 10.7.1: Consider an elastic string of length L whose ends are held fixed. T...
 10.7.2: Consider an elastic string of length L whose ends are held fixed. T...
 10.7.3: Consider an elastic string of length L whose ends are held fixed. T...
 10.7.4: Consider an elastic string of length L whose ends are held fixed. T...
 10.7.5: Consider an elastic string of length L whose ends are held fixed. T...
 10.7.6: Consider an elastic string of length L whose ends are held fixed. T...
 10.7.7: Consider an elastic string of length L whose ends are held fixed. T...
 10.7.8: Consider an elastic string of length L whose ends are held fixed. T...
 10.7.9: If an elastic string is free at one end, the boundary condition to ...
 10.7.10: Consider an elastic string of length L. The end x = 0 is held fixed...
 10.7.11: Suppose that the string in is started instead from the initial posi...
 10.7.12: Dimensionless variables can be introduced into the wave equation a2...
 10.7.13: 13 and 14 indicate the form of the general solution of the wave equ...
 10.7.14: 13 and 14 indicate the form of the general solution of the wave equ...
 10.7.15: A steel wire 5 ft in length is stretched by a tensile force of 50 l...
 10.7.16: Consider the wave equation a2 uxx = utt in an infinite onedimensio...
 10.7.17: Consider the wave equation a2 uxx = utt in an infinite onedimensio...
 10.7.18: By combining the results of 16 and 17, show that the solution of th...
 10.7.19: 19 and 20 indicate how the formal solution (20), (22) of Eqs. (1), ...
 10.7.20: 19 and 20 indicate how the formal solution (20), (22) of Eqs. (1), ...
 10.7.21: The motion of a circular elastic membrane, such as a drumhead, is g...
 10.7.22: The total energy E(t) of the vibrating string is given as a functio...
 10.7.23: Dispersive Waves. Consider the modified wave equation a2 utt + 2 u ...
 10.7.24: Dispersive Waves. Consider the modified wave equation a2 utt + 2 u ...
Solutions for Chapter 10.7: The Wave Equation: Vibrations of an Elastic String
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 10.7: The Wave Equation: Vibrations of an Elastic String
Get Full SolutionsElementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. Since 24 problems in chapter 10.7: The Wave Equation: Vibrations of an Elastic String have been answered, more than 13829 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 10.7: The Wave Equation: Vibrations of an Elastic String includes 24 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Column space C (A) =
space of all combinations of the columns of A.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.