AE 5332 – Professor Dora E. Musielak

Lecture 9

Review Textbook Sections 19.2 and 19.3

Vibrating String Problem:

(1)

(2)

(3)

Superposition solution Eq. (8) in textbook Section 19.2

(8)

Solution, Eq. (16) in textbook

(16)

for the case when the B.C.s are as (2), and where the coefficients and are

(18a)

(18b)

Eq. (17a):

Eq. (17b):

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Problem 1. Solve the wave equation with the boundary and initial conditions:

, ,

Solution: Since the second B.C. (at is not as defined in (2), we cannot use

Eq. (16) in the textbook to solve this problem.

Thus, we can either (1) solve as we did in class (Lecture 7, pages 4-6), deriving the

general solution; or (2) solve starting from the generic superposition general

solution Eq. (8) in the textbook. Your choice!

Solutions should be the same, of course.

My method (1): Let's first carry out the analysis as I do, and begin with the general

solutions of the ODEs for x and t,

Applying the B.C.s , mean that and , that is

So,

The last expression gives the eigenvalues

The corresponding eigenfunctions are

Thus, solutions of this vibrating string satisfying the B.C.s of this problem are

Professor

Solution

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where

since for all n, and the solution of this vibrating string reduces to

We shall compare this result with solution using textbook method. They must be identical!

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(2) The textbook approach: it begins with Eq. (8) in Section 19.2:

Apply the first B.C.

so

and

or

where

Apply the second B.C.

so

and the solution equation becomes

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Textbook

Solution

Apply the first I.C.

where

and the second I.C,

Thus, for all n, and the solution reduces to

Is this the same solution obtained with Professor's method?

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Problem 2. For a plucked string at its midpoint and then released from rest, , and

is given by

Using Equations (16) and (18a) in textbook, leads to Eq. (19)

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The solution becomes, Eq. (20),

The RHS is a superposition of distinct modes of vibration (spatial and temporal).

Of course, if

then the solution is, Eq. (24),

Example 3. Solve for of a vibrating string in which , and

Solution: In the textbook,

and since , the coefficient and the above eq. reduces to

where the coefficient is determined from (18b),

either by hand or using a computer program you determine

Since for we may think that for all n. But of course this is

incorrect, since the initial velocity of the string is as defined by .

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If you use l'Hôpital rule for , you should get

And thus the solution is

Alternative Method: You could easily solve this type of problem by comparing the

RHS and LHS of the equation (17 a and 17b) where you establish the I.C.. In this

case is the latter, because you know

Thus,

The textbook calls this comparison "matching terms". That is, look at the terms on

both sides of the equality and attempt to match them to determine the coefficient :

Observe that for the equality to be true,

and all other .

Therefore,

which is consistent with the result obtained by integration in (18b).

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Example 4. Solve for of a vibrating string in which , and

Solution: You can use the "matching terms" method to show that

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all other , and the solution is

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Example 5. Solve for a vibrating string in which , and

Solve on your own for practice!

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2-D Wave Equation - Vibrating Membrane

Textbook solutions

Example 5. Solve for for a vibrating membrane in which c = 1, and

Solution: You can use the "matching terms" method. Start from (16a), and apply the

first I.C. at

that is

for this equality to be true,

all other

and the solution reduces to

or more compactly

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Example 7. Solve for for a vibrating membrane in which c = 1, and

Solution: You can use the "matching terms" method to show that

all other , and the solution is

and the solution reduces to

or more compactly

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Question: What if the problem prescribes and also ? How would you

solve it? Before answering, review the analysis in the textbook (Section 19.3) that

led to equations (16a), (16b), and (22)

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Summary of Solution Analysis for a Vibrating Membrane from Lecture 8

We found a solution of the wave equation for a rectangular membrane of sides

and ,

(1)

that satisfies the B.C.

(2)

(zero deflection at the boundary edge of the membrane), and the I.C.s

(3)

(4)

And the solution we obtained, using separation of variables is

(15)

where, from (12), the temporal frequency of the vibrating membrane is

From (15) and I.C. (3)

(16)

This expression represents a double Fourier series for the expansion of a function

(19)

(20)

for

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Compare equations (15), (19), and (20) with those in the textbook (chapter 19, section 19.3):

Can you explain the differences?

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Traveling Wave

We can also write the solution of the vibrating string in closed form in terms of the function

,

using the extended function

The solution becomes, Eq. (30):

where has period 2L, and we call this a traveling wave.

Example 6: Let , and use Eq. (20) to compute the displacement at

specified values of and . Then use Eq. (30) to show that both results agree.

Solution: With Eq. (20), the displacement is

Evaluate the displacement at , :

Using Eq. (30), the displacement at , is

since has period , we can add and subtract 20 (or integer multiples of the

argument of without changing its value

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Also

and

and so forth

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Damping Vibrating String

Vibration of string is normally damped due to its motion through a medium.

Denote the damping force

where is a known constant.

For a vibrating string subject to a damping force, the governing equation is

(1)

Can we use the separation of variables method to solve (1)?

Let

And following the analytical procedure we used before we obtain the differential equations

which can be easily solved.

Lateral Spring Attached to Vibrating String

We consider a system consisting of a lateral distributed spring attached to a spring with a

stiffness per unit length , as sketched below.

The governing equation is

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(1)

where .

Subject to the B.C.s and I.C.s

Let and we can show that separation of variables yields two differential

equations

which can be easily solved.

It can be shown that the lateral spring will increase the frequency of vibration.