We now let
x
→
x

vt
to have a wave traveling to the right. I.e.,
f
(
x, t
) =
y
m
sin[
k
(
x

vt
)] =
y
m
sin(
kx

kvt
) =
y
m
sin(
kx

ω
t
)
(1)
where
ω
:=
kv
,
v
=
ω
k
, and
ω
is the angular frequency. At any
x
, the motion is a simple
harmonic motion.
Let
t
→
t
+
2
π
ω
. Then,
f
(
x, t
+
2
π
ω
) =
y
m
sin[
kx

ω
(
t
+
2
π
ω
)] =
y
m
sin(
kx

ω
t

2
π
) =
y
m
sin(
kx

ω
t
)
(2)
therefore,
T
=
2
π
ω
is the period.
Recall that
f
=
1
T
=
ω
2
π
⇒
ω
= 2
π
f
.
v
=
ω
k
=
2
π
k
×
ω
2
π
=
λ
T
so that the wave travels a distance
λ
over a time
T
.
Another view: Take a snapshot in time:
so that
k
λ
= 2
π
.
4
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Snapshot in space:
so that
ω
T
= 2
π
.
String Velocity (in the transverse case)
String Velocity: Not the same as wave velocity!
u
(
x, t
) =
∂
f
∂
t
x
=const
(3)
Because we’re looking at how fast each material point is moving: the velocity here is of a
specific piece of the string. If
f
(
x, t
) =
y
m
sin(
kx

ω
t
) then
u
(
x, t
) =
∂
f
∂
t
=

ω
y
m
cos(
kx

ω
t
).
Di
ff
erent points have di
ff
erent velocity at same time. Same point has di
ff
erent velocities at
di
ff
erent times.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 LIORBURKO
 Physics, Derivative, Force, Mass, Velocity, Fundamental physics concepts

Click to edit the document details