Do for a rectangular membrane

Assume from electrostatics the equations and (E = electric field, = charge density, = constant, = electrostatic potential). Show that the electrostatic potential satisfies Laplace’s equation (1.1) in a charge-free region and satisfies Poisson’s equation (1.2) in a region of charge density .

u = f(x vt) and u = f(x + vt) satisfy the wave equation, where f is any function with a second derivative. This is the dAlembert solution of the wave equation. (See Chapter 4, Section 11, Example 1.) The function f(x vt) represents a wave moving in the positive x direction and f(x + vt) represents a wave moving in the opposite direction. (b) Show that u(r, t) = (1/r)f(r vt) and u(r, t) = (1/r)f(r +vt) satisfy the wave equation in spherical coordinates. [Use the first term of (7.1) for 2u since here u is independent of and .] These functions represent spherical waves spreading out from the origin or converging on the origin.

Assume from electrodynamics the following equations which are valid in free space. (They are called Maxwells equations.) E = 0 B = 0 E = B t B = 1 c2 E t where E and B are the electric and magnetic fields, and c is the speed of light in a vacuum. From them show that any component of E or B satisfies the wave equation (1.4) with v = c.

Obtain the heat flow equation (1.3) as follows: The quantity of heat Q flowing across a surface is proportional to the normal component of the (negative) temperature gradient, (T )n. Compare Chapter 6, equation (10.4), and apply the discussion of flow of water given there to the flow of heat. Thus show that the rate of gain of heat per unit volume per unit time is proportional to T. But T /t is proportional to this gain in heat; thus show that T satisfies (1.3).

Show that the solutions of (2.5) can also be written as X = ( eikx, eikx, Y = ( sinh ky, cosh ky. Also show that these solutions are equivalent to (2.7) if k is real and equivalent to (2.18) if k is pure imaginary. (See Chapter 2, Section 12.) Also show that X = sin k(x a), Y = sinh k(y b) are solutions of (2.5).

Show that the series in (2.12) can be summed to get T = 200 arc tan sin (x/10) sinh (y/10) (with the arc tangent in radians). Use this formula to check the value T = 26.1 at x = y = 5. Hints for summing the series: Use sin (nx/10) = Im einx/10 to write the series as ImP odd n zn/n. (What is z?) Compare this with the series for ln[(1 + z)/(1 z)] (see Chapter 1, Problem 13.17). Then use (13.5) of Chapter 2.

Solve Problem 3 if the plate is cut off at height 1 and the temperature at y = 1 is held at 0. Answer: T = 4 X even n n (n2 1) sinh n sinh n(1 y) sin nx.

Find the steady-state temperature distribution in a rectangular plate 30 cm by 40 cm given that the temperature is 0 along the two long sides and along one short end; the other short end along the x axis has temperature T = ( 100, 0

Solve Problem 2 if the plate is cut off at height 10 and the temperature of the top edge is 0.

Find the steady-state temperature distribution in a metal plate 10 cm square if one side is held at 100 and the other three sides at 0. Find the temperature at the center of the plate. Answer: T = X odd n 400 n sinh n sinh n 10 (10 y) sin nx 10 , T(5, 5) 25.

Find the steady-state temperature distribution in the plate of Problem 10 if two adjacent sides are at 100 and the other two at 0. Hint: Use your solution of Problem 10. You should not have to do any calculationjust write down the answer!

Find the temperature distribution in a rectangular plate 10 cm by 30 cm if two adjacent sides are held at 100 and the other two sides at 0.

Find the steady-state temperature distribution in a rectangular plate covering the area 0

the temperature (assumed finite) as y in this problem is determined by the given temperature at y = 0. Let T = f(x) = x at y = 0, repeat your calculations above to find the temperature distribution, and find the value of T for large y. Dont forget the k = 0 term in the series!

Consider a finite plate, 10 cm by 30 cm, with two insulated sides, one end at 0 and the other at a given temperature T = f(x). Try f(x) = 100; f(x) = x. You should convince yourself that this problem cannot be done using just the solutions (2.7). To see what is wrong, go back to the differential equations (2.5) and solve them if k = 0. You should find solutions x, y, xy and constant [the constant is already contained in (2.7) for k = 0, but the other three solutions are not]. Now go back over each of the problems we have done so far and see why we could ignore these k = 0 solutions; then including the k = 0 solutions, finish the problem of the finite plate with insulated sides. For the case f(x) = x, the answer is: T = 1 6 (30 y) 40 2 X odd n 1 n2 sinh 3n sinh n 10 (30 y) cos nx 10 .

