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Page 1
14. Wave Motion
Introductory Exercise 14.1
1. A func tion, f can rep re sent wave
equa tion, if it sat isfy
¶
¶
=
¶
¶
2
2
2
2
2
f
t
v
f
x
For, y a t = sin , w
¶
¶
= - = -
2
2
2 2
y
t
a t y w w w sin
but,
¶
¶
=
2
2
0
y
x
So, y do not represent wave equation.
2. y x t ae
bx et
( , )
( )
=
- -
2
=
- -
ae
kx t ( ) w
2
Þ k b = and w = e Þ v
k
c
b
= =
w
3. y x t
x t
( , )
( )
=
+ +
1
1 4
2
w
rep r e sent the
given pulse, where,
y x
k x x
( , ) 0
1
1
1
1
2 2 2
=
+
=
+
Þ k = 1
and y x z
x x
( , )
( ) ( )
=
+ -
=
+ -
1
1 2
1
1 1
2 2
w
Þ w =
1
2
\ v
k
= = =
w 1 2
1
/
0.5 m/s
4. y
x t
a
b kx t
=
+ +
=
+ +
10
5 2
2 2
( ) ( ) w
Amplitude, y
a
b
max
= = =
10
5
2 m
and k = 1; w = 2
v
k
= =
w
2 m/s and is travelling in (–) x
direction.
5. y
kx t
=
- +
10
2
2
( ) w
y x
k x x
( , ) 0
10
2
10
2
2 2 2
=
+
=
+
Þ k = 1
w = = vk 2 m/s ´
-
1
1
m = 2 rad/s
Þ y
x t
=
- +
10
2 2
2
( )
Introductory Exercise 14.2
1. y x t ( , ) = 0.02 sin
x t
0.05 0.01
+
æ
è
ç
ö
ø
÷
m
= + A kx t sin ( ) w m
Þ A = 0.02 m, k =
1
0.05
m
-1
, w =
-
1
001
1
.
s
(a) v
k
= =
w 0.05
0.01
m/s = 5 m/s
(b) v
y
t
A kx t
p
=
¶
¶
= + w w cos ( )
v
p
( , ) 0.2 0.3 0.02
0.01
= ´
1
cos
0.2
0.5
0.3
0.01
+
æ
è
ç
ö
ø
÷
= + 2 4 30 cos ( )
= 2 34 cos
= - 2( 0.85)
= - 1.7 m/s
2. Yes, ( ) ( )
max
v A Ak
k
Ak v
p
= = × = w
w
3. l = 4 cm, v = 40 cm/s (given)
(a) n
l
= = =
v 40
4
10
cm/s
cm
Hz
Page 2
14. Wave Motion
Introductory Exercise 14.1
1. A func tion, f can rep re sent wave
equa tion, if it sat isfy
¶
¶
=
¶
¶
2
2
2
2
2
f
t
v
f
x
For, y a t = sin , w
¶
¶
= - = -
2
2
2 2
y
t
a t y w w w sin
but,
¶
¶
=
2
2
0
y
x
So, y do not represent wave equation.
2. y x t ae
bx et
( , )
( )
=
- -
2
=
- -
ae
kx t ( ) w
2
Þ k b = and w = e Þ v
k
c
b
= =
w
3. y x t
x t
( , )
( )
=
+ +
1
1 4
2
w
rep r e sent the
given pulse, where,
y x
k x x
( , ) 0
1
1
1
1
2 2 2
=
+
=
+
Þ k = 1
and y x z
x x
( , )
( ) ( )
=
+ -
=
+ -
1
1 2
1
1 1
2 2
w
Þ w =
1
2
\ v
k
= = =
w 1 2
1
/
0.5 m/s
4. y
x t
a
b kx t
=
+ +
=
+ +
10
5 2
2 2
( ) ( ) w
Amplitude, y
a
b
max
= = =
10
5
2 m
and k = 1; w = 2
v
k
= =
w
2 m/s and is travelling in (–) x
direction.
5. y
kx t
=
- +
10
2
2
( ) w
y x
k x x
( , ) 0
10
2
10
2
2 2 2
=
+
=
+
Þ k = 1
w = = vk 2 m/s ´
-
1
1
m = 2 rad/s
Þ y
x t
=
- +
10
2 2
2
( )
Introductory Exercise 14.2
1. y x t ( , ) = 0.02 sin
x t
0.05 0.01
+
æ
è
ç
ö
ø
÷
m
= + A kx t sin ( ) w m
Þ A = 0.02 m, k =
1
0.05
m
-1
, w =
-
1
001
1
.
