Mathematics 173 – Applied Mathematics 1 Lab/Problem Set 6 – November 18, 1999

Goals

In the lab portion of this assignment, we will compute and plot solutions to the 1D Wave Equation to visualize the corresponding motions of a string.

Background

Recall that last time we saw that the motions of a thin, highly stretched, perfectly elastic string can be modeled by the 1D Wave Equation: u_{tt} = c^{2}u_{xx}, where u is the vertical displacement from the equilibrium or rest position of the string.

Lab Questions

A) In class just now, we began to consider the solution of the boundary value problem:

\{ \begin{matrix}u_{tt} = c^{2}u_{xx} \\ u(0, t) = 0 \\ u(L, t) = 0 \\ u(x, 0) = \{ \begin{matrix}2ax/L & if 0 ≤ x ≤ L/2 \\ 2a(L − x)/L & if L/2 ≤ x ≤ L \end{matrix} \\ u_{t}(x, 0) = 0 \end{matrix}

1) Using Maple, verify our formula for the solution

u(x, t) = ∑_{n=1}^{∞} \frac{8a sin (\frac{nπ}{2} )}{n^{2}π^{2}} sin (\frac{nπx}{L}) cos (\frac{nπct}{L})

Take a = .1, L = 10, c = 1, and plot (partial sums of) this solution as functions of x, t. Describe the motion of the string. Look up the animate command in the plots package and see if you can use that to make an animation of the vibrating string.

2) Let F be the odd periodic extension of the initial displacement function of the string u(x, 0) above (still with a = .1, L = 10, c = 1). Plot F (x−ct) and F (x+ct) as functions of (x, t) using Maple. These are called travelling waves. Do you see why the name is appropriate?

3) What happens if you plot (F (x−ct)+F (x+ct))/2 in the interval [0, 10]? Compare with your result from part 1. (Note: You will want to use Maple’s piecewise mechanism to define the odd periodic extension. Of course, you cannot include an infinite number of different options in the definition. This means you will need to define a function that agrees with the odd periodic extension only on some interval, and then plot only (x, t) such that x + ct and x − ct are in that interval(!)) See problem D below for a general statement about what is happening here!

B) Repeat question A, parts 1,2 with the same initial string displacement function but now make u_{t}(x, 0) = x(L − x)/10.

C) Note that the solutions in A,B represent strings that keep oscillating in an exact periodic fashion forever. In the real world, that never happens because of additional forces that we ignored in the derivation of the wave equation – e.g. air resistance, frictional forces at the ends, etc. In this problem, we will consider a PDE modeling a string moving in the presence of damping force(s) proportional to the string’s velocity:

ρ_{0}u_{tt} = T_{0}u_{xx} − βu_{t}

where β > 0. In addition, we assume that β is relatively small:

β^{2} < 4π^{2}ρ_{0}T_{0}/L^{2}.

Here ρ_{0} is the mass density, and T_{0} is the tension as before. We want to solve the boundary value problem

\{ \begin{matrix}ρ_{0}u_{tt} = T_{0}u_{xx} − βu_{t} \\ u(0, t) = 0 \\ u(L, t) = 0 \\ u(x, 0) = \{ \begin{matrix}2ax/L & if 0 ≤ x ≤ L/2 \\ 2a(L − x)/L & if L/2 ≤ x ≤ L \end{matrix} \\ u_{t}(x, 0) = 0 \end{matrix}

1) Show that if we apply the separation of variables method, looking for product solutions u(x, t) = ϕ(x)h(t), then we get the ordinary differential equations

ρ_{0}h′′(t) + βh′(t) + λT_{0}h(t) = 0

and

ϕ′′(x) = −λϕ(x)

2) Show that ϕ(x) = sin (\frac{nπx}{L}), and λ = (\frac{nπ}{L} )^{2} for some n ≥ 1.

3) Show that if

w_{n} = \sqrt{\frac{T_{0}}{ρ_{0}} (\frac{nπ}{L} )^{2} − \frac{β^{2}}{4ρ_{0}^{2}}}

then

h(t) = e^{−βt/2ρ_{0}} cos(w_{n}t)

and

h(t) = e^{−βt/2ρ_{0}} sin(w_{n}t)

are both solutions of the equation for h(t). Then we can construct series solutions:

u(x, t) = e^{−βt/2ρ_{0}} ∑_{n=1}^{∞}(a_{n} cos(w_{n}t) + b_{n} sin(w_{n}t)) sin (\frac{nπx}{L})

Explain how to determine a_{n} and b_{n} to satisfy the initial conditions u(x, 0) = f(x) and u_{t}(x, 0) = g(x) for any given functions f, g.

D) Problem 4.4.7 from the text. Assignment

Individual write-ups due Thursday, December 2.