Problem 8 - Calculus Of Variations, With Applications To Physics And Engineering

Question

Given the inhomogeneous boundary condition \(w=g(x, y)\) on \(C\) for the membrane equation $$ \sigma \frac{\partial^{2} w}{\partial t^{2}}=\tau \nabla^{2} w $$ show that we may write \(w=u(x, y, t)+v(x, y)\) where \(u=0\) and \(v=g(x, y)\) on \(C\), \(\nabla^{2} v=0\) and \(\sigma \ddot{u}=\tau \nabla^{2} u\) in \(D .\) Thus we reduce a membrane problem with an inhomogeneous boundary condition to one having a homogeneous boundary condition, plus a solution of the two- dimensional Laplace's equation with boundary values prescribed. (The latter part of the problem is discussed in Chap. 12.)

Step-by-Step Solution

Answer

The solution of the membrane equation with the given inhomogeneous boundary condition can be expressed as the sum of two functions: \(u(x, y, t)\) and \(v(x, y)\) such that \(u\) satisfies the wave equation with a homogeneous boundary condition and \(v\) satisfies the Laplace's equation with the prescribed boundary conditions.

Step 1: Express w as the sum of u and v

Given the inhomogeneous boundary conditions, we can express \(w\) as \(w=u(x, y, t)+v(x, y)\). Here, \(u=0\) and \(v=g(x, y)\) on \(C\). Note that \(u\) depends on time \(t\) while \(v\) does not.

Step 2: Apply Laplace's equation to v

With \(w\) written in terms of \(u\) and \(v\), we apply Laplace's equation \(\nabla^{2}\) to \(v\) and obtain \(\nabla^{2}v = 0\). This equation means that \(v\) satisfies the two-dimensional Laplace's equation.

Step 3: Substitute w into the wave equation

Next, substitute the expression for \(w\) into the given wave equation. We get \(\sigma \frac{\partial^{2} (u+v)}{\partial t^{2}} = \tau \nabla^{2} (u+v)\).

Step 4: Separate the resultant equation into two parts

Notice that when we differentiate \(v\) with respect to \(t\), we get 0 because it does not depend on time. Similarly, when we apply Laplace operator on \(v\), we get 0 (as per step 2). Therefore, we can rewrite our equation separately for \(u\) and \(v\) to get \(\sigma \frac{\partial^{2}u}{\partial t^{2}} = \tau \nabla^{2}u\). This shows that \(u\) satisfies a wave equation with a homogeneous boundary condition since from the beginning, we have mentioned \(u=0\) on \(C\).

Step 5: Interpretation

In this way, we have reduced a membrane problem with an inhomogeneous boundary condition to two separate problems: one is the solution of the two-dimensional Laplace's equation with prescribed boundary values (represented by \(v\)), and the other is the solution of the wave equation with a homogenous boundary condition (represented by \(u\)).

Key Concepts

Boundary Conditions

Boundary conditions are critical in the calculus of variations, especially in solving differential equations. They specify the behavior of a solution at the boundary of its domain. In this problem, we have an inhomogeneous boundary condition given by \( w=g(x, y) \) on boundary \( C \). This means at every point \( (x, y) \) on the boundary \( C \), the function \( w \) equals \( g(x, y) \).

• **Inhomogeneous Boundary Conditions:** These conditions specify that the solution should match a certain given function at the boundary.
• **Homogeneous Boundary Conditions:** These conditions often set the function to zero on the boundary, simplifying the problem.

To transform an inhomogeneous boundary problem into a homogeneous one, we decompose the function \( w \) into two parts: \( w = u(x, y, t) + v(x, y) \). Here, \( u \) is set to zero on \( C \), making it satisfy a homogeneous boundary condition, while \( v \) takes on the inhomogeneous component \( g(x, y) \).
By doing this, we can handle each part separately: \( u \) as a wave function and \( v \) as a solution to Laplace's equation.

Laplace's Equation

Laplace's equation is a second-order partial differential equation, key in the study of potential theory in the calculus of variations. In two dimensions, it is written as:\[abla^2 v = \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0\]This equation describes a function \( v(x, y) \) that is harmonic, meaning no local minima or maxima form within a domain.

• **Solution Characteristics:** The solutions to Laplace's equation are called harmonic functions. They offer a smooth, stable solution without any perturbations or "spikes."
• **Boundary Conditions:** For Laplace's equation, besides solving the equation inside the domain, ensuring the function satisfies certain values on the boundary is crucial.

In our exercise, the part \( v(x, y) \) fulfills the inhomogeneous boundary condition \( v=g(x, y) \) along \( C \), and satisfies the condition \( abla^2 v = 0 \). This gives \( v \) a prescribed boundary value making it a solution to a version of Laplace's equation.

Wave Equation

The wave equation in calculus of variations describes the propagation of waves, such as sound or light waves, through a medium. The form it takes in our problem is:\[ \sigma \frac{\partial^2 u}{\partial t^2} = \tau abla^2 u \]Here, \( u(x, y, t) \) represents the part of \( w \) that satisfies this equation.

• **Time-Dependent Behavior:** The presence of \( \frac{\partial^2 u}{\partial t^2} \) signifies how the wave evolves over time at any point \( (x, y) \).
• **Space-Dependent Behavior:** The term \( abla^2 u \) represents how the wave spreads spatially through the medium.
• **Simplification:** By reducing the original problem to \( u=0 \) on \( C \), it results in a wave equation with homogeneous boundary conditions.

This means \( u \), which responds to the wave dynamics of the problem, can be analyzed separately, with \( v \) removing the complications introduced by boundary conditions, thus solving the problem more efficiently.