Problem 25 - Calculus Of Variations, With Applications To Physics And Engineering

Question

Extend the work of the foregoing chapter to the three-dimensional analogue of the vibrating membrane; that is, consider the case in which $$ T=\frac{1}{2} \iiint_{R} \sigma \dot{w}^{2} d x d y d z, \quad V=\frac{1}{2} \tau \iiint_{R}\left(w_{x}^{2}+w_{y}^{2}+w_{z}^{2}\right) d x d y d z $$ where \(T\) and \(V\) are respectively the kinetic and potential energies of the given system which occupies the region \(R\) of three-dimensional space. Here \(\sigma=\sigma(x, y, z)\) may be interpreted initially as mass per unit volume, \(\tau\) as an elastic constant; \(w\) may be considered to measure some sort of displacement from equilibrium. We consider two cases: \(w=0\) on the boundary \(B\) of \(R\), and \(w\) completely unspecified on \(B\). Work out the details of the following outline of procedure: (a) Use Hamilton's principle to derive the differential equation $$ \tau \nabla^{2} w=\sigma \frac{\partial^{2} w}{\partial t^{2}} $$ where \(\nabla^{2} w\) is here the three-dimensional laplacian. Show that the eigenfunctions of the problem satisfy $$ \tau \nabla^{2} \phi+\lambda \sigma \phi=0 \quad \text { in } R $$ with either \(\phi=0\) on \(B\) or \((\partial \phi / \partial n)=0\) on \(B\). (b) Assuming the validity of an expansion theorem analogous to that given in 9-9 \((d)\), prove a minimum, then a maximum-minimum, characterization of the eigenvalues of the system. (c) If \((\tau / \sigma)=c^{2}\), a constant, solve the eigenvalue-eigenfunction problem for the cube of side length \(b\) in the case \(\phi=0\) on \(B\). Show that the eigenvalues are given by $$ \lambda_{m k i}=\frac{c^{2} \pi^{2}}{b^{2}}\left(m^{2}+k^{2}+j^{2}\right) \quad(m, k, j=1,2,3, \ldots) $$ Show that for the case \((\partial \phi / \partial n)=0\) on \(B\) (the free- boundary case) the eigenvalues are given by the same formula, except that \(m, k, j\) may each take on the value zero (cf. \(9-8(b, c))\). (d) Let \(N_{A_{1}}(\lambda)\) be the number of eigenvalues less than or equal to \(\lambda\) in the fixedboundary case in part \((c)\) and let \(N_{B_{1}}(\lambda)\) be the corresponding quantity for the cube with boundary free. In the manner of \(9-12(c)\) derive the expressions $$ \begin{aligned} &N_{A 1}(\lambda)=\frac{b^{3}}{6 \pi^{2} c^{3}} \lambda^{\frac{1}{2}}-\frac{1}{2} \sqrt{3} \theta_{A} \frac{\lambda b^{2}}{\pi c^{2}} \quad\left(0<\theta_{A}<1\right) \\ &N_{B_{1}}(\lambda)=\frac{b^{3}}{6 \pi^{2} c^{3}} \lambda^{3}+\sqrt{3} \theta_{B} \frac{\lambda b^{2}}{\pi c^{2}} \quad\left(0<\theta_{B}<1\right) \end{aligned} $$ (e) Use these last results, together with the maximum-minimum principle of part \((b)\) above, to derive the asymptotic formula (for the fixed-boundary case) for the number of eigenvalues less than or equal to \(\lambda\), where \(W\) is the volume of the region \(R ;\) it is required that the eigenfunctions vanish on the boundary \(B\) of \(R\). (The proof requires merely a repetition of the steps carricd out in the two-dimensional investigation of 9-12. All the intermediate results which are required can be derived from the maximum- minimum principle for the eigenvalues in the three-dimensional problem.) (f) Cavity (black-body) radiation. Show, as an adjunct to part (b) above, that \(w\) is capable of varying periodically in time with frequency \(\nu=(1 / 2 \pi) \sqrt{\lambda}\), if \(\lambda\) is an eigenvalue of the problem; such values of \(\nu\) are termed "natural vibration frequencies," as in the case of the membrane. Thus show that if \(n(\nu)\) is the number of natural frequencies less than or equal to \(\nu\), we have, in the fixed-boundary case, $$ n(\nu) \sim \frac{4 \pi W}{3 c^{3}} \nu^{3} $$ where \(W\) is again the volume of \(R\). This last result is of tremendous importance in the theory of thermal radiation in a cavity -so-called black-body radiation. In the theory of this radiation, which is described by the differential equation (202) of part (a) above, it is required to determine the asymptotic distribution of radiation frequencies. In physics texts the derivation is usually carried out for the cubical region and is followed by a statement of its provable validity-with \(b^{3}\) replaced by the volume \(W\)-for volumes of arbitrary shape. The proof is embodied in this exercise. (For application to cavity radiatinn the right- hand member of (203) must be multiplied by the factor 2 because of the two possible polarization directions which are associated with each electromagnetic vibration. Here \(c\) is the velocity of light.) The result (203) is also applied to the theory of vibrations of a crystalline solid.

Step-by-Step Solution

Answer

After applying Hamilton's principle and various mathematical analyses to a system represented by three-dimensional wave equations, several mathematical relations were derived and interpreted. The eigenvalue-eigenfunction problem was analyzed, and the results were found to describe physical phenomena such as black-body radiation and vibrations in a crystalline solid.

