# 6.10: Problems

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## Exercise $$\PageIndex{1}$$

A rectangular plate $$0 \leq x \leq L\:, 0 \leq y \leq H$$ with heat diffusivity constant $$k$$ is insulated on the edges $$y=0, H$$ and is kept at constant zero temperature on the other two edges. Assuming an initial temperature of $$u(x, y, 0)=$$ $$f(x, y)$$, use separation of variables $$\mathrm{t}$$ find the general solution.

## Exercise $$\PageIndex{2}$$

Solve the following problem. $\begin{array}{r} u_{x x}+u_{y y}+u_{z z}=0, \quad 0<x<2 \pi, \quad 0<y<\pi, \quad 0<z<1, \\ u(x, y, 0)=\sin x \sin y, \quad u(x, y, z)=0 \text { on other faces. } \end{array}\nonumber$

## Exercise $$\PageIndex{3}$$

Consider Laplace’s equation on the unit square, $$u_{x x}+u_{y y}=0,0 \leq x, y \leq$$ 1. Let $$u(0, y)=0, u(1, y)=0$$ for $$0<y<1$$ and $$u_{y}(x, 0)=0$$ for $$0<y<1$$. Carry out the needed separation of variables and write down the product solutions satisfying these boundary conditions.

## Exercise $$\PageIndex{4}$$

Consider a cylinder of height $$H$$ and radius $$a$$.

1. Write down Laplace’s Equation for this cylinder in cylindrical coordinates.
2. Carry out the separation of variables and obtain the three ordinary differential equations that result from this problem.
3. What kind of boundary conditions could be satisfied in this problem in the independent variables?

## Exercise $$\PageIndex{5}$$

Consider a square drum of side $$s$$ and a circular drum of radius $$a$$.

1. Rank the modes corresponding to the first 6 frequencies for each.
2. Write each frequency (in $$\mathrm{Hz}$$ ) in terms of the fundamental (i.e., the lowest frequency.)
3. What would the lengths of the sides of the square drum have to be to have the same fundamental frequency? (Assume that $$c=1.0$$ for each one.)

## Exercise $$\PageIndex{6}$$

We presented the full solution of the vibrating rectangular membrane in Equation 6.1.26. Finish the solution to the vibrating circular membrane by writing out a similar full solution.

## Exercise $$\PageIndex{7}$$

A copper cube $$10.0 \mathrm{~cm}$$ on a side is heated to $$100^{\circ} \mathrm{C}$$. The block is placed on a surface that is kept at $$0^{\circ} \mathrm{C}$$. The sides of the block are insulated, so the normal derivatives on the sides are zero. Heat flows from the top of the block to the air governed by the gradient $$u_{z}=-10^{\circ} \mathrm{C} / \mathrm{m}$$. Determine the temperature of the block at its center after 1.o minutes. Note that the thermal diffusivity is given by $$k=\frac{K}{\rho c_{p}}$$, where $$K$$ is the thermal conductivity, $$\rho$$ is the density, and $$c_{p}$$ is the specific heat capacity.

## Exercise $$\PageIndex{8}$$

Consider a spherical balloon of radius $$a$$. Small deformations on the surface can produce waves on the balloon’s surface.

1. Write the wave equation in spherical polar coordinates. (Note: $$\rho$$ is constant!)
2. Carry out a separation of variables and find the product solutions for this problem.
3. Describe the nodal curves for the first six modes.
4. For each mode determine the frequency of oscillation in $$\mathrm{Hz}$$ assuming $$c=1.0 \mathrm{~m} / \mathrm{s}$$.

## Exercise $$\PageIndex{9}$$

Consider a circular cylinder of radius $$R=4.00 \mathrm{~cm}$$ and height $$H=20.0$$ $$\mathrm{cm}$$ which obeys the steady state heat equation $u_{r r}+\frac{1}{r} u_{r}+u_{z z} .\nonumber$ Find the temperature distribution, $$u(r, z)$$, given that $$u(r, 0)=0^{\circ} \mathrm{C}, u(r, 20)=$$ $$20^{\circ} \mathrm{C}$$, and heat is lost through the sides due to Newton’s Law of Cooling $\left[u_{r}+h u\right]_{r=4}=0,\nonumber$ for $$h=1.0 \mathrm{~cm}^{-1}$$.

## Exercise $$\PageIndex{10}$$

The spherical surface of a homogeneous ball of radius one in maintained at zero temperature. It has an initial temperature distribution $$u(\rho, 0)=$$ $$100^{\circ} \mathrm{C}$$. Assuming a heat diffusivity constant $$k$$, find the temperature throughout the sphere, $$u(\rho, \theta, \phi, t)$$.

## Exercise $$\PageIndex{11}$$

Determine the steady state temperature of a spherical ball maintained at the temperature $u(x, y, z)=x^{2}+2 y^{2}+3 z^{2}, \quad \rho=1 .\nonumber$ [Hint - Rewrite the problem in spherical coordinates and use the properties of spherical harmonics.]

## Exercise $$\PageIndex{12}$$

A hot dog initially at temperature $$50^{\circ} \mathrm{C}$$ is put into boiling water at $$100^{\circ} \mathrm{C}$$. Assume the hot dog is $$12.0 \mathrm{~cm}$$ long, has a radius of $$2.00 \mathrm{~cm}$$, and the heat constant is $$2.0 \times 10^{-5} \mathrm{~cm}^{2} / \mathrm{s}$$.

1. Find the general solution for the temperature. [Hint: Solve the heat equation for $$u(r, z, t)=T(r, z, t)-100$$, where $$T(r, z, t)$$ is the temperature of the hot dog.]
2. Indicate how one might proceed with the remaining information in order to determine when the hot dog is cooked; i.e., when the center temperature is $$80^{\circ} \mathrm{C}$$.

This page titled 6.10: Problems is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Russell Herman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.