15.10:
a) The relevant expressions are
)
cos(
)
,
(
t
ω
kx
A
t
x
y

=
)
(
sin
ωt
kx
ωA
t
y
v
y

=
∂
∂
=
).
(
cos
2
2
2
kx
ωt
A
ω
t
v
t
y
a
y
y


=
∂
∂
=
∂
∂
=
b) (Take
A
,
k
and
ω
to be positive. At
,
0
=
t
x
the wave is represented by (19.7(a)); point
(i) in the problem corresponds to the origin, and points (ii)(vii) correspond to the points
in the figure labeled 17.) (i)
,
)
0
cos(
ωA
ωA
v
y
=
=
and the particle is moving upward (in
the positive
y
direction).
,
0
)
0
sin(
2
=

=
A
ω
a
y
and the particle is instantaneously not
accelerating. (ii)
,
2
)
4
cos(
ωA
π
ωA
v
y
=

=
and the particle is moving up.
,
2
)
4
sin(
2
2
A
ω
π
A
ω
a
y
=


=
and the particle is speeding up.
(iii)
,
0
)
2
cos(
=

=
π
ωA
v
y
and the particle is instantaneously at rest.
,
)
2
sin(
2
2
A
ω
π
A
ω
a
y
=


=
and the particle is speeding up.
(iv)
,
2
)
4
3
cos(
ωA
π
ωA
v
y

=

=
and the particle is moving down.
,
2
)
4
3
sin(
2
2
A
ω
π
A
ω
a
y
=


=
and the particle is slowing down (
y
v
is becoming less
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