Physics Formulas for Class 11 and Class 12
Formulae Sheet for Waves
1 Waves Motion
General equation of wave:
2y x2
=
1 v2
2y t2
.
Notation: Amplitude A, Frequency , Wavelength , Period T , Angular Frequency , Wave Number k,
1 2
2
T = = , v = , k =
Progressive wave travelling with speed v:
y = f (t - x/v), +x; y = f (t + x/v), -x
Progressive sine wave:
y A
2
x
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 = T /? Transmitted power: Pav = 22?vA22
Interference:
y1 = A1 sin(kx - t), y2 = A2 sin(kx - t + ) y = y1 + y2 = A sin(kx - t + )
A = A12 + A22 + 2A1A2 cos
tan = A2 sin A1 + A2 cos
=
2n,
constructive;
(2n + 1), destructive.
2A cos kx
Standing Waves:
x AN A N A
/4
y1 = A1 sin(kx - t), y2 = A2 sin(kx + t)
y = y1 + y2 = (2A cos kx) sin t
x=
n
+
1 2
2
,
nodes;
n = 0, 1, 2, . . .
n
2
,
antinodes. n = 0, 1, 2, . . .
L
String fixed at both ends: N
N
ANA
/2
1. Boundary conditions: y = 0 at x = 0 and at x = L
2.
Allowed
Freq.:
L
=
n
2
,
=
n 2L
T ?
,
n
=
1, 2, 3, . . ..
3.
Fundamental/1st
harmonics:
0
=
1 2L
T ?
4.
1st
overtone/2nd
harmonics:
1
=
2 2L
T ?
concepts-of- | pg. 1
5.
2nd
overtone/3rd
harmonics:
2
=
3 2L
T ?
6. All harmonics are present.
L
String fixed at one end:
N
A
A
N
/2
1. Boundary conditions: y = 0 at x = 0
2.
Allowed Freq.:
L=
(2n
+
1)
4
,
=
2n+1 4L
0, 1, 2, . . ..
3.
Fundamental/1st
harmonics:
0
=
1 4L
T ?
T ?
,
n
=
4.
1st
overtone/3rd
harmonics:
1
=
3 4L
T ?
5.
2nd
overtone/5th
harmonics:
2
=
5 4L
T ?
6. Only odd harmonics are present.
Sonometer:
1 L
,
T,
1? .
=
n 2L
T ?
3 Sound Waves Displacement wave: s = s0 sin (t - x/v) Pressure wave: p = p0 cos (t - x/v), p0 = (B/v)s0 Speed of sound waves:
B
Y
P
vliquid =
,
vsolid =
,
vgas =
Intensity:
I
=
22 v
B
s0
2
2
=
p0 2 v 2B
=
p0 2 2v
Standing longitudinal waves:
p1 = p0 sin (t - x/v), p2 = p0 sin (t + x/v) p = p1 + p2 = 2p0 cos kx 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/1st
harmonics:
0
=
v 4L
4.
1st
overtone/3rd
harmonics:
1
= 30
=
3v 4L
5.
2nd
overtone/5th
harmonics:
2
= 50
=
5v 4L
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Formulae Sheet for Waves
6. Only odd harmonics are present.
Open organ pipe:
A
N
L
A
N
A
1. Boundary condition: y = 0 at x = 0
Allowed
freq.:
L
=
n
2
,
=
n
v 4L
,
n
=
1, 2, . . .
2.
Fundamental/1st
harmonics:
0
=
v 2L
3.
1st
overtone/2nd
harmonics:
1
= 20
=
2v 2L
4.
2nd
overtone/3rd
harmonics:
2
= 30
=
3v 2L
5. All harmonics are present.
l2 + d l1 + d
Resonance column:
l1
+
d
=
2
,
l2
+
d
=
3 4
,
v = 2(l2 - l1)
Beats: two waves of almost equal frequencies 1 2
p1 = p0 sin 1(t - x/v), p2 = p0 sin 2(t - x/v) p = p1 + p2 = 2p0 cos (t - x/v) sin (t - x/v) = (1 + 2)/2, = 1 - 2 (beats freq.)
Doppler Effect:
=
v v
+ -
uo us
0
where, v is the speed of sound in the medium, u0 is the speed of the observer w.r.t. the medium, considered positive when it moves towards the source and negative when it moves away from the source, and us is the speed of the source w.r.t. the medium, considered positive when it moves towards the observer and negative when it moves away from the observer.
4 Light Waves
Plane
Wave:
E = E0 sin (t -
x v
),
I
= I0
Spherical
Wave:
E
=
aE0 r
sin (t -
r v
),
I
=
I0 r2
Young's double slit experiment
concepts-of- | pg. 2
Path
difference:
x =
dy D
S1
P
y
d
S2
D
Phase
difference:
=
2
x
Interference Conditions: for integer n,
=
2n,
constructive;
(2n + 1), destructive,
x =
n,
constructive;
n
+
1 2
,
destructive
Intensity:
I = I1 + I2 + 2 I1I2 cos ,
Imax =
2
I1 + I2 , Imin =
2
I1 - I2
I1
=
I2
:
I
=
4I0 cos2
2
,
Imax
=
4I0,
Imin
=
0
Fringe
width:
w
=
D d
Optical path: x = ?x
Interference of waves transmitted through thin film:
n,
constructive;
x = 2?d =
n
+
1 2
,
destructive.
y
Diffraction from a single slit:
b
y
D
For Minima: n = b sin b(y/D)
Resolution:
sin =
1.22 b
Law of Malus: I = I0 cos2
I0
I
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