USSC3002 Oscillations and Waves Lecture 4 The Wave Equation

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USSC3002 Oscillations and Waves Lecture 4 The
Wave Equation

• 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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DERIVATION OF WAVE EQUATION
We derive the equation for the transverse
vibrations of a string of uniform linear density
Ignoring gravity and assuming that

• every point on the string is displaced (only)
• in the vertical direction from its equilibrium
• position on the x-axis,

(ii) displacement y(x,t) satisfies
(iii) and the tension T is positive and constant.

we proceed to derive the wave equation.
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DERIVATION OF WAVE EQUATION
This figure shows a small length ds of string.
Hence
1.
2.
3.
4.
5.
6.
7.
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ALTERNATE DERIVATION
from transverse vibrations of N particles on a
string
1.
2.
3.
4.
5.
6.
7.
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NORMAL MODES OF N PARTICLES
on a string of length d have the form
and satisfy
therefore
and
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HARMONIC SOLUTION
Given a solution

of the wave equation
hence
we see that
where
therefore
This is a sum of waves moving left, right with
speed
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GENERAL FORMULA
Given a solution
we define a function
therefore the wave equation for y(x,t) implies
that
for some functions f and g, hence
which is consistent with the harmonic solution.
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DALEMBERTS FORMULA
If y solves the wave equation and the initial
conditions
then
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TUTORIAL 4

• (HRWalker, p. 164) The potential
• energy of a diatomic molecule is

where r is the separation of the
two atoms of the molecule and A and B are
positive constants and it is associated with the
force that binds the two atoms together. (i) Find
the equilibrium separation, that is, the distance
between the atoms at which the force on each atom
is zero. Is the force repulsive or attractive if
their separation is (ii) smaller, (iii) larger
than the equilibrium separation? (iv) compute the
frequency of vibrations of the two atoms for very
small displacements from equilbrium.
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TUTORIAL 4
2. Complete all the details of each step in the
derivation of the wave equation in vufoil 3.
3. Derive the wave equation if T is not constant
(this can happen if body forces are present).
4. Use MATLAB to compute and plot the eigenvalues
eigenvectors of the vibration matrix for N
particles.
5. Show DAlemberts formula for harmonic
solutions.
6. Read the handouts and use Greens Theorem to
do problems 17-20 on page 1090.
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