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Chapter 13: Q4P (page 647)

Question:Find the characteristic frequencies for sound vibration in a rectangular box (say a room) of sides a, b, c. Hint: Separate the wave equation in three dimensions in rectangular coordinates. This problem is like Problem 3 but for three dimensions instead of two. Discuss degeneracy (see Problem 3).

Short Answer

Expert verified

The solution if the membrane is square is given below.

Step by step solution

01

Given Information.

Characteristic frequencies are given as below.

02

Definition of Laplace’ equation.

The total of the second-order partial derivatives of , the unknown function, with respect to the Cartesian coordinates equals , according to Laplace's equation.

The wave equation is

03

Use wave equation.

Start from a wave equation.

Put a solution of the form mentioned below in the above equation Z(z )=F(x, y, z ) T (t) .
Dividing by F(x, y, z) T ( t ) .

Both sides are a function of a different variable and they must be equal to some constant if they are to be equal.

Write the derived two-equation.

Write the solution of the time equation.

04

Use the boundary condition.

05

Solve further.

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Most popular questions from this chapter

Write the Schrödinger equation (3.22) if $ψ$ is a function ofx, and $V = \frac{1}{2} m ω^{2} x^{2}$ (this is a one-dimensional harmonic oscillator). Find the solutions $ψ_{n} (x)$ and the energy eigenvalues $E_{n}$ . Hints: In Chapter 12, equation (22.1) and the first equation in (22.11), replace xby $α x$ where $α = \sqrt{m ω / ℏ}$ . (Don't forget appropriate factors of $α$ for the $x^{'}$ 's in the denominators of $D = \frac{d}{d x}$ and $ψ^{''} = \frac{d^{2} ψ}{d x^{2}}$ .) Compare your results for equation (22.1) with the Schrödinger equation you wrote above to see that they are identical if $E_{n} = (n + \frac{1}{2}) ℏ ω$ . Write the solutions $ψ_{n} (x)$ of the Schrödinger equation using Chapter 12, equations (22.11) and (22.12).

Find the steady-state temperature distribution in a spherical shell of inner radius 1 and outer radius 2 if the inner surface is held at $0 °$ and the outer surface has its upper half at $100 °$ and its lower half at role="math" localid="1664359640240" $0 °$ . Hint: r = 0 is not in the region of interest, so the solutions $r^{- l - 1}$ in (7.9) should be included. Replace $c_{l} r^{l}$ in (7.11) by $(c l r^{l} + b l r^{- l - 1})$ .

Using the formulas of Chapter 12, Section 5, sum the series in (8.20) to get (8.21).

Repeat Problem 17 for a membrane in the shape of a circular sector of angle $60 °$ .

Find the steady-state temperature distribution inside a sphere of radius 1 when the surface temperatures are as given in Problems 1 to 10.

$| cosθ |$ .

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