s
(a) v
k
= =
w 0.05
0.01
m/s = 5 m/s
(b) v
y
t
A kx t
p
=
¶
¶
= + w w cos ( )
v
p
( , ) 0.2 0.3 0.02
0.01
= ´
1
cos
0.2
0.5
0.3
0.01
+
æ
è
ç
ö
ø
÷
= + 2 4 30 cos ( )
= 2 34 cos
= - 2( 0.85)
= - 1.7 m/s
2. Yes, ( ) ( )
max
v A Ak
k
Ak v
p
= = × = w
w
3. l = 4 cm, v = 40 cm/s (given)
(a) n
l
= = =
v 40
4
10
cm/s
cm
Hz
(b) D D f
p
l
=
2
x
= ´
2
4
p
cm
2.5 cm =
5
4
p
rad
(c) D D D t
T
= =
2
1
2 p
f
pn
f
=
´
´
1
2 10 3 p
p
=
1
60
s
(d) v v
p p
= ( )
max
= - = - A A w p n 2
= - ´ ´
-
2 2 10
1
p cm s
= - 40 p cm/s
= - 1.26 cm/s
4. (a)
y A t kx = - sin ( ) w
= × -
æ
è
ç
ö
ø
÷ A v t x sin
2 2 p
l
p
l
= ´ -
æ
è
ç
ö
ø
÷ 0.05
0.4 0.4
sin 12
2 2 p p
t x
= - 0.05 sin ( ) 60 5 p p t x
(b) y ( , 0.25 0.15)
= ´ - ´ 0.05 0.15 0.25) sin (60 5 p p
= - 0.05 .25 sin ( ) 9 1 p p
= 0.05 7.75 sin ( ) p = 0.05 1.75 sin ( ) p
= - 0.0354 m = - 3.54 cm
(c) D D
Df
t
T
= = =
2 60 p
f
w
p
p
0.25
=
1
240
s = 4.2 ms
Introductory Exercise 14.3
1. v
T T
m l
Tl
m
= = =
m /
=
´
=
500 2 100 5
3 0.06
= 129.1 m/s
2. v
T T
A
= =
× m r
=
´ ´
=
-
0.98
9.8 10 10
10
3 6
m/s
Introductory Exercise 14.4
1. I
P
r
= =
´
=
4
1
4 1
1
4
2 2
p p p
W
m
W m
2
( )
/
2. For line source, I
rl
=
r
p 2
Þ I
r
µ
1
and as I A µ
2
Þ A
r
µ
1
2 | Waves & Motion
x
y
Page 3
14. Wave Motion
Introductory Exercise 14.1
1. A func tion, f can rep re sent wave
equa tion, if it sat isfy
¶
¶
=
¶
¶
2
2
2
2
2
f
t
v
f
x
For, y a t = sin , w
¶
¶
= - = -
2
2
2 2
y
t
a t y w w w sin
but,
¶
¶
=
2
2
0
y
x
So, y do not represent wave equation.
2. y x t ae
bx et
( , )
( )
=
- -
2
=
- -
ae
kx t ( ) w
2
Þ k b = and w = e Þ v
k
c
b
= =
w
3. y x t
x t
( , )
( )
=
+ +
1
1 4
2
w
rep r e sent the
given pulse, where,
y x
k x x
( , ) 0
1
1
1
1
2 2 2
=
+
=
+
Þ k = 1
and y x z
x x
( , )
( ) ( )
=
+ -
=
+ -
1
1 2
1
1 1
2 2
w
Þ w =
1
2
\ v
k
= = =
w 1 2
1
/
0.5 m/s
4. y
x t
a
b kx t
=
+ +
=
+ +
10
5 2
2 2
( ) ( ) w
Amplitude, y
a
b
max
= = =
10
5
2 m
and k = 1; w = 2
v
k
= =
w
2 m/s and is travelling in (–) x
direction.
5. y
kx t
=
- +
10
2
2
( ) w
y x
k x x
( , ) 0
10
2
10
2
2 2 2
=
+
=
+
Þ k = 1
w = = vk 2 m/s ´
-
1
1
m = 2 rad/s
Þ y
x t
=
- +
10
2 2
2
( )
Introductory Exercise 14.2
1. y x t ( , ) = 0.02 sin
x t
0.05 0.01
+
æ
è
ç
ö
ø
÷
m
= + A kx t sin ( ) w m
Þ A = 0.02 m, k =
1
0.05
m
-1
, w =
-
1
001
1
.
s
(a) v
k
= =
w 0.05
0.01
m/s = 5 m/s
(b) v
y
t
A kx t
p
=
¶
¶
= + w w cos ( )
v
p
( , ) 0.2 0.3 0.02
0.01
= ´
1
cos
0.2
0.5
0.3
0.01
+
æ
è
ç
ö
ø
÷
= + 2 4 30 cos ( )
= 2 34 cos
= - 2( 0.85)
= - 1.7 m/s
2. Yes, ( ) ( )
max
v A Ak
k
Ak v
p
= = × = w
w
3. l = 4 cm, v = 40 cm/s (given)
(a) n
l
= = =
v 40
4
10
cm/s
cm
Hz
(b) D D f
p
l
=
2
x
= ´
2
4
p
cm
2.5 cm =
5
4
p
rad
(c) D D D t
T
= =
2
1
2 p
f
pn
f
=
´
´
1
2 10 3 p
p
=
1
60
s
(d) v v
p p
= ( )
max
= - = - A A w p n 2
= - ´ ´
-
2 2 10
1
p cm s
= - 40 p cm/s
= - 1.26 cm/s
4. (a)
y A t kx = - sin ( ) w
= × -
æ
è
ç
ö
ø
÷ A v t x sin
2 2 p
l
p
l
= ´ -
æ
è
ç
ö
ø
÷ 0.05
0.4 0.4
sin 12
2 2 p p
t x
= - 0.05 sin ( ) 60 5 p p t x
(b) y ( , 0.25 0.15)
= ´ - ´ 0.05 0.15 0.25) sin (60 5 p p
= - 0.05 .25 sin ( ) 9 1 p p
= 0.05 7.75 sin ( ) p = 0.05 1.75 sin ( ) p
= - 0.0354 m = - 3.54 cm
(c) D D
Df
t
T
= = =
2 60 p
f
w
p
p
0.25
=
1
240
s = 4.2 ms
Introductory Exercise 14.3
1. v
T T
m l
Tl
m
= = =
m /
=
´
=
500 2 100 5
3 0.06
= 129.1 m/s
2. v
T T
A
= =
× m r
=
´ ´
=
-
0.98
9.8 10 10
10
3 6
m/s
Introductory Exercise 14.4
1. I
P
r
= =
´
=
4
1
4 1
1
4
2 2
p p p
W
m
W m
2
( )
/
2. For line source, I
rl
=
r
p 2
Þ I
r
µ
1
and as I A µ
2
Þ A
r
µ
1
2 | Waves & Motion
x
y
AIEEE Corner
¢ Sub je c tive Ques ti ons (Level 1)
1. y x t ( , ) cos = 6.50 mm 2p
p
28.0 cm 0.0360 s
-
æ
è
ç
ç
ö
ø
÷
÷
t
= -
æ
è
ç
ö
ø
÷
A
x t
T
cos 2p
l
Þ A = 6.50 mm, l = 28.0 cm,
n = = =
-
1 1
1
T 0.036
s 27.78 Hz
v = = ´ =
-
nl 28.0 cm 27.78s cm/s
1
778
= 7.78 m/s
The wave is travelling along ( ) + ve x-axis.