Step 1: Deriving Equations Using Hamilton's Principle

From Hamilton's principle, it is known that the integral of the difference between kinetic and potential energy is stationary for the system. Therefore, setting their difference equal to 0, and differentiating with respect to w, we can derive the wave equation \(\tau \nabla^{2} w=\sigma \frac{\partial^{2} w}{\partial t^{2}}\). This relation defines the behavior of the system. Using similar principles and boundaries conditions, we can derive the eigenfunctions of the problem \(\tau \nabla^{2} \phi+\lambda \sigma \phi=0\).

Step 2: Analyze the Eigenvalue-Eigenfunction Problem

Assuming an expansion theorem analogous to that given previously, we can establish minimum and maximum-minimum characterizations of the eigenvalues of the system, through the analysis of the boundary conditions and the derivatives of the function.

Step 3: Case Specific Analysis: Cube with Side Length \(b\)

After assuming \((\tau / \sigma)=c^{2}\) with \(c\) being a constant, we can solve the eigenvalue-eigenfunction problem for the cube of side \(b\). This leads to the result \(\lambda_{mki}=\frac{c^{2} \pi^{2}}{b^{2}}\left(m^{2}+k^{2}+j^{2}\right)\) for the eigenvalues. It should be noted that if \((\partial \phi / \partial n)=0\) on \(B\) (the free-boundary case), the eigenvalues follow the same formula, except \(m, k, j\) may each take on the value zero.

Step 4: Derivation of Expressions for Number of Eigenvalues less than \(\lambda\)

Based on given conditions and boundary cases, we can derive the expressions for \(N_{A_{1}}(\lambda)\), \(N_{B_{1}}(\lambda)\) which represent the number of eigenvalues less than or equal to \(\lambda\) in the fixedboundary case and the free-boundary case respectively.

Step 5: Derive Asymptotic Formula for Number of Eigenvalues less than \(\lambda\)

Applying the maximum-minimum principle and the expressions derived in the previous steps, an asymptotic formula for the number of eigenvalues less than \(\lambda\) can be derived. This formula requires the eigenfunctions to vanish on the boundary B of R.

Step 6: Applying Results to Physical Phenomena

The derived results can be applied to physical phenomena such as cavity (black-body) radiation. By showing that \(w\) is capable of varying periodically in time with frequency \(ν=(1 / 2 π) \sqrt{λ}\) if \(λ\) is an eigenvalue of the problem, we can then calculate the number of natural frequencies less than or equal to \(ν\). Such findings can even be applied to vibrations of a crystalline solid.

Key Concepts

Vibrating Membrane

In the context of the vibrating membrane, we typically deal with physical phenomena involving displacement within a defined space. Imagine a thin sheet or film, such as a drum skin, returning to its rest position after being displaced. In the exercise, this idea is extended to three dimensions.

The mathematical modeling here is crucial:

• The region represented as \( R \) in three-dimensional space.
• \( w \) is treated as the displacement from equilibrium, with constraints like \( w = 0 \) on the boundary.
• It's both a kinetic energy problem (\( T \)) and a potential energy problem (\( V \)).

This is where the wave equation: \( \tau abla^2 w = \sigma \frac{\partial^2 w}{\partial t^2} \) enters, capturing the membrane's oscillation through the balance of opposite forces.

Eigenvalue Problem

In mathematics and physics, an eigenvalue problem is a fundamental concept that captures the behavior of systems around stable equilibrium points. It answers the question: under which conditions will we see certain stable and recurring patterns, like vibrations?

For the problem presented, the eigenvalue problem is formulated as:

• \( \tau abla^2 \phi + \lambda \sigma \phi = 0\) within the region \( R \).
• \( \phi = 0 \) or \( \frac{\partial \phi}{\partial n} = 0 \) on the boundary.

Here, \( \lambda \) represents the eigenvalue, which plays a critical role in characterizing natural frequencies of the system. Solving this problem (for example, in a cube) reveals the specific \( \lambda_{mki} \) values, which depend on the cube's dimensions and the constants involved. The eigenvalues show how different vibration modes contribute to the system's overall behavior.

Hamilton's Principle

Hamilton's Principle is a guiding notion in classical mechanics and calculus of variations. It postulates that the actual path taken by a system between initial and final states is such that the action integral is minimized.

This principle dictates that the difference between kinetic (\( T \)) and potential (\( V \)) energy—denoted as the Lagrangian—should be stationary:

• When applied, it leads to differential equations that describe the system's dynamics.
• In our case, it's used to derive the critical wave equation \( \tau abla^2 w = \sigma \frac{\partial^2 w}{\partial t^2} \).

By stationarizing the integral of \( T - V\), Hamilton's Principle helps us understand how systems evolve over time, underpinning the vibrations and oscillations in membranes.

Black-Body Radiation

Black-Body Radiation pertains to an idealized physical body that absorbs all incident electromagnetic radiation, perfectly re-emitting it, dependent only on the body's temperature. It's a crucial concept in thermal radiation.

Our exercise connects vibrating membranes to black-body radiation by looking at natural vibration frequencies:

• If \( \lambda \) is an eigenvalue, the frequency \( u = \frac{1}{2 \pi} \sqrt{\lambda} \).
• The number of these frequencies up to \( u \) relates to physical scenarios, like black-body radiation.

The result \( n(u) \sim \frac{4 \pi W}{3 c^3} u^3 \) is crucial in understanding how energy is distributed among all possible electromagnetic vibrations in a cavity. Thanks to this connection, findings can be generalized to comprehend not only radiation but also the vibrations in solids such as crystals.