Important Formulas: Wave Motion

 Page 1


1 Waves Motion
General equation of wave:
@
2
y
@x
2
=
1
v
2
@
2
y
@t
2
.
Notation: Amplitude A, Frequency , Wavelength , Pe-
riod T , Angular Frequency !, Wave Number k,
T =
1

=
2
!
; v =; k =
2

Progressive wave travelling with speed v:
y =f(tx=v); +x; y =f(t +x=v); x
Progressive sine wave:

2

x
y
A
y =A sin(kx!t) =A sin(2 (x=t=T ))
2 Waves on a String
Speed of waves on a string with mass per unit length 
and tension T : v =
p
T=
Transmitted power: P
av
= 2
2
vA
2

2
Interference:
y
1
=A
1
sin(kx!t); y
2
=A
2
sin(kx!t +)
y =y
1
+y
2
=A sin(kx!t +)
A =
q
A
1
2
+A
2
2
+ 2A
1
A
2
cos
tan =
A
2
sin
A
1
+A
2
cos
 =

2n; constructive;
(2n + 1); destructive:
Standing Waves:
2Acoskx
A N A N A
x
=4
y
1
=A
1
sin(kx!t); y
2
=A
2
sin(kx +!t)
y =y
1
+y
2
= (2A coskx) sin!t
x =
 
n +
1
2



2
; nodes; n = 0; 1; 2;:::
n

2
; antinodes. n = 0; 1; 2;:::
String xed at both ends:
L
N
A N A
N
=2
1. Boundary conditions: y = 0 at x = 0 and at x =L
2. Allowed Freq.: L =n

2
;  =
n
2L
q
T

; n = 1; 2; 3;:::.
3. Fundamental/1
st
harmonics: 
0
=
1
2L
q
T

4. 1
st
overtone/2
nd
harmonics: 
1
=
2
2L
q
T

5. 2
nd
overtone/3
rd
harmonics: 
2
=
3
2L
q
T

6. All harmonics are present.
String xed at one end:
L
N
A N
A
=2
1. Boundary conditions: y = 0 at x = 0
2. Allowed Freq.: L = (2n + 1)

4
;  =
2n+1
4L
q
T

; n =
0; 1; 2;:::.
3. Fundamental/1
st
harmonics: 
0
=
1
4L
q
T

4. 1
st
overtone/3
rd
harmonics: 
1
=
3
4L
q
T

5. 2
nd
overtone/5
th
harmonics: 
2
=
5
4L
q
T

6. Only odd harmonics are present.
Sonometer: /
1
L
, /
p
T , /
1
p

.  =
n
2L
q
T

3 Sound Waves
Displacement wave: s =s
0
sin!(tx=v)
Pressure wave: p =p
0
cos!(tx=v); p
0
= (B!=v)s
0
Speed of sound waves:
v
liquid
=
s
B

; v
solid
=
s
Y

; v
gas
=
s

P

Intensity: I =
2
2
B
v
s
0
2

2
=
p0
2
v
2B
=
p0
2
2v
Standing longitudinal waves:
p
1
=p
0
sin!(tx=v); p
2
=p
0
sin!(t +x=v)
p =p
1
+p
2
= 2p
0
coskx sin!t
Closed organ pipe:
L
1. Boundary condition: y = 0 at x = 0
2. Allowed freq.: L = (2n + 1)

4
;  = (2n + 1)
v
4L
; n =
0; 1; 2;:::
3. Fundamental/1
st
harmonics: 
0
=
v
4L
4. 1
st
overtone/3
rd
harmonics: 
1
= 3
0
=
3v
4L
5. 2
nd
overtone/5
th
harmonics: 
2
= 5
0
=
5v
4L
 
Page 2


1 Waves Motion
General equation of wave:
@
2
y
@x
2
=
1
v
2
@
2
y
@t
2
.
Notation: Amplitude A, Frequency , Wavelength , Pe-
riod T , Angular Frequency !, Wave Number k,
T =
1

=
2
!
; v =; k =
2

Progressive wave travelling with speed v:
y =f(tx=v); +x; y =f(t +x=v); x
Progressive sine wave:

2

x
y
A
y =A sin(kx!t) =A sin(2 (x=t=T ))
2 Waves on a String
Speed of waves on a string with mass per unit length 
and tension T : v =
p
T=
Transmitted power: P
av
= 2
2
vA
2

2
Interference:
y
1
=A
1
sin(kx!t); y
2
=A
2
sin(kx!t +)
y =y
1
+y
2
=A sin(kx!t +)
A =
q
A
1
2
+A
2
2
+ 2A
1
A
2
cos
tan =
A
2
sin
A
1
+A
2
cos
 =

