 16.16.1: Let y(x,t)= sin(nx/L) cos(nct/L) for 0 x L. Show that y satisfies t...
 16.16.23: In each of 1 through 6, solve the wave equation on the real line fo...
 16.16.29: In each of 1 through 5, solve the problem for wave equation on the ...
 16.16.34: Use the Laplace transform to write the solution (in terms of f (t))...
 16.16.39: In each of 1 through 6, write the dAlembert solution for the proble...
 16.16.57: Let c = R = 1, f (r) = 1 r, and g(r) = 0. Approximate the coefficie...
 16.16.60: Approximate the vertical deflections of the center of a circular me...
 16.16.2: Show that z(x, y,t)=sin(nx) cos(my) cos n2 + m2t satisfies the two...
 16.16.24: In each of 1 through 6, solve the wave equation on the real line fo...
 16.16.30: In each of 1 through 5, solve the problem for wave equation on the ...
 16.16.35: Use the Laplace transform to write the solution (in terms of f (t))...
 16.16.40: In each of 1 through 6, write the dAlembert solution for the proble...
 16.16.58: Repeat with f (r) = 1 r 2 .
 16.16.61: Use the general solution derived in this section to prove the plaus...
 16.16.3: Let f be a twicedifferentiable function of one variable. Show that...
 16.16.25: In each of 1 through 6, solve the wave equation on the real line fo...
 16.16.31: In each of 1 through 5, solve the problem for wave equation on the ...
 16.16.36: Use the Laplace transform to solve 2 y t 2 = c2 2 y x 2 At for x > ...
 16.16.41: In each of 1 through 6, write the dAlembert solution for the proble...
 16.16.59: Repeat with f (r) = sin(r).
 16.16.4: Show that y(x,t) = sin(x) cos(ct) + 1 c cos(x)sin(ct) satisfies the...
 16.16.26: In each of 1 through 6, solve the wave equation on the real line fo...
 16.16.32: In each of 1 through 5, solve the problem for wave equation on the ...
 16.16.37: Use the Laplace transform to find the solution y(x,t) = 1 2 ( fo(x ...
 16.16.42: In each of 1 through 6, write the dAlembert solution for the proble...
 16.16.5: Formulate an initialboundary value problem for vibrations of a rec...
 16.16.27: In each of 1 through 6, solve the wave equation on the real line fo...
 16.16.33: In each of 1 through 5, solve the problem for wave equation on the ...
 16.16.38: Use the Laplace transform to solve: y t 2 = c2 y x 2 Axt for x > 0,...
 16.16.43: In each of 1 through 6, write the dAlembert solution for the proble...
 16.16.6: Formulate an initialboundary value problem for the motion of an el...
 16.16.28: In each of 1 through 6, solve the wave equation on the real line fo...
 16.16.44: In each of 1 through 6, write the dAlembert solution for the proble...
 16.16.7: In each of 1 through 8, solve the initialboundary value problem usi...
 16.16.45: In each of 7 through 12, solve the problem utt = c2 ux x + F(x,t) f...
 16.16.8: In each of 1 through 8, solve the initialboundary value problem usi...
 16.16.46: In each of 7 through 12, solve the problem utt = c2 ux x + F(x,t) f...
 16.16.9: In each of 1 through 8, solve the initialboundary value problem usi...
 16.16.47: In each of 7 through 12, solve the problem utt = c2 ux x + F(x,t) f...
 16.16.10: In each of 1 through 8, solve the initialboundary value problem usi...
 16.16.48: In each of 7 through 12, solve the problem utt = c2 ux x + F(x,t) f...
 16.16.11: In each of 1 through 8, solve the initialboundary value problem usi...
 16.16.49: In each of 7 through 12, solve the problem utt = c2 ux x + F(x,t) f...
 16.16.12: In each of 1 through 8, solve the initialboundary value problem usi...
 16.16.50: In each of 7 through 12, solve the problem utt = c2 ux x + F(x,t) f...
 16.16.13: In each of 1 through 8, solve the initialboundary value problem usi...
 16.16.51: In each of 13 through 18, write the solution of the problem utt = u...
 16.16.14: In each of 1 through 8, solve the initialboundary value problem usi...
 16.16.52: In each of 13 through 18, write the solution of the problem utt = u...
 16.16.15: Solve the initialboundary value problem 2 y t 2 = 3 2 y x 2 + 2x f...
 16.16.53: In each of 13 through 18, write the solution of the problem utt = u...
 16.16.16: Solve 2 y t 2 = 9 2 y x 2 + x 2 for 0 < x < 4,t > 0 y(0,t) = y(4,t)...
 16.16.54: In each of 13 through 18, write the solution of the problem utt = u...
 16.16.17: Solve 2 y t 2 = 2 y x 2 cos(x) for 0 < x < 2,t > 0 y(0,t) = y(2,t) ...
 16.16.55: In each of 13 through 18, write the solution of the problem utt = u...
 16.16.18: Transverse vibrations in a homogeneous rod of length are modeled by...
 16.16.56: In each of 13 through 18, write the solution of the problem utt = u...
 16.16.19: Solve the telegraph equation 2 u t 2 + A u t + Bu = c2 2 u x 2 for ...
 16.16.20: (a) Write a series solution for 2 y t 2 = 9 2 y x 2 + 5x 3 for 0 < ...
 16.16.21: (a) Write a series solution for 2 y t 2 = 9 2 y x 2 ex for 0 < x < ...
 16.16.22: (a) Write a series solution for 2 y t 2 = 9 2 y x 2 + cos(x) for 0 ...
Solutions for Chapter 16: Wave Motion on an Interval
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 16: Wave Motion on an Interval
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Since 61 problems in chapter 16: Wave Motion on an Interval have been answered, more than 15692 students have viewed full stepbystep solutions from this chapter. Chapter 16: Wave Motion on an Interval includes 61 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).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Iterative method.
A sequence of steps intended to approach the desired solution.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = 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.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.