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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