Problem 13 - Applied Partial Differential Equations

Question

Consider the one-dimensional infinite space wave equation with a periodic source of frequency \(\omega\) : $$ \frac{\partial^{2} \psi}{\partial t^{2}}=c^{2} \frac{\partial^{2} \psi}{\partial x^{2}}+g(x) e^{-i \omega} $$ (a) Show that a particular solution \(\phi-u(x) c^{-\text {ine }}\) of \((8.3 .46)\) is obtained if \(u\) satisfies a nonhomogeneous Helmholtz equation $$ \frac{d^{2} u}{d x^{2}}+k^{2} u=f(x) $$ *(b) The Green's function \(G\left(x, x_{0}\right)\) satisfies $$ \frac{d^{2} G}{d x^{2}}+k^{2} G=\delta\left(x-x_{0}\right) $$ Determine this infinite space Green's function so that the corresponding \(\phi(x, t)\) is an outward propagating wave. (c) Determine a particular solution of (8.3.46) above.

Step-by-Step Solution

Answer

The particular solution of the one-dimensional infinite space wave equation is given by a real function u(x) that satisfies a nonhomogeneous Helmholtz equation. The corresponding Green's function is given by \(-\frac{1}{2k} \exp(-k|x-x_0|)\). A specific solution to the wave equation can thus be obtained from the integral \(\int dx_0 G(x,x_0) g(x_0)\).

Step 1: Show that u satisfies the given equation

We start with the given wave equation \(\frac{\partial^{2} \psi}{\partial t^{2}}=c^{2} \frac{\partial^{2}\psi}{\partial x^{2}}+g(x) e^{-i \omega} \) . Setting \(\psi = \phi-u(x) e^{-i \omega t}\), where \(\phi\) and \(u(x)\) are both real, and substituting this result back into the wave equation, we collect the real parts and get \[c^{2} \frac{\partial^{2} u}{\partial x^{2}} = g(x)\] which is a nonhomogeneous Helmholtz equation.

Step 2: Determine the Green's function

The Green's function \(G\left(x, x_{0}\right)\) satisfies the equation \[\frac{d^{2} G}{d x^{2}}+k^{2} G=\delta\left(x-x_{0}\right)\] Solving this differential equation with the delta function right side, we obtain \[G(x, x_0) = -\frac{1}{2k} \exp(-k|x-x_0|)\]

Step 3: Determine a particular solution of the initial wave equation

With the function \(u(x)\) from step 1 and the Green's function \(G(x, x_0)\) from step 2, we can then obtain a particular solution of the initial wave equation. We have \[\phi(x, t) = \int dx_0 G(x,x_0) g(x_0)\]

Step 4: Calculation

Putting in the specific solution for the Green's function from step 2 \(G(x, x_0) = -\frac{1}{2k} \exp(-k|x-x_0|)\) into the solution we found in step 3, we finally get the particular solution of the wave equation from the beginning of the exercise. The exact form will depend on the function \(g(x_0)\) that we have not specified yet.

Key Concepts

Green's Function

In mathematics, a Green's function is a powerful tool used to solve differential equations. It's especially useful when dealing with linear differential equations, such as those appearing in wave and Helmholtz equations. It essentially represents the response of the system to a point source or impulse. In our situation, the Green's function, denoted as \(G(x, x_0)\), satisfies the equation:

• \(\frac{d^{2} G}{d x^{2}}+k^{2} G=\delta(x-x_0)\)

Where \(\delta(x-x_0)\) is the Dirac delta function, representing a point source at \(x_0\). The Green's function helps us express complex functions as integrals and provides a method to find particular solutions by convolving \(G\) with source terms in equations.

Helmholtz Equation

The Helmholtz equation is a fundamental partial differential equation often encountered in physics. It describes how physical quantities like electromagnetic fields, sound, or heat propagate in space. The form we are dealing with is:

• \(\frac{d^{2} u}{d x^{2}}+k^{2} u=f(x)\)

This is a nonhomogeneous equation where \(k\) is related to the wave number and represents how disturbances behave in an infinite medium. Solving it requires finding a particular solution \(u(x)\) that satisfies the equation for a given source term \(f(x)\).

Particular Solution

A particular solution is a specific solution to a nonhomogeneous differential equation that fits the given conditions or sources. In our problem, we need to find a particular solution for the wave equation.The general form involves both the homogeneous and particular solutions:

• \(\phi = u(x)e^{-i \omega t}\)

To determine \(u(x)\), we solve the corresponding Helmholtz equation. The particular solution incorporates the behavior of external forces or sources described by \(g(x)\).

Infinite Space

Infinite space refers to problems that extend without boundaries in one or more dimensions. In our example, the wave equation is considered in one-dimensional infinite space. This assumption simplifies the analysis by allowing us to ignore complications introduced by edges or boundaries. Working in infinite space requires:

• Solutions that are applicable in all regions.
• No need for boundary conditions.

This is why techniques like Green's functions are particularly useful, capturing the behavior in an unbounded domain.

Periodic Source

A periodic source oscillates at a regular frequency, introducing a repeating disturbance into a system. Here, the source is given by \(g(x) e^{-i \omega}\). This drives the wave equation by providing consistent energy input at frequency \(\omega\).Periodic sources are crucial because they:

• Introduce sinusoidal elements into solutions.
• Lead to steady-state solutions reflecting the driving frequency.

Understanding these sources helps in analyzing resonance and stability within the system.

Frequency

Frequency, denoted by \(\omega\), is a crucial concept in analyzing oscillatory behavior. It defines how often a wave oscillates per unit time and directly influences how solutions behave.In this context:

• It sets the temporal variation in the wave equation.
• Solutions involving \(\omega\) describe steady states influenced by this frequency.

Frequency plays a key role in resonance phenomena and system responses, making it a fundamental element in the study of waves.

Outward Propagating Wave

An outward propagating wave moves away from a source, typically depicting an energy or disturbance spreading in space. For the described wave, this behavior is modeled as solutions that capture how such dynamics unfold over an infinite medium. For outward propagation:

• Boundary conditions might not be needed in infinite space.
• Green's functions help show how waves emanate from point sources.

Understanding this concept is vital for fields like acoustics, optics, and electromagnetic theory, where wave propagation is central.