Chapter 14

Find the steady-state temperature \(u(r, \theta)\) in a circular plate of radius 1 if the temperature on the circumference is as given. $$ u(1, \theta)=\left\\{\begin{array}{ll} u_{0}, & 0<\theta<\pi \\ 0, & \pi<\theta<2 \pi \end{array}\right. $$

In Problems 1 and 2 , find the steady-state temperature \(u(r, \theta)\) in a circular plate of radius \(c\) if the temperature on the circumference is as given. $$ u(c, \theta)=\left\\{\begin{array}{lc} u_{0}, & 0<\theta<\pi \\ -u_{0}, & \pi<\theta<2 \pi \end{array}\right. $$

In Problems \(1-4\), find the steady-state temperature \(u(r, \theta)\) in a circular plate of radius 1 if the temperature on the circumference is as given. $$ u(1, \theta)= \begin{cases}u_{0}, & 0<\theta<\pi \\ 0, & \pi<\theta<2 \pi\end{cases} $$

Find the steady-state temperature \(u(r, \theta)\) in a circular plate of radius 1 if the temperature on the circumference is as given. $$ u(1, \theta)=\left\\{\begin{array}{ll} \theta, & 0<\theta<\pi \\ \pi-\theta, & \pi<\theta<2 \pi \end{array}\right. $$

In Problems 1 and 2 , find the steady-state temperature \(u(r, \theta)\) in a circular plate of radius \(c\) if the temperature on the circumference is as given. $$ u(c, \theta)=\left\\{\begin{array}{lr} 1, & 0<\theta<\pi / 2 \\ 0, & \pi / 2<\theta<3 \pi / 2 \\ 1, & 3 \pi / 2<\theta<2 \pi \end{array}\right. $$

In Problems \(1-4\), find the steady-state temperature \(u(r, \theta)\) in a circular plate of radius 1 if the temperature on the circumference is as given. $$ u(1, \theta)= \begin{cases}\theta, & 0<\theta<\pi \\ \pi-\theta, & \pi<\theta<2 \pi\end{cases} $$

Find the steady-state temperature \(u(r, \theta)\) in a circular plate of radius 1 if the temperature on the circumference is as given. $$ u(1, \theta)=2 \pi \theta-\theta^{2}, 0<\theta<2 \pi $$

In Problems 3 and 4 , find the steady-state temperature \(u(r, \theta)\) in a semicircular plate of radius 1 if boundary conditions are as given. $$ \begin{aligned} &u(1, \theta)=u_{0}\left(\pi \theta-\theta^{2}\right), 0<\theta<\pi \\ &u(r, 0)=0, \quad u(r, \pi)=0,0

In Problems \(1-4\), find the steady-state temperature \(u(r, \theta)\) in a circular plate of radius 1 if the temperature on the circumference is as given. $$ u(1, \theta)=2 \pi \theta-\theta^{2}, 0<\theta<2 \pi $$

Find the steady-state temperature \(u(r, \theta)\) in a circular plate of radius 1 if the temperature on the circumference is as given. $$ u(1, \theta)=\theta, 0<\theta<2 \pi $$

If the boundaries \(\theta=0\) and \(\theta=\pi\) of a semicircular plate of radius 2 are insulated, we then have $$ \left.\frac{\partial u}{\partial \theta}\right|_{\theta=0}=0,\left.\quad \frac{\partial u}{\partial \theta}\right|_{\theta=\pi}=0, \quad 0

Find the steady-state temperature \(u(r, \theta)\) in a semicircular plate of radius \(c\) if the boundaries \(\theta=0\) and \(\theta=\pi\) are insulated and \(u(c, \theta)=f(\theta), 0<\theta<\pi\).

Find the steady-state temperature \(u(r, \theta)\) in a semicircular plate of radius 1 if the boundary-conditions are $$ \begin{aligned} &u(1, \theta)=u_{0}, \quad 0<\theta<\pi \\ &u(r, 0)=0, \quad u(r, \pi)=u_{0}, \quad 0

In Problems \(5-8\), find the steady-state temperature \(u(r, z)\) in a finite cylinder defined by \(0 \leq r \leq 1,0 \leq z \leq 1\) if the boundary conditions are as given. \(u(1, z)=z, \quad 0

The steady-state temperatare in a hemisphere of radius \(c\) is detrimined from $$ \begin{gathered} \frac{\partial^{2} u}{\partial r^{2}}+\frac{2}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}+\frac{\cot \theta}{r^{2}} \frac{\partial u}{\partial \theta}=0,0

In Problems 5-8, find the steady-state temperature \(u(r, z)\) in a finite cylinder defined by \(0 \leq r \leq 1,0 \leq z \leq 1\) if the boundary conditions are as given. $$ \begin{aligned} &u(1, z)=u_{0}, \quad 0

