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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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):
2
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
3
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!
■
(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
4
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?
■
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)
5
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 .
6
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).
■
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
■
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
9
■
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
■
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)
10
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
11
Compare equations (15), (19), and (20) with those in the textbook (chapter 19, section 19.3):
Can you explain the differences?
12
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
13
Also
and
and so forth
14
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
15
(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.