DC Pandey Solutions: Wave Motion - Notes | Study Physics Class 11 - NEET

 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