In Problems \(5-8\), find the steady-state temperature \(u(r, z)\) in a finite cylinder defined by \(0 \leq r \leq 1,0 \leq z \leq 1\) if the boundary conditions are as given. \(u(1, z)=0, \quad 0

The temperature in a circular plate of radius \(c\) is determined from the boundary-value problem $$ \begin{aligned} &k\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}\right)=\frac{\partial u}{\partial t}, \quad 00 \\ &u(c, t)=0, \quad t>0 \\ &u(r, 0)=f(r), \quad 0

If the boundary conditions for an annular plate defined by \(1

Find the steady-state temperature \(u(r, \theta)\) in the annular plate shown in Figure 14.1.4 if \(a=1, b=2\), and $$ u(1, \theta)=75 \sin \theta, \quad u(2, \theta)=60 \cos \theta, \quad 0<\theta<2 \pi . $$

Solve the boundary-value problem involving spherical vibcations: $$ \begin{aligned} &a^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{2}{r} \frac{\partial u}{\partial r}\right)=\frac{\partial^{2} u}{\partial t^{2}}, 00 \\ &u(c, t)=0, \quad t>0 \\ &u(r, 0)=f(r),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=g(r), \quad 0

A conducting sphere of radius \(c\) is grounded and placed in a uniform electric field that has intensity \(E\) in the \(z\) -direction. The potential \(u(r, \theta)\) outside the sphere is determined from the boundary-value problem \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{2}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}+\frac{\cot \theta}{r^{2}} \frac{\partial u}{\partial \theta}=0, \quad r>c, 0<\theta<\pi\) \(u(c, \theta)=0, \quad 0<\theta<\pi\) \(\lim _{\sim \infty} u(r, \theta)=-E z=-E r \cos \theta\) Show that $$ u(r, \theta)=-E r \cos \theta+E \frac{c^{3}}{r^{2}} \cos \theta $$

Suppose \(x_{k}\) is a positive zero of \(J_{0}\). Show that a solution of the boundary-value problem $$ a^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}\right)=\frac{\partial^{2} u}{\partial t^{2}}, \quad 00 $$ $$ \begin{aligned} &u(1, t)=0, \quad t>0 \\ &u(r, 0)=u_{0} J_{0}\left(x_{k} r\right),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0, \quad 0

A conducting sphere of radius \(c\) is grovnded and placed in a uniform electric field that has intensity \(E\) in the \(z\)-direction. The potential \(u(r, \theta)\) outside the sphere is determined from the boundary-value problem $$ \begin{aligned} &\frac{\partial^{2} u}{\partial r^{2}}+\frac{2}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}+\frac{\cot \theta}{r^{2}} \frac{\partial u}{\partial \theta}=0, \quad r>c, 0<\theta<\pi \\\ &u(c, \theta)=0, \quad 0<\theta<\pi \\ &\lim _{\rightarrow \infty} u(r, \theta)=-E z=-E r \cos \theta . \end{aligned} $$ Show that $$ u(r, \theta)=-E r \cos \theta+E \frac{c^{3}}{r^{2}} \cos \theta $$

You are ashed to find a product solution \(u(r, \theta, \phi)=R(r) \Theta(\theta) \Phi(\phi)\) of Helmholtz's partial differential equation \(\nabla^{2} u+k^{2} u=0\) where the Laplacian \(\nabla^{2} u\) is defined in (2). (a) Proceed as in Example 1 but using \(u(r, \theta, \phi)=\) \(R(r) \Theta(\theta) \Phi(\phi)\) and the separation constant \(n(n+1)\) to show that the radial dependence of the solution \(u\) is defined by the equation $$ r^{2} \frac{d^{2} R}{d r^{2}}+2 r \frac{d R}{d r}+\left[k^{2} r^{2}-n(n+1)\right] R=0 $$ (b) Now use the second separation constant \(m^{2}\) to show that the remaining separated equations are $$ \begin{aligned} &\frac{d^{2} \Phi}{d \phi^{2}}+m^{2} \Phi=0 \\ &\left.\frac{d^{2} \theta}{d \theta^{2}}+\frac{\cos \theta}{\sin \theta} \frac{d \theta}{d \theta}+\left[n(n+1)-\frac{m^{2}}{\sin ^{2} \theta}\right]\right]=0 . \end{aligned} $$ (c) Use the substitution \(x=\cos \theta\) to show that the second differeatial equation in part (b) becomes \(\left(1-x^{2}\right) \frac{d^{2} \Theta}{d x^{2}}-2 x \frac{d \theta}{d x}+\left[n(n+1)-\frac{m^{2}}{1-x^{2}}\right] Q=0\)