2. y t
x
= -
æ
è
ç
ö
ø
÷
5 30
240
sin p
= -
æ
è
ç
ö
ø
÷
5 30
8
sin p
p
t x = - A t kx sin ( ) w
(a) y( , ) sin 2 0 5 3 0
8
2 = ´ - ´
æ
è
ç
ö
ø
÷
p
p
= - = - = - 5
4
5
2
35 sin
p
3.5 cm
(b) l
p p
p /
= = =
2 2
8
16
k
cm
(c) v
k
= = =
w p
p
30
8
240
/
cm/s
(d) n
w
p
p
p
= = =
2
30
2
15 Hz
3. y x t = -
- -
3 314
1 1
cm 3.14 cm s sin ( )
= =
- -
3 100
1 1
cm cm s sin ( ) p p x t
= - A kx t sin ( ) w
(a) ( )
max
v A
p
= = ´
-
w p 3 100
1
cm s
= = = 300 3 p p cm/s m/s 9.4 m/ s
(b) a y = - = - ´
-
w p
2 1 2
100 3 ( ) s cm
sin ( ) 6 111 p p -
= - - = 300 105 0 p p sin ( )
4. (a) D D D x
v
= f = =
´
´
l
p
n
p
f
p
p
2 2
350
500 2 3
/
= = =
7
60
7
50
p
p
m m 0.166
(b) D D D f
p
pn = =
2
2
T
t t = ´ ´
-
2 500 10
3
p
= = ° p 180
5. y x t
kx t
( , )
( )
=
+ +
6
3
2
w
y x
k x x
( , ) 0
6
3
6
3
2 2 2
=
+
=
+
Þ k =
-
1
1
m
Þ w = = ´ =
-
vk 4.5 m/s 4.5 rad/s 1
1
m
Þ y x t
x t
( , )
( )
=
- +
6
3
2
4.5
6. y
x t
= -
æ
è
ç
ö
ø
÷
1.0
2.0 0.01
sin p
= -
æ
è
ç
ö
ø
÷
1.0 sin
4.0 0.02
2p
x t
= -
æ
è
ç
ö
ø
÷
A
x t
T
sin 2p
l
(a) A = 1.0 mm, l = = 4.0 cm, 0.02 T s
(b) v
y
t
A
x t
T
p
= = - -
æ
è
ç
ö
ø
÷
¶
¶
w p
l
cos 2
= - -
æ
è
ç
ö
ø
÷
2
2
p
p
l
A
T
x t
T
cos
= -
´
-
æ
è
ç
ç
ö
ø
÷
÷
2
2
p
p
1.0 mm
0.02 4.0 0.02 s s
cos
x t
= - -
æ
è
ç
ç
ö
ø
÷
÷
p
p
10
m/s
2.0 cm 0.01
cos
x t
s
v
p
( ) 1.0 cm, 0.01s =
- -
æ
è
ç
ö
ø
÷
p
p
10
1
2
m/s
0.01
0.01
cos
= - =
p p
10 2
0 m/ s m/ s cos
(c) v
p
( ) 3.0, 0.01
= - -
æ
è
ç
ö
ø
÷
p
p
1 0
3
2
1 c os = 0 m/ s
Waves & Motion | 3
Page 4
14. Wave Motion
Introductory Exercise 14.1
1. A func tion, f can rep re sent wave
equa tion, if it sat isfy
¶
¶
=
¶
¶
2
2
2
2
2
f
t
v
f
x
For, y a t = sin , w
¶
¶
= - = -
2
2
2 2
y
t
a t y w w w sin
but,
¶
¶
=
2
2
0
y
x
So, y do not represent wave equation.
2. y x t ae
bx et
( , )
( )
=
- -
2
=
- -
ae
kx t ( ) w
2
Þ k b = and w = e Þ v
k
c
b
= =
w
3. y x t
x t
( , )
( )
=
+ +
1
1 4
2
w
rep r e sent the
given pulse, where,
y x
k x x
( , ) 0
1
1
1
1
2 2 2
=
+
=
+
Þ k = 1
and y x z
x x
( , )
( ) ( )
=
+ -
=
+ -
1
1 2
1
1 1
2 2
w
Þ w =
1
2
\ v
k
= = =
w 1 2
1
/
0.5 m/s
4. y
x t
a
b kx t
=
+ +
=
+ +
10
5 2
2 2
( ) ( ) w
Amplitude, y
a
b
max
= = =
10
5
2 m
and k = 1; w = 2
v
k
= =
w
2 m/s and is travelling in (–) x
direction.
