1. USSC3002 Oscillations and Waves Lecture 11 Continuous Systems
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore
Tel (65) 1

2. 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 2

3. 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 3

4. 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
Question 1. What should the EL equations be ? 4

5. 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 5

6. VARIATIONAL PROBLEMS FOR MULTIPLE INTEGRALSIf
Green’s theorem 
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) 6

7. EL EOM FOR A VIBRATING MEMBRANE
A vibrating membrane with constant area density
and tension
with vertical displacement
having small
and with Dirichlet boundary conditions
has Lagrangian
whose density is
and EL EOM is
where the Laplacian 7

8. 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 8

9. 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 9

10. 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
that
and since
it follows that if
solves
the minimization problem then 10

11. EIGENFUNCTIONS OF THE LAPLACIAN
Problem 2. Find
that minimizes
and
with constraints
The solution
must satisfy
where
and
are Lagrange multipliers.
Clearly
hence
Question 4 
Continuing we construct an orthonormal basis
for
consisting of
eigenfunctions of
Also each 11

12. EXAMPLES
Example 1.
is a rectangle
Example 2.
is a unit disc.
eigenfunctions 12

13. NORMAL MODES
for zero boundary conditions on a bounded domain
Wave Equation
Heat Equation
Question 5. How can
and
be determined ? 13

14. FOURIER MODES
can be use to expand solutions in
Wave Equation
Heat Equation
Question 6. How can
be determined ? 14

15. REFLECTION AT A CHANGE OF DENSITY
Consider a solution of the wave equation
for transverse displacements
on an infinite string,
with constant tension
but whose linear density
and
for
for
that has the form
Question 8. What is the physical significance ? 15

16. 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 ? 16

17. 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 x+dx
is stretched or compressed by the factor
so by
Hook’s law results in tension
at the point x where
is the constant tension. The
net force on a length
(with mass ) is 17

18. 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 18

19. TUTORIAL 11.
Problem 1. Fix an angle
define the rotation operator
that maps a function
to the 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 19

20. TUTORIAL 11.
Problem 4. Let
be a bounded region with
boundary
and let
be continuous.
Prove that the function
that minimizes
subject to the constraint
satisfies
on the interior of
(ie it is harmonic)
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 20