USSC3002 Oscillations and Waves Lecture 11 Continuous Systems

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USSC3002 Oscillations and Waves Lecture 11
Continuous Systems

• Wayne M. Lawton
• Department of Mathematics
• National University of Singapore
• 2 Science Drive 2
• Singapore 117543

Email matwml_at_nus.edu.sg http//www.math.nus/matwm
l Tel [phone]
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POTENTIAL ENERGY OF A TAUT STRING

Consider a string without boundary that moves in
the x-y plane. If the string displacement is
given by a function y f(t,x) with support
-b(t),b(t) and small then its length
increases by
Therefore, if the string is taut and has
tension then its elastic potential energy
increases by
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KINETIC ENERGY OF A STRING

The kinetic energy of the string equals
where
is its linear density (mass per length).
Therefore its Lagrangian equals
where the Lagrangian density is defined by
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CONFIGURATION AND STATE VARIABLES
We recall that the Euler-Lagrange equations for
a physical system with conservative forces and a
finite dimensional configuration variable q and

Lagrangian
can be expressed as
where we interpret the partial derivatives to
be vector valued Frechet derivatives. This
suggests that for the vibrating string we treat
the Lagrangian to be a function of infinite
dimensional configuration variable f and velocity
variable
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Question 1. What should the EL equations be ?
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EULER-LAGRANGE EOM FOR A STRING

are
where
is the Frechet
derivative of L with respect to the velocity
hence
it is a linear functional whose value at a
function g is
and
so the EOM
is the wave equation
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VARIATIONAL PROBLEMS FOR MULTIPLE INTEGRALS

Greens theorem ?
If
where
is a bounded planar region and
is its
boundary oriented counterclockwise () direction.
If g vanishes on the boundary then the Frechet
derivative of L is represented by a density
function
(note the new notation)
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EL EOM FOR A VIBRATING MEMBRANE
A vibrating membrane with constant area density

with vertical displacement
and tension
having small
and with Dirichlet boundary conditions
has Lagrangian
whose density is
and EL EOM is
where the Laplacian
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FUNCTION SPACES
Consider a bounded planar region

and define the following space of functions
Question 2. Show that the scalar product
satisfies the Schwartz inequality
and with equality holding iff either g 0 or
We define the norm
and orthogonality
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FUNCTION SPACES
Define the Sobolev space (there are many others)

and its subspace
Question 3. Show that the scalar product
exists and satisfies the Schwartz inequality.
Question 4. Show that if
then
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EIGENFUNCTIONS OF THE LAPLACIAN
Problem 1. Find
that minimizes

subject to the constraint
The solution must satisfy
where the partials denote Frechet derivatives and
is a Lagrange multiplier. Question 3 implies
and since
that
it follows that if
solves
the minimization problem then
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EIGENFUNCTIONS OF THE LAPLACIAN
Problem 2. Find
that minimizes

and
with constraints
The solution
must satisfy
and
are Lagrange multipliers.
where
Clearly
hence
Question 4 ?
Continuing we construct an orthonormal basis
for
consisting of
eigenfunctions of
Also each
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EXAMPLES
Example 1.
is a rectangle

Example 2.
is a unit disc.
eigenfunctions
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NORMAL MODES

for zero boundary conditions on a bounded domain
Wave Equation
Heat Equation
Question 5. How can
and
be determined ?
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FOURIER MODES

can be use to expand solutions in
Wave Equation
Heat Equation
Question 6. How can
be determined ?
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REFLECTION AT A CHANGE OF DENSITY

Consider a solution of the wave equation
for transverse displacements
on an infinite string,
but whose linear density
with constant tension
and
for
for

that has the form
Question 8. What is the physical significance ?
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REFLECTION AT A CHANGE OF DENSITY

Question 9. Why does
Question 10. Why does
Question 11. Why does

These two boundary conditions give
We define coefficients of reflection
transmission
Question 12. What is their physical meaning ?
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LONGITUDINAL WAVES IN BARS
In a longitudinal wave the displacement
is in the

same direction as the wave as shown below
hence a small length dx of the bar between x and
xdx

is stretched or compressed by the factor
so by
Hooks law results in tension
at the point x where
is the constant tension. The
net force on a length
(with mass
)
is
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WAVES IN ELASTIC SOLIDS

The displacements are described by a vector
function
of a coordinate vector
Tension is described by the stress tensor
that is linearly related to the strain tensor

For an isotropic material with Lame constants
and the wave equations are
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TUTORIAL 11.
Problem 1. Fix an angle
define the rotation operator

to the function
that maps a function
defined by
Show that if f is twice differentiable then
Problem 2. Show that if f is twice differentiable
then

where
Problem 3. Use this polar coordinate expression
for
and the properties of the Bessel functions on
vufoil 12
to derive the following differential equations
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TUTORIAL 11.
Problem 4. Let
be a bounded region with

boundary
be continuous.
and let
that minimizes
Prove that the function
satisfies
subject to the constraint
(ie it is harmonic)
on the interior of

and satisfies the Dirichlet boundary conditions.
Then prove the solution of this Laplace problem
is unique.
Problem 5. Derive the solution for the reflection
problem on vufoil 15 if the incident wave has the
form
Problem 6. Use equations in vufoil 18 to compute
speeds of
if u depends only on t and
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