5. y
kx t
=
- +
10
2
2
( ) w
y x
k x x
( , ) 0
10
2
10
2
2 2 2
=
+
=
+
Þ k = 1
w = = vk 2 m/s ´
-
1
1
m = 2 rad/s
Þ y
x t
=
- +
10
2 2
2
( )
Introductory Exercise 14.2
1. y x t ( , ) = 0.02 sin
x t
0.05 0.01
+
æ
è
ç
ö
ø
÷
m
= + A kx t sin ( ) w m
Þ A = 0.02 m, k =
1
0.05
m
-1
, w =
-
1
001
1
.
s
(a) v
k
= =
w 0.05
0.01
m/s = 5 m/s
(b) v
y
t
A kx t
p
=
¶
¶
= + w w cos ( )
v
p
( , ) 0.2 0.3 0.02
0.01
= ´
1
cos
0.2
0.5
0.3
0.01
+
æ
è
ç
ö
ø
÷
= + 2 4 30 cos ( )
= 2 34 cos
= - 2( 0.85)
= - 1.7 m/s
2. Yes, ( ) ( )
max
v A Ak
k
Ak v
p
= = × = w
w
3. l = 4 cm, v = 40 cm/s (given)
(a) n
l
= = =
v 40
4
10
cm/s
cm
Hz
(b) D D f
p
l
=
2
x
= ´
2
4
p
cm
2.5 cm =
5
4
p
rad
(c) D D D t
T
= =
2
1
2 p
f
pn
f
=
´
´
1
2 10 3 p
p
=
1
60
s
(d) v v
p p
= ( )
max
= - = - A A w p n 2
= - ´ ´
-
2 2 10
1
p cm s
= - 40 p cm/s
= - 1.26 cm/s
4. (a)
y A t kx = - sin ( ) w
= × -
æ
è
ç
ö
ø
÷ A v t x sin
2 2 p
l
p
l
= ´ -
æ
è
ç
ö
ø
÷ 0.05
0.4 0.4
sin 12
2 2 p p
t x
= - 0.05 sin ( ) 60 5 p p t x
(b) y ( , 0.25 0.15)
= ´ - ´ 0.05 0.15 0.25) sin (60 5 p p
= - 0.05 .25 sin ( ) 9 1 p p
= 0.05 7.75 sin ( ) p = 0.05 1.75 sin ( ) p
= - 0.0354 m = - 3.54 cm
(c) D D
Df
t
T
= = =
2 60 p
f
w
p
p
0.25
=
1
240
s = 4.2 ms
Introductory Exercise 14.3
1. v
T T
m l
Tl
m
= = =
m /
=
´
=
500 2 100 5
3 0.06
= 129.1 m/s
2. v
T T
A
= =
× m r
=
´ ´
=
-
0.98
9.8 10 10
10
3 6
m/s
Introductory Exercise 14.4
1. I
P
r
= =
´
=
4
1
4 1
1
4
2 2
p p p
W
m
W m
2
( )
/
2. For line source, I
rl
=
r
p 2
Þ I
r
µ
1
and as I A µ
2
Þ A
r
µ
1
2 | Waves & Motion
x
y
AIEEE Corner
¢ Sub je c tive Ques ti ons (Level 1)
1. y x t ( , ) cos = 6.50 mm 2p
p
28.0 cm 0.0360 s
-
æ
è
ç
ç
ö
ø
÷
÷
t
= -
æ
è
ç
ö
ø
÷
A
x t
T
cos 2p
l
Þ A = 6.50 mm, l = 28.0 cm,
n = = =
-
1 1
1
T 0.036
s 27.78 Hz
v = = ´ =
-
nl 28.0 cm 27.78s cm/s
1
778
= 7.78 m/s
The wave is travelling along ( ) + ve x-axis.