Show that there is only one function u which takes given values on the (closed) boundary of a region and satisfies Laplaces equation 2u = 0 in the interior of the region. Hints: Suppose u1 and u2 are both solutions with the same boundary conditions so that U = u1 u2 = 0 on the boundary. In Greens first identity (Chapter 6, Problem 10.16), let == U to show that U 0. Thus show U 0 everywhere inside the region.

Do Problem 6.6 in 3 dimensional rectangular coordinates. That is, solve the particle in a box problem for a cube.

Separate the time-independent Schrodinger equation (3.22) in spherical coordinates assuming that V = V (r) is independent of and . (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 Schrodinger 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 [V (r) E]. Show that the , solutions are spherical harmonics as in (7.10) and Problem 16. Show that the r equation with k = l(l + 1) is [compare (7.6)] 1 R d dr r2 dR dr 2Mr2 h2 [V (r) E) = l(l + 1).

Find the eigenfunctions and energy eigenvalues for a particle in a spherical box r

Write the Schrodinger equation (3.22) if is a function of x, and V = 1 2m2x2 (this is a one-dimensional harmonic oscillator). Find the solutions n(x) and the energy eigenvalues En. Hints: In Chapter 12, equation (22.1) and the first equation in (22.11), replace x by x where = pm/h. (Dont forget appropriate factors of for the xs in the denominators of D = d/dx and = d2/dx2.) Compare your results for equation (22.1) with the Schrodinger equation you wrote above to see that they are identical if En = (n + 1 2 )h. Write the solutions n(x) of the Schrodinger equation using Chapter 12, equations (22.11) and (22.12). 2

Write the Schrodinger equation (3.22) if is a function of x, and V = 1 2m2x2 (this is a one-dimensional harmonic oscillator). Find the solutions n(x) and the energy eigenvalues En. Hints: In Chapter 12, equation (22.1) and the first equation in (22.11), replace x by x where = pm/h. (Dont forget appropriate factors of for the xs in the denominators of D = d/dx and = d2/dx2.) Compare your results for equation (22.1) with the Schrodinger equation you wrote above to see that they are identical if En = (n + 1 2 )h. Write the solutions n(x) of the Schrodinger equation using Chapter 12, equations (22.11) and (22.12). 2

Find the energy eigenvalues and eigenfunctions for the hydrogen atom. The potential energy is V (r) = e2/r in Gaussian units, where e is the charge of the electron and r is in spherical coordinates. Since V is a function of r only, you know from Problem 18 that the eigenfunctions are R(r) times the spherical harmonics Y m l (, ), so you only have to find R(r). Substitute V (r) into the R equation in Problem 18 and make the following simplifications: Let x = 2r/, y = rR; show that then r = x/2, R(r) = 2 xy(x), d dr = 2 d dx, d dr r2 dR dr = 2 xy. Let 2 = 2ME/h2 (note that for a bound state, E is negative, so 2 is positive) and = Me2/h2, to get the first equation in Problem 22.26 of Chapter 12. Do this problem to find y(x), and the result that is an integer, say n. [Caution: not the same n as in equation (22.26)]. Hence find the possible values of (these are the radii of the Bohr orbits), and the energy eigenvalues. You should have found proportional to n; let = na, where a is the value of when n = 1, that is, the radius of the first Bohr orbit. Write the solutions R(r) by substituting back y = rR, and x = 2r/(na), and find En from .

Find the steady-state temperature in the region between two spheres r = 1 and r = 2 if the surface of the outer sphere has its upper half held at 100 and its lower half at 100 and these temperatures are reversed for the inner sphere. Hint: See Problem 7.14. Here you will need to find two Legendre series (when r = 1 and when r = 2) and solve for al and bl.

Find the general solution for the steady-state temperature in Figure 2.2 if the boundary temperatures are the constants T = A, T = B, etc., on the four sides, and the rectangle covers the area 0

The Klein-Gordon equation is 2u = (1/v2)2u/t2 + 2u. This equation is of interest in quantum mechanics, but it also has a simpler application. It describes, for example, the vibration of a stretched string which is embedded in an elastic medium. Separate the one-dimensional Klein-Gordon equation and find the characteristic frequencies of such a string. Answer: n = v 2 p(n/l)2 + (/)2.

Find the characteristic frequencies of a circular membrane which satisfies the KleinGordon equation (Problem 25). Hint: Separate the equation in two dimensions in polar coordinates.

Do Problem 26 for a rectangular membrane

Find the steady-state temperature in a semi-infinite plate covering the region x > 0, 0 y 1, if the edges along the x axis and y axis are insulated (see Problem 2.14) and the top edge is held at u(x, 1) = ( 100, 0 1. Hint: Look for a solution as a Fourier integral. Leave your answer as an integral (just as we usually give answers as series.)