This value is
y
max
= 2
A
. This value is attained at (
x
n
, t
m
) for which
cos
kx
n
+
φ
2
¶
= 1
,
x
n
=
1
k
2
nπ

φ
2
¶
,
n
= 0
,
1
,
2
,
.....
sin
ωt
m
+
φ
2
¶
=

1
,
t
m
=
(4
m
+ 3)
π

φ
2
ω
,
m
= 0
,
1
,
2
,
.....
or at (
x
n
, t
m
) for which
cos
kx
n
+
φ
2
¶
=

1
,
x
n
=
1
k
•
(2
n
+ 1)
π

φ
2
‚
,
n
= 0
,
1
,
2
,
.....
sin
ωt
m
+
φ
2
¶
= 1
,
t
m
=
(4
m
+ 1)
π

φ
2
ω
,
m
= 0
,
1
,
2
,
.....
Note that the
x
n
are the locations of the
antinodes
of the resultant wave.
(6) The resultant wave is a
standing
wave, and therefore does not travel in either direction.
(7) The nodal positions are at
x
j
, where

2
A
cos
kx
j
+
φ
2
¶
sin
ωt
+
φ
2
¶
= 0
cos
kx
j
+
φ
2
¶
= 0
kx
j
+
φ
2
=
(2
j
+ 1)
π
2
,
j
= 0
,
1
,
2
,
....
x
j
=
(2
j
+ 1)
π

φ
2
k
,
j
= 0
,
1
,
2
,
....
(8) The antinode positions are at
x
n
, where
cos
kx
n
+
φ
2
¶
=
±
1
kx
n
+
φ
2
=
nπ,
n
= 0
,
1
,
2
,
.....
x
n
=
2
nπ

φ
2
k
,
n
= 0
,
1
,
2
,
.....
Note that this is the same set of
x
n
as obtained by combining the two sets from the answer
to question (5).
(9) The distance
d
between successive nodes, or between successive antinodes, is half a
wavelength, which is
2