2. y t
x
= -
æ
è
ç
ö
ø
÷
5 30
240
sin p
= -
æ
è
ç
ö
ø
÷
5 30
8
sin p
p
t x = - A t kx sin ( ) w
(a) y( , ) sin 2 0 5 3 0
8
2 = ´ - ´
æ
è
ç
ö
ø
÷
p
p
= - = - = - 5
4
5
2
35 sin
p
3.5 cm
(b) l
p p
p /
= = =
2 2
8
16
k
cm
(c) v
k
= = =
w p
p
30
8
240
/
cm/s
(d) n
w
p
p
p
= = =
2
30
2
15 Hz
3. y x t = -
- -
3 314
1 1
cm 3.14 cm s sin ( )
= =
- -
3 100
1 1
cm cm s sin ( ) p p x t
= - A kx t sin ( ) w
(a) ( )
max
v A
p
= = ´
-
w p 3 100
1
cm s
= = = 300 3 p p cm/s m/s 9.4 m/ s
(b) a y = - = - ´
-
w p
2 1 2
100 3 ( ) s cm
sin ( ) 6 111 p p -
= - - = 300 105 0 p p sin ( )
4. (a) D D D x
v
= f = =
´
´
l
p
n
p
f
p
p
2 2
350
500 2 3
/
= = =
7
60
7
50
p
p
m m 0.166
(b) D D D f
p
pn = =
2
2
T
t t = ´ ´
-
2 500 10
3
p
= = ° p 180
5. y x t
kx t
( , )
( )
=
+ +
6
3
2
w
y x
k x x
( , ) 0
6
3
6
3
2 2 2
=
+
=
+
Þ k =
-
1
1
m
Þ w = = ´ =
-
vk 4.5 m/s 4.5 rad/s 1
1
m
Þ y x t
x t
( , )
( )
=
- +
6
3
2
4.5
6. y
x t
= -
æ
è
ç
ö
ø
÷
1.0
2.0 0.01
sin p
= -
æ
è
ç
ö
ø
÷
1.0 sin
4.0 0.02
2p
x t
= -
æ
è
ç
ö
ø
÷
A
x t
T
sin 2p
l
(a) A = 1.0 mm, l = = 4.0 cm, 0.02 T s
(b) v
y
t
A
x t
T
p
= = - -
æ
è
ç
ö
ø
÷
¶
¶
w p
l
cos 2
= - -
æ
è
ç
ö
ø
÷
2
2
p
p
l
A
T
x t
T
cos
= -
´
-
æ
è
ç
ç
ö
ø
÷
÷
2
2
p
p
1.0 mm
0.02 4.0 0.02 s s
cos
x t
= - -
æ
è
ç
ç
ö
ø
÷
÷
p
p
10
m/s
2.0 cm 0.01
cos
x t
s
v
p
( ) 1.0 cm, 0.01s =
- -
æ
è
ç
ö
ø
÷
p
p
10
1
2
m/s
0.01
0.01
cos
= - =
p p
10 2
0 m/ s m/ s cos
(c) v
p
( ) 3.0, 0.01
= - -
æ
è
ç
ö
ø
÷
p
p
1 0
3
2
1 c os = 0 m/ s
Waves & Motion | 3
v
p
( , ) cos 5.0 cm 0.01 ms s = - -
æ
è
ç
ö
ø
÷
p
p
10
5
2
1
= 0 m/s
v
p
( ) cos 7.0 cm, 0.01s m/s = - -
æ
è
ç
ö
ø
÷
p
p
10
7
2
1
= 0 m/s
(d) v
p
( ) 1.0 cm, 0.011s
= -
p
10
m/s
cos p
1
2
-
æ
è
ç
ö
ø
÷
0.011
0.01
= - -
æ
è
ç
ö
ø
÷
p
p
10
1
12
cos 1.1
= -
p
p
10
cos 0.6 = - =
p p
10
3
5
cos 9.7 cm/ s
v
p
( ) 1.0 cm, 0.012 s
= - -
æ
è
ç
ö
ø
÷
p
10
1
2
m / s
0 . 0 1 2
0 . 0 1
cos
= - -
p
p
10
cos ( ) 0.5 1.2
= - =
p
p
10
cos 0.7 18.5 cm/s
v
p
( ) 1.0 cm, 0.013 s = -
p
10
m/s
cos cos p
p
p
1
2 10
-
æ
è
ç
ö
ø
÷
= -
0.013
0.01
0.8
= 25.4 cm/s
7. (a) k = = =
-
2 2
40 20
1
p
l
p p
cm
cm
= 0.157 rad/cm
T = = =
1 1
8 n
s s 0.125
w pn p = = 2 16 rad/s = 50.26 rad/s
v = = ´ =
-
nl 8 40 320
1
s cm cm/ s
(b) y x t A kx t ( , ) cos ( ) = - w
= - 15.0 cm 0.157 50.3 cos ( ) x t
8. A = 0.06m and 2.5 cm l = 20
Þ l = =
20
8
2.5
cm cm
n
l
= =
v 300
8
m/s
cm
= 3750 Hz
y A kx t = - sin ( ) w = 0.06m
sin
2
2 3750
p
p
0.08
x t - ´
æ
è
ç
ö
ø
÷
= -
- -
0.06 78.5 23561.9 m m s sin ( )
1 1
x t
9. (a) n
l
= = =
v 8.00 m/s
0.32 m
25 Hz
T = = =
1 1
15 n
s 0.043 Hz
k = = =
2 2 p
l
p
0.32
19.63 rad/m
m
(b) y A kx t A
x t
T
= + = +
æ
è
ç
ö
ø
÷
cos ( ) cos w p
l
2
= +
æ
è
ç
ç
ö
ø
÷
÷
0.07
0.32 0.04 s
m
m
cos 2p
x t
(c) y = +
æ
è
ç
ö
ø
÷
0.07
0.36
0.32
0.15
0.04
m cos 2p
= +
æ
è
ç
ö
ø
÷
0.07 m cos 2
9
8
30
8
p
= 0.07 m cos
39
4
p
= -
æ
è
ç
ö
ø
÷
0.07 m cos 10
4
p
p
= 0.07 m cos
p
4
= 0.0495 m
(d) D D
D
t
T
= f =
f
=
+
´ 2 2
4
2 25 p pn
p p
p
/
= =
3
200
0.015 s s
10. v
T T
A
Mg
A
= = =
m r r
=
´
´ ´ ´
-
2
8920 10
3 2
9.8
3.14 1.2 ( )
=
´ ´
´ ´
=
2 10
22
4
9.8
89.2 3.14 1.44
m/s
11. l n µ µ µ T M
Þ
l
l
2
1
2
1
=
M
M
= = =
8
2
4 2.
