- Consider a uniformly beaded string with
beads that is similar
to that pictured in Figure 19, except that each end of the string
is attached to a massless ring that slides (in the
-direction)
on a frictionless rod.
Demonstrate that the normal modes of the system take the form
- Consider a uniformly beaded string with
beads that is similar
to that pictured in Figure 19, except that the left end of the string is
fixed, and the right end is attached to a massless ring which slides (in the
-direction)
on a frictionless rod. Find the normal
modes and normal frequencies of the system.
- The preceding figure shows the left and right extremities of a linear LC network consisting of
identical inductors of inductance
, and
identical capacitors of capacitance
. Let the instantaneous current
flowing through the
th inductor be
, for
. Demonstrate from
Kirchhoff's circuital laws that the currents evolve in time according to the
coupled equations
- Suppose that the outermost two capacitors in the circuit considered in the previous exercise are short-circuited. Find the new normal frequencies of the system.
- A uniform string of length
, tension
, and mass per unit length
, is stretched between two immovable walls. Suppose that the string is
initially in its equilibrium state. At
it is
struck by a hammer in such a manner as to impart an impulsive
velocity
to a small segment of length
centered on the mid-point.
Find an expression for the subsequent motion of the string. Plot the motion
as a function of time in a similar fashion to Figure 29, assuming that
.
- A uniform string of length
, tension
, and mass per unit length
, is stretched between two massless rings, attached to its ends, that
slide (in the
-direction) along frictionless rods. Demonstrate that, in this case,
the most general solution to the wave equation takes the form

where , , and

Suppose that the string is initially in its equilibrium state. At it is struck by a hammer in such a manner as to impart an impulsive velocity to a small segment of length centered on the mid-point. Find an expression for the subsequent motion of the string. - The linear LC circuit considered in Exercise 3 can be thought of as a discrete
model of a uniform lossless transmission line (e.g., a co-axial cable). In this interpretation,
represents
, where
. Moreover,
, and
, where
and
are the capacitance per unit length and the
inductance per unit length of the line, respectively. Show that, in the limit
,
the evolution equation for the coupled currents given in Exercise 3 reduces to
the wave equation

in the transmission line limit. Hence, demonstrate that the voltage in a transmission line satisfies the wave equation - Consider a uniform string of length
, tension
, and mass per unit length
that is stretched between two immovable walls. Show that the total energy
of the string, which is the sum of its kinetic and potential energies, is