91Ó°ÊÓ

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

${\mathbf{sin}}^{\mathbf{2}}\mathbf{賦´Ç²õ賦´Ç²õ}\mathbf{2}\mathbf{Ï•}\mathbf{−}\mathbf{³¦´Ç²õθ}$(See problem 9).

A plate in the shape of a quarter circle has boundary temperatures as shown. Find the interior steady-state temperature $\mathbf{u}(\mathbf{r},\mathbf{θ})$. (See Problem 5.12.)

Separate the time-independent Schrödinger equation (3.22) in spherical coordinates assuming that $\mathit{V}\mathbf{=}\mathit{V}\mathbf{\left(}\mathbf{r}\mathbf{\right)}$is independent of $\mathit{θ}$and $\mathit{ϕ}$. (If V depends only on r , then we are dealing with central forces, for example, electrostatic or gravitational forces.) Hints: You may find it helpful to replace the mass m in the Schrödinger equation by M when you are working in spherical coordinates to avoid confusion with the letter m in the spherical harmonics (7.10). Follow the separation of (7.1) but with the extra term $\mathbf{[}\mathbf{V}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{−}\mathbf{E}\mathbf{]}\mathit{Ψ}$. Show that the $\mathit{θ}\mathbf{,}\mathit{ϕ}$solutions are spherical harmonics as in (7.10) and Problem 16. Show that the r equation with $\mathit{k}\mathbf{=}\mathit{l}\mathbf{(}\mathbf{l}\mathbf{+}\mathbf{1}\mathbf{)}$is [compare (7.6)].

$\frac{\mathbf{1}}{\mathbf{R}}\frac{\mathbf{d}}{\mathbf{d}\mathbf{r}}\mathbf{\left(}{\mathbf{r}}^{\mathbf{2}}\frac{\mathbf{d}\mathbf{R}}{\mathbf{d}\mathbf{r}}\mathbf{\right)}\mathbf{−}\frac{\mathbf{2}\mathbf{M}{\mathbf{r}}^{\mathbf{2}}}{{\mathbf{h}}^{\mathbf{2}}}\mathbf{[}\mathbf{V}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{−}\mathbf{E}\mathbf{]}\mathbf{=}\mathit{l}\mathbf{(}\mathbf{l}\mathbf{+}\mathbf{1}\mathbf{)}$

A long conducting cylinder is placed parallel to thez-axis in an originally uniform electric field in the negativexdirection. The cylinder is held at zero potential. Find the potential in the region outside the cylinder.

Find the steady-state temperature distribution in a rectangular plate covering the area $\mathbf{0}<\mathbf{x}<\mathbf{1}$, $\mathbf{0}<\mathbf{y}<\mathbf{2}$, if $\mathit{T}\mathbf{=}\mathbf{0}$for $\mathit{x}\mathbf{=}\mathbf{0}$, $\mathit{x}\mathbf{=}\mathbf{1}$, $\mathit{y}\mathbf{=}\mathbf{2}$, and $\mathit{T}\mathbf{=}\mathbf{1}\mathbf{−}\mathit{x}$for$\mathit{y}\mathbf{=}\mathbf{0}$.

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).

A long wire occupying the x-axis is initially at rest. The end x = 0 is oscillated up and down so that $\mathbf{y}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}\mathbf{2}\mathbf{sin}\mathbf{3}\mathbf{t}\mathbf{,}\mathbf{\text{ â¶Ä‰â¶Ä‰}}\mathbf{t}\mathbf{>}\mathbf{0}$. Find the displacement $\mathbf{y}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}$. The initial and boundary conditions are $\mathbf{y}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}\mathbf{2}\mathbf{sin}\mathbf{\left(}\mathbf{3}\mathbf{t}\mathbf{\right)}$, $\mathbf{y}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{=}\mathbf{0}$, ${\raisebox{1ex}{$\mathbf{∂}\mathbf{y}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{∂}\mathbf{t}$}\right.\mathbf{|}}_{\mathbf{t}\mathbf{=}\mathbf{0}}\mathbf{=}\mathbf{0}$. Take Laplace transforms of these conditions and of the wave equation with respect to t as in Example 1. Solve the resulting differential equation to get $\mathbf{Y}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{p}\mathbf{)}\mathbf{=}\frac{\mathbf{6}{\mathbf{e}}^{\mathbf{−}\mathbf{(}\mathbf{p}\mathbf{/}\mathbf{v}\mathbf{)}\mathbf{x}}}{{\mathbf{p}}^{\mathbf{2}}\mathbf{+}\mathbf{9}}$. Use L3 and L28 to find

role="math" localid="1664430675935" $\mathbf{y}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}\{\begin{array}{l}\begin{array}{cc}2\sin 3(t−\frac{{\displaystyle x}}{{\displaystyle v}}),& x<\mathrm{vt}\end{array}\\ \begin{array}{cc}0,& x>\mathrm{vt}\end{array}\end{array}\begin{array}{}\end{array}$.

Question: A square membrane of side l is distorted into the shape

$\mathbf{f}\mathbf{}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{x}\mathbf{}\mathbf{y}\mathbf{}\mathbf{(}\mathbf{l}\mathbf{-}\mathbf{x}\mathbf{)}\mathbf{(}\mathbf{l}\mathbf{-}\mathbf{y}\mathbf{)}$

and released. Express its shape at subsequent times as an infinite series. Hint: Use a double Fourier series as in Problem 5.9.

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

$\left\{\begin{array}{l}\begin{array}{ccc}\mathbf{³¦´Ç²õθ}\mathbf{,}& \mathbf{0}\mathbf{<}\mathbf{θ}\mathbf{<}\mathbf{Ï€}\mathbf{/}\mathbf{2}\mathbf{,}& \mathbf{\text{thatis,upperhemisphere,}}\end{array}\\ \begin{array}{ccc}\mathbf{0}\mathbf{,}& \mathbf{Ï€}\mathbf{/}\mathbf{2}\mathbf{<}\mathbf{θ}\mathbf{<}\mathbf{Ï€}\mathbf{,}& \mathbf{\text{thatis,lowerhemisphere}}\end{array}\end{array}\right.$.