In Problems 13 and 14, you are asled to find a product solution \(u(r, \theta, \phi)=R(r) \Theta(\theta) \Phi(\phi)\) of Helmholtz's partial differential equation \(\nabla^{2} u+k^{2} u=0\) where the Laplacian \(\nabla^{2} u\) is defined in (2). (a) Proceed as in Example 1 but using \(u(r, \theta, \phi)=\) \(R(r) \Theta(\theta) \Phi(\phi)\) and the separation constant \(n(n+1)\) to show that the radial dependence of the solution \(u\) is defined by the equation $$ r^{2} \frac{d^{2} R}{d r^{2}}+2 r \frac{d R}{d r}+\left[k^{2} r^{2}-n(n+1)\right] R=0 $$ (b) Now use the second separation constant \(m^{2}\) to show that the cemaining separated equations are $$ \begin{aligned} &\frac{d^{2} \Phi}{d \phi^{2}}+m^{2} \Phi=0 \\ &\frac{d^{2} \theta}{d \theta^{2}}+\frac{\cos \theta}{\sin \theta} \frac{d O}{d \theta}+\left[n(n+1)-\frac{m^{2}}{\sin ^{2} \theta}\right] \Theta=0 \end{aligned} $$ (c) Use the substitution \(x=\cos \theta\) to show that the second differential equation in part (b) becomes $$ \left(1-x^{2}\right) \frac{d^{2} \theta}{d x^{2}}-2 x \frac{d O}{d x}+\left[n(n+1)-\frac{m^{2}}{1-x^{2}}\right] \Theta=0 $$

Solve the boundary-value problem $$ \begin{aligned} &\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{\partial^{2} u}{\partial z^{2}}=0, \quad 0

In Problems 13 and 14, use the substitution \(u(r, t)=v(r, t)+\psi(r)\) to solve the given boundary-value problem.$$ \begin{aligned} &\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\beta=\frac{\partial u}{\partial t}, \quad 00, \quad \beta \text { a constant } \\ &u(1, t)=0, \quad t>0 \\ &u(r, 0)=0, \quad 0

Solve the boundary-value problem $$ \begin{aligned} &\frac{\partial^{2} u}{\partial r^{2}}+\frac{2}{r} \frac{\partial u}{\partial r}=\frac{\partial^{2} u}{\partial t^{2}}, \quad 00 \\ &\left.\frac{\partial u}{\partial r}\right|_{r=1}=0, \quad t>0 \\ &u(r, 0)=f(r),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=g(r), \quad 0

In this problem we consider the general case- that is, with \(\theta\) dependence - of the vibrating circular membrane of radius \(c\) : $$ \begin{aligned} &a^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}\right)=\frac{\partial^{2} u}{\partial t^{2}}, \quad 00 \\ &u(c, \theta, t)=0, \quad 0<\theta<2 \pi, t>0 \end{aligned} $$ \(u(r, \theta, 0)=f(r, \theta), \quad 0

The function \(u(x)=Y_{0}(\alpha a) J_{0}(\alpha x)-J_{0}(\alpha a) Y_{0}(\alpha x), a>0\) is a solution of the parametric Bessel equation $$ x^{2} \frac{d^{2} u}{d x^{2}}+x \frac{d u}{d x}+\alpha^{2} x^{2} u=0 $$ on the interval \([a, b]\). If the eigenvalues \(\lambda_{n}=\alpha_{n}^{2}\) are defined by the positive roots of the equation $$ Y_{0}(\alpha a) J_{0}(\alpha b)-J_{0}(\alpha a) Y_{0}(\alpha b)=0, $$ show that the functions $$ \begin{aligned} &u_{m}(x)=Y_{0}\left(\alpha_{m} a\right) J_{0}\left(\alpha_{m} x\right)-J_{0}\left(\alpha_{m} a\right) Y_{0}\left(\alpha_{m} x\right) \\ &u_{n}(x)=Y_{0}\left(\alpha_{n} a\right) J_{0}\left(\alpha_{n} x\right)-J_{0}\left(\alpha_{n} a\right) Y_{0}\left(\alpha_{n} x\right) \end{aligned} $$ are orthogonal with respect to the weight function \(p(x)=x\) on the interval \([a, b]\); that is, $$ \int_{a}^{b} x u_{m}(x) u_{n}(x) d x=0, m \neq n . $$

Consider an idealized drum consisting of a thin membrane stretched over a circular frame of radius 1 . When such a drum is struck at its center, one hears a sound that is frequently described as a dull thud rather than a melodic tone. We can model a single drumbeat using the boundary-value problem solved in Example \(1 .\) (a) Find the solution \(u(r, t)\) given in (9) when \(c=1, f(r)=0\), and $$ g(r)= \begin{cases}-v_{0}, & 0 \leq r