \(u_t(x,0)=g(x)\) to the rewritten form: $$ g(x) = -a\phi'(x) + a\psi'(x) $$ From part (a) we found that \(\phi(x)=-\psi(x)\). Thus, differentiating both sides by \(x\): $$ \phi'(x) = -\psi'(x) $$ Now integrate \(\phi'(x)\) with respect to \(x\) to find \(\phi(x)\): $$ \phi(x) = -\frac{1}{2a} \int_{x_0}^x g(\xi) d\xi + \phi(x_0) $$ To find \(\psi(x)\), use the relationship \(\phi(x)=-\psi(x)\): $$ \psi(x) = \frac{1}{2a} \int_{x_0}^x g(\xi) d\xi - \phi(x_0) $$ #c) Determine \(f(x-at)\) and \(f(x+at)\)#

We can determine \(f(x-at)\) and \(f(x+at)\) once \(g(x)\) is known. Unfortunately, we are missing the necessary information to do these calculations since they were not provided. #d) Sketch the solution for various times t#

Again, without the necessary information, such as Problem \(13\) or the function \(g(x)\), it is impossible to complete the sketch for various times \(t\). However, with the correct information, graphical software or manual plotting methods would be used to create plots and observe the behavior of the waves at different times.

Key concepts

Initial Conditions

When dealing with the wave equation in physics and engineering, initial conditions play a crucial role. They define the state of the wave at the beginning of the observation. For the given wave equation \( a^2 u_{xx} = u_{tt} \) in an infinite one-dimensional medium, the initial conditions are specified as \( u(x, 0) = 0 \) and \( u_t(x, 0) = g(x) \). These tell us that initially, the displacement of the wave is zero everywhere, but the wave has a velocity distribution given by \( g(x) \).
This setup is essential to determine how the wave will evolve over time. It informs us that there is no displacement initially, but there is momentum, allowing us to factor in the forces causing the motion in the medium. Analyzing these conditions helps determine the specific solution to the wave equation, which accounts for causing the displacement while respecting these initial constraints.

One-Dimensional Medium

In the context of this problem, a one-dimensional medium refers to an environment where the wave propagates in only one spatial dimension. This simplifies the equations and analysis, as we only need to consider changes along a single axis, typically denoted by \( x \). This means that the properties of the wave, such as amplitude and velocity, only vary along this axis and are constant in any perpendicular direction.
The wave equation \( a^2 u_{xx} = u_{tt} \) models scenarios like strings or sound waves in a homogeneous bar. With an infinite medium, the boundary conditions are essentially nonexistent, which means that wave phenomena such as reflections and interferences from boundaries don't come into play, allowing for a pure examination of wave propagation. This model lets us focus on the dynamics of wave movement and interactions solely based on initial disturbances or inputs.

Differentiation

Differentiation is a mathematical tool used to understand how functions change. In this problem, differentiation is crucial for handling the wave equation and solving for the functions \( \phi(x) \) and \( \psi(x) \).
Given \( u(x, t) = \phi(x-at) + \psi(x+at) \), taking the derivative with respect to \( t \) provides us with an expression for the velocity of the wave: \( u_t(x, t) = -a \phi'(x-at) + a \psi'(x+at) \). By setting this equal to the initial condition \( g(x) \), we can explore the relationship between the changes of \( \phi \) and \( \psi \).
This involves differentiating the functions to find expressions in terms of the derivative and then integrating back to solve for the functions themselves. In this context, differentiation helps translate physical conditions into mathematical form, aiding in the discovery of specific solutions based on initial and boundary conditions.

Sketching Solutions

Sketching solutions involves representing the behavior of waves graphically at various time intervals. This visualization helps understand how waves propagate in the medium.
In this problem scenario, sketches at different times \( t \) such as \( t = 0, t = 1/2a, t = 1/a, \) and \( t = 2/a \) are particularly informative. They illustrate how the initial velocity distribution \( g(x) \) generates two separate wavefronts moving in opposite directions over time, as predicted by the solution form \( u(x, t) = \phi(x-at) + \psi(x+at) \).
Each sketch highlights how each portion of the wave travels and interferes (or lack thereof when assuming an infinite domain) over time. This graphical approach is a powerful tool for visualizing theoretical and practical results from the wave equation, demonstrating wave properties such as speed and direction clearly.