I have included a Wave Equation problem with parts a-c, that has variable tension. It involves separation of variables, the Sturm-Liouville system, and an application to the "Rayleigh Quotient" involving the Eigenvalues. I have included notes on the Sturm-Liouville system with examples and properties. Please refer to these notes to maintain continuity for the solution needed. Thank you for your time and consideration in these matters.

1.) Given the wave equation below (with variable tension τ(x) = x) and appropriate Boundary Conditions:

(a) Apply Separation of Variables to the PDE to get 2 ODE's.

(b) Show that the Eigenvalue problem is a singular Sturm-Liouville system; identify the functions p(x), w(x), and q(x). Do not try to solve it.

(c) Based on the equation below, called the "Rayleigh quotient", do you expect this problem to have any negative Eigenvalues? How about a zero Eigenvalue? Be sure to use specific p(x), w(x), q(x) and Boundary Conditions for this problem.
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1. Two waves are traveling through the same container of nitrogen gas. Wave A has a
wavelength of 1.5 m. Wave B has a wavelength of 4.5 m. The speed of wave B must be________ the speed of wave A.
a. one-ninth
b. one-third
c. the same as
d. three times larger than
2. As the wavelength of a wave in a uniform medium inc

A wave causes a displacement y that is given in meters according to y=(0.45) sin (8.0 * 3.14t - 3.14x), where t and x are expressed in seconds and meters.
a.) What is the amplitude of the wave?
b.) what is the frequency of the wave?
c.) What is the wavelength of the wave?
d.) What is the speed of the wave?
e.) Is th

The equation for a wave moving along a straight wire is: (1) y= 0.5 sin (6 x - 4t)
To look at the motion of the crest, let y = ym= 0.5 m, thus obtaining an equation with only two variables, namely x and t.
a. For y= 0.5, solve for x to get (2) x(t) then take a (partial) derivative of x(t) to get the rate of change of

The diagrams attached represent the polarization states of light. In each case the wave is traveling along the x-axis in the positive x direction.
i) Which diagram represents linear polarized light at 45 degrees?
ii) Which diagram represents left circular light? Explain.
iii) Which diagram represents un-polarized light?

Uxx means second derivative with respect to x
Uyy means second derivative with respect to y
Uxx + Uyy = 0, 0 < x < pi, 0 < y < 1
Ux(0,y) = 0 = U(pi,y), 0 < y < 1
U(x,0) = 1, U(x,1) = 0, 0 < x < pi
Please show all work including how eigenvalues and eigenvectors are derived.
Thank you

As shown in ATTACHMENT #1, a wave is traveling toward +x on a wire. The motion of a point at x1= .45 m is shown. From the diagram, initial value of y is .12 meters and is increasing so initial slope is positive. The amplitude is .20 m, and the period is .5 sec.
From this information, develop the equation y(x,t) of the wave,

The vertical displacement of a string is given by y(x,t) = (6.00mm) cos[(3.25m to the power of (-1)) x - (7.22 s to the power of (-1))t]. What is the wavelength of the wave?

One end of a long wire under tension is moved up and down sending a wave along it. Assume the wire lies along an x axis with the moving end at the origin. The equation giving displacement y, in meters, of points on the wave is:
(1) y= (.06 m) sin (5 x - 25 t)
a. From given constants, calculate the value and units of the quan

The question attached is from this page.
http://farside.ph.utexas.edu/teaching/em/lectures/node48.html
Please answer with vector notation. The question is only about (450) so I don't believe that you have to read through all of it to answer.

1) A transverse traveling wave on a cord is represented by D = 0.48 sin (5.6x +84t) where D and x are in meters and t in seconds. For this wave determine (a) the wavelength (b) the frequency (c) velocity (magnitude and direction) (d) amplitude and (e) maximum and minimum speeds of particles of the cord.