2n; constructive;
(2n + 1); destructive:
Standing Waves:
2Acoskx
A N A N A
x
=4
y
1
=A
1
sin(kx!t); y
2
=A
2
sin(kx +!t)
y =y
1
+y
2
= (2A coskx) sin!t
x =
 
n +
1
2



2
; nodes; n = 0; 1; 2;:::
n

2
; antinodes. n = 0; 1; 2;:::
String xed at both ends:
L
N
A N A
N
=2
1. Boundary conditions: y = 0 at x = 0 and at x =L
2. Allowed Freq.: L =n

2
;  =
n
2L
q
T

; n = 1; 2; 3;:::.
3. Fundamental/1
st
harmonics: 
0
=
1
2L
q
T

4. 1
st
overtone/2
nd
harmonics: 
1
=
2
2L
q
T

5. 2
nd
overtone/3
rd
harmonics: 
2
=
3
2L
q
T

6. All harmonics are present.
String xed at one end:
L
N
A N
A
=2
1. Boundary conditions: y = 0 at x = 0
2. Allowed Freq.: L = (2n + 1)

4
;  =
2n+1
4L
q
T

; n =
0; 1; 2;:::.
3. Fundamental/1
st
harmonics: 
0
=
1
4L
q
T

4. 1
st
overtone/3
rd
harmonics: 
1
=
3
4L
q
T

5. 2
nd
overtone/5
th
harmonics: 
2
=
5
4L
q
T

6. Only odd harmonics are present.
Sonometer: /
1
L
, /
p
T , /
1
p

.  =
n
2L
q
T

3 Sound Waves
Displacement wave: s =s
0
sin!(tx=v)
Pressure wave: p =p
0
cos!(tx=v); p
0
= (B!=v)s
0
Speed of sound waves:
v
liquid
=
s
B

; v
solid
=
s
Y

; v
gas
=
s

P

Intensity: I =
2
2
B
v
s
0
2

2
=
p0
2
v
2B
=
p0
2
2v
Standing longitudinal waves:
p
1
=p
0
sin!(tx=v); p
2
=p
0
sin!(t +x=v)
p =p
1
+p
2
= 2p
0
coskx sin!t
Closed organ pipe:
L
1. Boundary condition: y = 0 at x = 0
2. Allowed freq.: L = (2n + 1)

4
;  = (2n + 1)
v
4L
; n =
0; 1; 2;:::
3. Fundamental/1
st
harmonics: 
0
=
v
4L
4. 1
st
overtone/3
rd
harmonics: 
1
= 3
0
=
3v
4L
5. 2
nd
overtone/5
th
harmonics: 
2
= 5
0
=
5v
4L
 
6. Only odd harmonics are present.
Open organ pipe:
L
A
N
A
N
A
1. Boundary condition: y = 0 at x = 0
Allowed freq.: L =n

2
;  =n
v
4L
; n = 1; 2;:::
2. Fundamental/1
st
harmonics: 
0
=
v
2L
3. 1
st
overtone/2
nd
harmonics: 
1
= 2
0
=
2v
2L
4. 2
nd
overtone/3
rd
harmonics: 
2
= 3
0
=
3v
2L
5. All harmonics are present.
Resonance column:
l1 +d
l2 +d
l
1
+d =

2
; l
2
+d =
3
4
; v = 2(l
2
l
1
)
Beats: two waves of almost equal frequencies !
1
!
2
p
1
=p
0
sin!
1
(tx=v); p
2
=p
0
sin!
2
(tx=v)
p =p
1
+p
2
= 2p
0
cos 
!(tx=v) sin!(tx=v)
! = (!
1
+!
2
)=2; 
! =!
1
!
2
(beats freq.)
Doppler Eect:
 =
v +u
o
vu
s

0
where, v is the speed of sound in the medium, u
0
is
the speed of the observer w.r.t. the medium, consid-
ered positive when it moves towards the source and
negative when it moves away from the source, and u
s
is the speed of the source w.r.t. the medium, consid-
ered positive when it moves towards the observer and
negative when it moves away from the observer.
4 Light Waves
Plane Wave: E =E
0
sin!(t
x
v
); I =I
0
Spherical Wave: E =
aE0
r
sin!(t
r
v
); I =
I0
r
2
Young's double slit experiment
Path dierence: 
x =
dy
D
S1
P
S2
d
y
D

Phase dierence:  =
2


x
Interference Conditions: for integer n,
 =

2n; constructive;
(2n + 1); destructive;

x =

n; constructive;

n +
1
2


; destructive
Intensity:
I =I
1
+I
2
+ 2
p
I
1
I
2
cos;
I
max
=

p
I
1
+
p
I
2

2
; I
min
=

p
I
1

p
I
2

2
I
1
=I
2
:I = 4I
0
cos
2 
2
; I
max
= 4I
0
; I
min
= 0
Fringe width: w =
D
d
Optical path: 
x
0
=
x
Interference of waves transmitted through thin lm:

x = 2d =

n; constructive;

n +
1
2


; destructive:
Diraction from a single slit: 
b
y
y
D
For Minima: n =b sinb(y=D)
Resolution: sin =
1:22
b
Law of Malus: I =I
0
cos
2

I0 I