Þ l l
2 1
2 =
= 0.12 m.
4 | Waves & Motion
Page 5
14. Wave Motion
Introductory Exercise 14.1
1. A func tion, f can rep re sent wave
equa tion, if it sat isfy
¶
¶
=
¶
¶
2
2
2
2
2
f
t
v
f
x
For, y a t = sin , w
¶
¶
= - = -
2
2
2 2
y
t
a t y w w w sin
but,
¶
¶
=
2
2
0
y
x
So, y do not represent wave equation.
2. y x t ae
bx et
( , )
( )
=
- -
2
=
- -
ae
kx t ( ) w
2
Þ k b = and w = e Þ v
k
c
b
= =
w
3. y x t
x t
( , )
( )
=
+ +
1
1 4
2
w
rep r e sent the
given pulse, where,
y x
k x x
( , ) 0
1
1
1
1
2 2 2
=
+
=
+
Þ k = 1
and y x z
x x
( , )
( ) ( )
=
+ -
=
+ -
1
1 2
1
1 1
2 2
w
Þ w =
1
2
\ v
k
= = =
w 1 2
1
/
0.5 m/s
4. y
x t
a
b kx t
=
+ +
=
+ +
10
5 2
2 2
( ) ( ) w
Amplitude, y
a
b
max
= = =
10
5
2 m
and k = 1; w = 2
v
k
= =
w
2 m/s and is travelling in (–) x
direction.
5. y
kx t
=
- +
10
2
2
( ) w
y x
k x x
( , ) 0
10
2
10
2
2 2 2
=
+
=
+
Þ k = 1
w = = vk 2 m/s ´
-
1
1
m = 2 rad/s
Þ y
x t
=
- +
10
2 2
2
( )
Introductory Exercise 14.2
1. y x t ( , ) = 0.02 sin
x t
0.05 0.01
+
æ
è
ç
ö
ø
÷
m
= + A kx t sin ( ) w m
Þ A = 0.02 m, k =
1
0.05
m
-1
, w =
-
1
001
1
.
s
(a) v
k
= =
w 0.05
0.01
m/s = 5 m/s
(b) v
y
t
A kx t
p
=
¶
¶
= + w w cos ( )
v
p
( , ) 0.2 0.3 0.02
0.01
= ´
1
cos
0.2
0.5
0.3
0.01
+
æ
è
ç
ö
ø
÷
= + 2 4 30 cos ( )
= 2 34 cos
= - 2( 0.85)
= - 1.7 m/s
2. Yes, ( ) ( )
max
v A Ak
k
Ak v
p
= = × = w
w
3. l = 4 cm, v = 40 cm/s (given)
(a) n
l
= = =
v 40
4
10
cm/s
cm
Hz
(b) D D f
p
l
=
2
x
= ´
2
4
p
cm
2.5 cm =
5
4
p
rad
(c) D D D t
T
= =
2
1
2 p
f
pn
f
=
´
´
1
2 10 3 p
p
=
1
60
s
(d) v v
p p
= ( )
max
= - = - A A w p n 2
= - ´ ´
-
2 2 10
1
p cm s
= - 40 p cm/s
= - 1.26 cm/s
4. (a)
y A t kx = - sin ( ) w
= × -
æ
è
ç
ö
ø
÷ A v t x sin
2 2 p
l
p
l
= ´ -
æ
è
ç
ö
ø
÷ 0.05
0.4 0.4
sin 12
2 2 p p
t x
= - 0.05 sin ( ) 60 5 p p t x
(b) y ( , 0.25 0.15)
= ´ - ´ 0.05 0.15 0.25) sin (60 5 p p
= - 0.05 .25 sin ( ) 9 1 p p
= 0.05 7.75 sin ( ) p = 0.05 1.75 sin ( ) p
= - 0.0354 m = - 3.54 cm
(c) D D
Df
t
T
= = =
2 60 p
f
w
p
p
0.25
=
1
240
s = 4.2 ms
Introductory Exercise 14.3
1. v
T T
m l
Tl
m
= = =
m /
=
´
=
500 2 100 5
3 0.06
= 129.1 m/s
2. v
T T
A
= =
× m r
=
´ ´
=
-
0.98
9.8 10 10
10
3 6
m/s
Introductory Exercise 14.4
1. I
P
r
= =
´
=
4
1
4 1
1
4
2 2
p p p
W
m
W m
2
( )
/
2. For line source, I
rl
=
r
p 2
Þ I
r
µ
1
and as I A µ
2
Þ A
r
µ
1
2 | Waves & Motion
x
y
AIEEE Corner
¢ Sub je c tive Ques ti ons (Level 1)
1. y x t ( , ) cos = 6.50 mm 2p
p
28.0 cm 0.0360 s
-
æ
è
ç
ç
ö
ø
÷
÷
t
= -
æ
è
ç
ö
ø
÷
A
x t
T
cos 2p
l
Þ A = 6.50 mm, l = 28.0 cm,
n = = =
-
1 1
1
T 0.036
s 27.78 Hz
v = = ´ =
-
nl 28.0 cm 27.78s cm/s
1
778
= 7.78 m/s
The wave is travelling along ( ) + ve x-axis.
