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Find the steady-state temperature distribution in a plate with the boundary temperatures$\mathbf{T}\mathbf{=}\mathbf{30}\mathbf{掳}$for $\mathit{x}\mathbf{=}\mathbf{0}$and $\mathit{y}\mathbf{=}\mathbf{3}$;$\mathbf{T}\mathbf{=}\mathbf{20}\mathbf{掳}$for $\mathit{y}\mathbf{=}\mathbf{0}$and $\mathit{x}\mathbf{=}\mathbf{5}$. Hint: Subtract$\mathbf{20}\mathbf{掳}$from all temperatures and solve the problem; then add $\mathbf{20}\mathbf{掳}$. (Also see Problem 2.)

A semi-infinite bar is initially at temperature $\mathbf{100}\mathbf{掳}$for $\mathbf{0}\mathbf{<}\mathit{x}\mathbf{<}\mathbf{1}$, and 0 for x > 1 . Starting at t = 0 , the end x = 0 is maintained at $\mathbf{0}\mathbf{掳}$and the sides are insulated. Find the temperature in the bar at time t , as follows. Separate variables in the heat flow equation and get elementary solutions ${\mathit{e}}^{\mathbf{鈭�}{\mathbf{伪}}^{\mathbf{2}}{\mathbf{k}}^{\mathbf{2}}\mathbf{t}}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathbf{k}\mathbf{x}\mathbf{\right)}$and ${\mathit{e}}^{\mathbf{鈭�}{\mathbf{伪}}^{\mathbf{2}}{\mathbf{k}}^{\mathbf{2}}\mathbf{t}}\mathbf{cos}\mathbf{\left(}\mathbf{k}\mathbf{x}\mathbf{\right)}$. Discard the cosines since u = 0 at x = 0 . Look for a solution $\mathit{u}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}{鈭�}_0^{\mathrm{鈭�}}B\mathbf{\left(}\mathbf{k}\mathbf{\right)}{\mathit{e}}^{\mathbf{鈭�}{\mathbf{伪}}^{\mathbf{2}}{\mathbf{k}}^{\mathbf{2}}\mathbf{t}}\mathbf{sin}\mathbf{\left(}\mathbf{k}\mathbf{x}\mathbf{\right)}\mathit{d}\mathit{k}$and proceed as in Example 2. Leave your answer as an integral.

Do the two-dimensional analog of the problem in Example 1. A 鈥減oint charge鈥� in a plane means physically a uniform charge along an infinite line perpendicular to the plane; a 鈥渃ircle鈥� means an infinitely long circular cylinder perpendicular to the plane. However, since all cross-sections of the parallel line and cylinder are the same, the problem is a two-dimensional one. Hint: The potential must satisfy Laplace鈥檚 equation in charge-free regions. What are the solutions of the two-dimensional Laplace equation?

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

$\mathbf{5}{\mathbf{cos}}^{\mathbf{3}}\mathbf{胃}\mathbf{鈭�}\mathbf{3}{\mathbf{sin}}^{\mathbf{2}}\mathbf{胃}$.

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 bar of length l is initially at $\mathbf{0}\mathbf{掳}$.From $\mathit{t}\mathbf{=}\mathbf{0}$on, the ends are held at $\mathbf{20}\mathbf{掳}$. Find $\mathit{u}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}$for$\mathit{t}\mathbf{>}\mathbf{0}$.

A long wire occupying the x-axis is initially at rest. The end x = 0 is oscillated up and down so that $\mathbf{y}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}\mathbf{2}\mathbf{sin}\mathbf{3}\mathbf{t}\mathbf{,}\mathbf{\text{鈥夆赌夆赌�}}\mathbf{t}\mathbf{>}\mathbf{0}$. 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{鈭�}\mathbf{y}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{鈭�}\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 $\mathbf{Y}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{p}\mathbf{)}\mathbf{=}\frac{\mathbf{6}{\mathbf{e}}^{\mathbf{鈭�}\mathbf{(}\mathbf{p}\mathbf{/}\mathbf{v}\mathbf{)}\mathbf{x}}}{{\mathbf{p}}^{\mathbf{2}}\mathbf{+}\mathbf{9}}$. Use L3 and L28 to find

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

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

$\mathbf{\left|}\mathbf{肠辞蝉胃}\mathbf{\right|}$.

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 method of images for problem 4.