Physics 498/MMA
Handout 8
Fall 2007
Mathematical Methods in Physics I
http://w3.physics.uiuc.edu/
∼
mstone5
Prof. M. Stone
2117 ESB
University of Illinois
1) Pantograph Drag
: A highspeed train picks up its electrical power via a pantograph
from an overhead line.
The locomotive travels at speed
U
and the pantograph exerts a
constant vertical force
F
on the power line.
F
U
Rheil Cymru
A highspeed train.
We make the usual small amplitude approximations and assume (not unrealistically) that
the line is supported in such a way that its vertical displacement obeys an inhomogeneous
KleinGordon equation
ρ
¨
y

Ty
00
+
ρ
Ω
2
y
=
Fδ
(
x

Ut
)
,
with
c
=
p
T/ρ
, the velocity of propagation of shortwavelength transverse waves on the
overhead cable.
a) Assume that
U < c
and solve for the steady state displacement of the cable about the
pickup point. (Hint: the disturbance is timeindependent when viewed from the train.)
b) Now assume that
U > c
. Again find an expression for the displacement of the cable.
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 Fall '07
 Stone
 Partial Differential Equations, Force, Power, Partial differential equation, Heat Diffusion Equation, inhomogeneous KleinGordon equation

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