2. y t
x
= -
æ
è
ç
ö
ø
÷
5 30
240
sin p
= -
æ
è
ç
ö
ø
÷
5 30
8
sin p
p
t x = - A t kx sin ( ) w
(a) y( , ) sin 2 0 5 3 0
8
2 = ´ - ´
æ
è
ç
ö
ø
÷
p
p
= - = - = - 5
4
5
2
35 sin
p
3.5 cm
(b) l
p p
p /
= = =
2 2
8
16
k
cm
(c) v
k
= = =
w p
p
30
8
240
/
cm/s
(d) n
w
p
p
p
= = =
2
30
2
15 Hz
3. y x t = -
- -
3 314
1 1
cm 3.14 cm s sin ( )
= =
- -
3 100
1 1
cm cm s sin ( ) p p x t
= - A kx t sin ( ) w
(a) ( )
max
v A
p
= = ´
-
w p 3 100
1
cm s
= = = 300 3 p p cm/s m/s 9.4 m/ s
(b) a y = - = - ´
-
w p
2 1 2
100 3 ( ) s cm
sin ( ) 6 111 p p -
= - - = 300 105 0 p p sin ( )
4. (a) D D D x
v
= f = =
´
´
l
p
n
p
f
p
p
2 2
350
500 2 3
/
= = =
7
60
7
50
p
p
m m 0.166
(b) D D D f
p
pn = =
2
2
T
t t = ´ ´
-
2 500 10
3
p
= = ° p 180
5. y x t
kx t
( , )
( )
=
+ +
6
3
2
w
y x
k x x
( , ) 0
6
3
6
3
2 2 2
=
+
=
+
Þ k =
-
1
1
m
Þ w = = ´ =
-
vk 4.5 m/s 4.5 rad/s 1
1
m
Þ y x t
x t
( , )
( )
=
- +
6
3
2
4.5
6. y
x t
= -
æ
è
ç
ö
ø
÷
1.0
2.0 0.01
sin p
= -
æ
è
ç
ö
ø
÷
1.0 sin
4.0 0.02
2p
x t
= -
æ
è
ç
ö
ø
÷
A
x t
T
sin 2p
l
(a) A = 1.0 mm, l = = 4.0 cm, 0.02 T s
(b) v
y
t
A
x t
T
p
= = - -
æ
è
ç
ö
ø
÷
¶
¶
w p
l
cos 2
= - -
æ
è
ç
ö
ø
÷
2
2
p
p
l
A
T
x t
T
cos
= -
´
-
æ
è
ç
ç
ö
ø
÷
÷
2
2
p
p
1.0 mm
0.02 4.0 0.02 s s
cos
x t
= - -
æ
è
ç
ç
ö
ø
÷
÷
p
p
10
m/s
2.0 cm 0.01
cos
x t
s
v
p
( ) 1.0 cm, 0.01s =
- -
æ
è
ç
ö
ø
÷
p
p
10
1
2
m/s
0.01
0.01
cos
= - =
p p
10 2
0 m/ s m/ s cos
(c) v
p
( ) 3.0, 0.01
= - -
æ
è
ç
ö
ø
÷
p
p
1 0
3
2
1 c os = 0 m/ s
Waves & Motion | 3
v
p
( , ) cos 5.0 cm 0.01 ms s = - -
æ
è
ç
ö
ø
÷
p
p
10
5
2
1
= 0 m/s
v
p
( ) cos 7.0 cm, 0.01s m/s = - -
æ
è
ç
ö
ø
÷
p
p
10
7
2
1
= 0 m/s
(d) v
p
( ) 1.0 cm, 0.011s
= -
p
10
m/s
cos p
1
2
-
æ
è
ç
ö
ø
÷
0.011
0.01
= - -
æ
è
ç
ö
ø
÷
p
p
10
1
12
cos 1.1
= -
p
p
10
cos 0.6 = - =
p p
10
3
5
cos 9.7 cm/ s
v
p
( ) 1.0 cm, 0.012 s
= - -
æ
è
ç
ö
ø
÷
p
10
1
2
m / s
0 . 0 1 2
0 . 0 1
cos
= - -
p
p
10
cos ( ) 0.5 1.2
= - =
p
p
10
cos 0.7 18.5 cm/s
v
p
( ) 1.0 cm, 0.013 s = -
p
10
m/s
cos cos p
p
p
1
2 10
-
æ
è
ç
ö
ø
÷
= -
0.013
0.01
0.8
= 25.4 cm/s
7. (a) k = = =
-
2 2
40 20
1
p
l
p p
cm
cm
= 0.157 rad/cm
T = = =
1 1
8 n
s s 0.125
w pn p = = 2 16 rad/s = 50.26 rad/s
v = = ´ =
-
nl 8 40 320
1
s cm cm/ s
(b) y x t A kx t ( , ) cos ( ) = - w
= - 15.0 cm 0.157 50.3 cos ( ) x t
8. A = 0.06m and 2.5 cm l = 20
Þ l = =
20
8
2.5
cm cm
n
l
= =
v 300
8
m/s
cm
= 3750 Hz
y A kx t = - sin ( ) w = 0.06m
sin
2
2 3750
p
p
0.08
x t - ´
æ
è
ç
ö
ø
÷
= -
- -
0.06 78.5 23561.9 m m s sin ( )
1 1
x t
9. (a) n
l
= = =
v 8.00 m/s
0.32 m
25 Hz
T = = =
1 1
15 n
s 0.043 Hz
k = = =
2 2 p
l
p
0.32
19.63 rad/m
m
(b) y A kx t A
x t
T
= + = +
æ
è
ç
ö
ø
÷
cos ( ) cos w p
l
2
= +
æ
è
ç
ç
ö
ø
÷
÷
0.07
0.32 0.04 s
m
m
cos 2p
x t
(c) y = +
æ
è
ç
ö
ø
÷
0.07
0.36
0.32
0.15
0.04
m cos 2p
= +
æ
è
ç
ö
ø
÷
0.07 m cos 2
9
8
30
8
p
= 0.07 m cos
39
4
p
= -
æ
è
ç
ö
ø
÷
0.07 m cos 10
4
p
p
= 0.07 m cos
p
4
= 0.0495 m
(d) D D
D
t
T
= f =
f
=
+
´ 2 2
4
2 25 p pn
p p
p
/
= =
3
200
0.015 s s
10. v
T T
A
Mg
A
= = =
m r r
=
´
´ ´ ´
-
2
8920 10
3 2
9.8
3.14 1.2 ( )
=
´ ´
´ ´
=
2 10
22
4
9.8
89.2 3.14 1.44
m/s
11. l n µ µ µ T M
Þ
l
l
2
1
2
1
=
M
M
= = =
8
2
4 2.
