Math Question BVP problem and Differential Equation

Math 4337: Homework 05:

1D Wave Equation

Problem 1. Solve the wave equation

utt = uxx

subject to boundary and initial conditions of the form

u (0, t) = 0

ux (1, t) = 0

u (x, 0) = 0

ut (x, 0) = 1 .

This would describe a string that is attached to something on the left side, connected to a frictionless hoop/rail system on the right side, and suddenly hit from below in such a way as to impart a constant velocity to the string.

Problem 2.1 The wave equation is typically derived for a long string under tension. However, it overlooks a major additional e�ect associated with this system. Unless the string is vibrating in a near-vacuum, then the string also loses momentum due to air resistance (even in a vacuum, it would slowly lose energy due to internal resistances). A crude model for this e�ect is to add a restoring force proportional to velocity. When this e�ect is added to the model, we obtain the equation

utt + δut = c2uxx

where δ is a damping coe�cient, and c is again the wave speed. Consider a plucked harp string subject to this equation that satis�es the following boundary and initial conditions:

u (0, t) = 0

u (2, t) = 0

u (x, 0) =

{ x 0 ≤ x < 1

2− x 1 ≤ x ≤ 2

ut (x, 0) = 0

.

Solve this equation, and compare the decay rates of the various modes � do the modes decay at di�erent rates as is seen in the di�usion equation? [Hints. You'll need to recall several things from ODEs here. (a) Solutions to a constant-coe�cient ODE take the form ert, which then leads to a characteristic equation for r. (b) For second-order ODEs like the ones we encounter here, r satis�es a quadratic equation, and can have complex roots a+ bi. (c) In that case, the associated homogeneous solutions are {eat cos (bt) , eat sin (bt)}.]

1 See Haberman 5/e: 4.4.3

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Problem 3. A jump rope could be crudely modeled by the equation

ρutt = τuxx u (0, t) = sin (ωt)

u (1, t) = sin (ωt)

where ρ is the rope density, τ is the tension, and ω is the forcing frequency. In general there are initial conditions associated with the problem as well, but in this problem we will only be interested in the particular solution. We will assume that damping causes the initial conditions to be forgotten (however, we will *not* include the damping explicitly, as it makes the algebra much messier).

(a) Split the problem into two parts u (x, t) = uB (x, t)+uI (x, t), and solve the boundary problem by direct integration.

(b) Identify the interior problem, noting that the time-dependent boundary conditions of the boundary problem produce an interior problem with a non-homogeneous forcing.

(c) Solve the interior problem by eigenfunction expansion. [Note: When you have reduced your problem to an ODE by means of the transform, you may ignore the homogeneous solutions (set both integration constants to zero), because we are ignoring the initial conditions.]

Now, the goal of a jump rope is to get the whole rope to oscillate more or less as a unit. That is, we want the shape of the solution to be dominated by the �rst mode sin (πx), without lots of wiggles due to the subsequent terms in the expansion.

(d) Note that the coe�cients of your eigenfunction expansion depend on {ρ, τ , ω}. For a jump rope, ρ is likely to be �xed. Why? On the other hand, τ and ω are likely to be adjustable. Why?

(e) Find a simple relation between τ and ω that induces resonance in the �rst mode sin (πx), and thus optimizes the jump rope experience.

Problem 4. (Optional Challenge) The simplest model of a long bridge is to describe it as a single beam, subject to the beam equation

ρutt + EIuxxxx = 0

where ρ is the density of the beam, and EI describes the beam's sti�ness. With four derivatives in x, this problem requires four boundary conditions. One common set of boundary conditions for beam problems is called clamped boundary conditions:

u (0, t) = 0

ux (0, t) = 0

u (L, t) = 0

ux (L, t) = 0

.

(a) Find the eigenvalues, eigenfunctions, and transform associated with the operator

L [u] = −uxxxx u (0, t) = 0

u (L, t) = 0

ux (0, t) = 0

ux (L, t) = 0

This is rather messy, and the eigenvalues must be determined numerically. Construct a graph involving the functions cos () and cosh−1 () that graphically illustrates the location of the eigenvalues.

(b) Assuming the λn have been numerically determined, apply the transform, solve the resulting ODE, and express the general solution in terms of an eigenfunction expansion. Using numerically-obtained values of the �rst few eigenvalues, state the natural oscillation frequency (in time) of the �rst few spatial modes.

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(c) Suppose that the wind can exert a forcing on the bridge with any time frequency less than a value of ωmax. What is the longest bridge that will not be subject to resonance phenomena (where the oscillation amplitude of any mode goes to in�nity)? What might be some strategies for building a longer bridge?