Þ l l
2 1
2 =
= 0.12 m.
4 | Waves & Motion
12. T x L x g v x
T x
( ) ( ) , ( )
( )
= - = m
m
= - g L x ( )
dx
g L x
dt
( ) -
= ;
Let, L x y - =
dx dy = -
-
=
ò
dy
g y
t
L
0
\ t
g
y
=
-
1
12
1
0
/
= 2
L
g
13. (a) dm R T d w q
2
2 = sin
m q w q R d R T d 2 2
2
=
Þ w
m
2 2
R
T
=
\ Wave speed, v
T
R R = = =
m
w w
2 2
(b) Kink remains stationary when rope
and kink moves in opposite sence
ie . ., if rope is rotating anticlockwise
then kink has to move clockwise.
14. x is be ing mea sured from lover end of the
string
\ m x dm x dx x
x
( ) = = =
ò ò
m m
0
0
0
2
1
2
\ v x
T x m x g
( )
( ) ( )
= =
m m
= =
1
2
1
2
0
2
0
m
m
x g
x
gx
Þ
dx
gx
dt
l t
1
2
0 0
ò ò
=
Þ t
g
l = ´
2
2
0
\ t
l
g
=
8
0
15. m = =
dm
dx
kx
Þ M dm kx dx kL = = =
ò ò
0
2
2
1
2
Þ k
M
L
=
2
2
v x
T T
kx
TL
Mx
dx
dt
( ) = = = =
m
2
2
\ t dt
Mx
TL
dx
M
TL
L
L
= = =
+
ò ò
+
2 2
1
2
1
2
0
2
1
2
1
= =
2
3
2 2
3
2
3
2
ML
TL
ML
T
16. (a) v
T Mg
= =
m m
=
´
=
1.5 9.8
0.055
16.3 m/s
(b) l
n
= = =
v 16.3 m/s
0.136
120 / s
m
(c) l µ µ µ v T M i e . ., if M is
doubled both speed and wavelength
increases by a factor of 2 .
17. E I At a vAt = = 2
2 2 2
p n r
= 2
2 2 2
p n r a A v t ( ) ( . )
= 2
2 2 2
p n m a l .
= 2
2 2 2
p n a m
= ´ ´ ´ ´
-
2 120 10
2 2 3 2
( ) ( ) ( ) 3.14 0.16
´ ´
-
80 10
3
= ´
-
582 10
6
J = = 582 mJ 0.58 mJ
18. P
E
t
IA a A = = = 2
2 2 2
p n rn = 2
2 2 2
p n m a v
= 2
2 2 2
p n m a T
= ´ ´ 2 60
2 2
( ( ) 3.14)
´ ´ ´ ´
- -
( ) 6 10 80 5 10
2 2 2
= ´ ´ = 4 60
2
( ) 3.14 0.06 511.6 W
Waves & Motion | 5
R
T
T
dq
dq
x
Read More
FAQs on DC Pandey Solutions: Wave Motion - DC Pandey Solutions for JEE Physics
| 1. What is wave motion? |  |
Ans. Wave motion refers to the transfer of energy through the oscillation of particles or disturbances in a medium. It can be described as the propagation of a disturbance from one point to another without the actual transfer of matter.
| 2. What are the types of waves in wave motion? | |
Ans. There are two main types of waves in wave motion: mechanical waves and electromagnetic waves. Mechanical waves require a medium to propagate, such as sound waves and water waves, while electromagnetic waves can travel through a vacuum, such as light waves and radio waves.
| 3. How is wave motion described mathematically? | |
Ans. Wave motion can be described mathematically using equations such as the wave equation or the general solution to the wave equation. These equations relate the wave's properties, such as wavelength, frequency, amplitude, and velocity, to each other.
| 4. What is the principle of superposition in wave motion? | |
Ans. The principle of superposition states that when two or more waves meet at a point in space, the resulting displacement at that point is the algebraic sum of the individual displacements of the waves. This principle allows for the interference of waves, resulting in phenomena such as constructive and destructive interference.
| 5. How is wave motion related to everyday phenomena? | |
Ans. Wave motion is related to various everyday phenomena. For example, sound waves are responsible for our ability to hear, ocean waves create tides and surf, and light waves allow us to see. Additionally, wave motion is utilized in various technologies, such as radar, wireless communication, and medical imaging.
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