Waves-FORMULA SHEETS | PHYSICS

Formula Sheet

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Waves

- All the formulas in one go

GENERAL EQUATION OF WAVE MOTION :

\frac{δ ^{2} y}{δ t ^{2}} = v^{2} \frac{δ ^{2} y}{δ x ^{2}}

y (x, t) = f (t \mp \frac{x}{v})

where,

y (x, t)

should be finite everywhere.

\Rightarrow

f (t + \frac{x}{v})

represents wave travelling in

- v e

x-axis.

\Rightarrow

f (t - \frac{x}{v})

represents wave travelling in

+ v e

x-axis.

y = A s i n (ω t \pm k x + ϕ)

here,

A

is the amplitude of wave.

ω

is the angular frequency of wave.

k

is the wave number.

(ω t \pm k x)

is the phase.

ϕ

is the phase difference.

TERMS RELATED TO WAVE MOTION ( FOR 1-D PROGRESSIVE SINE WAVE)

Wave number (or propagation constant) (k) :

k = 2 π / λ = \frac{ω}{v}

r a d i a n m^{- 1}

here,

λ =

wavelength

ω =

angular frequency

v =

linear velocity

Phase of wave : The argument of harmonic function $(ω t \pm k x + ϕ)$ is called phase of the wave.

Phase difference (

Δ ϕ

) : Difference in phases of two particles at any
time

t

Δ ϕ = \frac{2 π}{λ} Δ x

also,

Δ ϕ = \frac{2 π}{T} Δ t

here,

λ =

wavelength

T =

time period of wave

Speed of longitudinal wave

Speed of longitudinal wave in a medium is given by $v = \frac{E}{ρ}$

where,

E

is the modulus of elasticity,

ρ

is the density of the medium.

Speed of longitudinal wave in a solid in the form of rod is given by $v = \frac{Y}{ρ}$

where,

Y

is the Young's modulus of the solid,

ρ

is the density of the solid.

Speed of longitudinal wave in fluid is given by $v = \frac{B}{ρ}$

where,

B

is the bulk modulus,

ρ

is the density of the fluid.
Newton's formula
Newton assumed that propagation of sound wave in gas is an isothermal process. Therefore, according to Newton, speed of sound in gas is given by

v = \frac{P}{ρ}

, where

P

is the pressure of the gas and

ρ

is the density of the gas.
Laplace's correction
Laplace assumed that propagation of sound wave in gas in an adiabatic process. Therefore, according to Laplace, speed of sound in a gas is given by

v = \frac{γ P}{ρ}

where,

γ

is the adiabatic coefficient.

SPEED OF TRANSVERSE WAVE ALONG A STRING/WIRE.

v = \frac{T}{μ}

where,

T

is tension along a string,

μ

is the mass per unit length.

POWER TRANSMITTED ALONG THE STRING BY A SINE WAVE

Average power

(P) = 2 π^{2} f^{2} A^{2} μ v

Intensity

(I) = \frac{P}{s} = 2 π^{2} f^{2} A^{2} ρ v

Here,

f =

frequency,

A =

amplitude,

μ =

mass per unit length of string,

ρ =

density of the string and

v =

linear velocity.

Principle of Superposition

When two or more waves transverse through the same medium, the displacement of any particle of the medium is the sum of the displacement that the individual waves would give it.

y = Σ y_{i} (x, t)

Interference of waves

If two sinusoidal waves of the same amplitude and wavelength travel in the same direction they interface to produce a resultant sinusoidal wave travelling in that direction with resultant wave given by the relation

y^{'} (x, t) = [2 A_{m} c o s (\frac{Δ ϕ}{2})] s i n (ω t - k x + \frac{Δ ϕ}{2})

where

Δ ϕ

is the phase difference between two waves.

If $Δ ϕ = 0$ then interference would be fully constructive.
If $Δ ϕ = π$ the waves would be out of phase and there interference would be destructive.

REFLECTION AND REFRACTION OF WAVES

y_{i} = A_{i} s i n (ω t - k_{1} x)

If incident from rarer to denser medium $(v_{2} < v_{1})$

Transmitted Wave

y_{t} = A_{t} s i n (ω t - k_{2} x)

Reflected Wave

y_{r} = - A_{r} s i n (ω t + k_{1} x)

If incident from denser to rarer medium. $(v_{2} > v_{1})$

Transmitted Wave

y_{t} = A_{t} s i n (ω t - k_{2} x)

Reflected Wave

y_{r} = A_{r} s i n (ω t + k_{1} x)

Amplitude of reflected & transmitted waves.

A_{r} = \frac{∣ k _{1} - k _{2} ∣}{k _{1} + k _{2}} A_{i}

and

A_{t} = \frac{2 k _{1}}{k _{1} + k _{2}} A_{i}

STANDING/STATIONARY WAVES :

y_{1} = A s i n (ω t - k x + ϕ_{1})

y_{2} = A s i n (ω t + k x + ϕ_{2})

y_{1} + y_{2} = (2 A c o s (k x + \frac{ϕ _{2} + ϕ _{1}}{2})) s i n (ω t + \frac{ϕ _{1} + ϕ _{2}}{2})

The quantity

[2 A c o s (k x + \frac{ϕ _{2} + ϕ _{1}}{2})]

represents resultant amplitude at

x

.
At some position resultant amplitude is zero these are called nodes.
At some positions resultant amplitude is 2A, these are called anti-nodes.

Distance between successive nodes or antinodes= $λ / 2$
Distance between successive nodes and antinodes = $λ / 4$
All the particles in same segment (portion between two successive nodes) vibrate in same phase.
The particles in two consecutive segments vibrate in opposite phase.
Since nodes are permanently at rest so energy can not be transmitted across these.

VIBRATIONS OF STRINGS ( STANDING WAVE)

(a) Fixed at both ends :
Fixed ends will be nodes. So waves for which

L = \frac{n λ}{2}

λ = \frac{2 L}{n}

where

n = 1, 2, 3, . . . . .

v = \frac{T}{μ}

f_{n} (f r e q u e n c y) = \frac{n}{2 L} \frac{T}{μ}

n = no. of loops

(b) String free at one end :

For fundamental mode

L = λ / 4

First overtone

L = \frac{3 λ}{4}

. Hence,

λ = \frac{4 L}{3}

so,

f_{1} = \frac{3}{4 L} \frac{T}{μ}

(First overtone)

Second overtone

f_{2} = \frac{5}{4 L} \frac{T}{μ}

So,

f_{n} = \frac{( n + \frac{1}{2} )}{2 L} \frac{T}{μ} = \frac{2 n + 1}{4 L} \frac{T}{μ}

Vibrations in an organ pipe

1. Open Organ pipe (both ends open)

The open ends of the tube becomes antinodes because the particles at the open end can oscillate freely.
If there are $(n + 1)$ antinodes in all, length of tube, $l = \frac{n λ}{2}$
So, Frequency of oscillations is $f = \frac{n v}{2 l}$

2. Closed organ pipe (One end closed)

The open end becomes antinode and closed end become a node.
If there are $n$ nodes and $n$ antinodes, $L = (2 n - 1) λ / 4$
So, frequency of oscillations is $f = \frac{v}{λ} = \frac{( 2 n - 1 ) v}{4 L}$
There are only odd harmonics in a tube closed at one end.

Beats

Beats are formed by the superposition of two waves of slightly different frequencies moving in the same direction. The resultant effect heard in this case at any fixed position will consist of alternate loud and weak sounds.

Beat per second = f_{1} - f_{2}

here,

f_{1}

is the frequency of first wave and

f_{2}

is the frequency of second wave.

Doppler Effect

According to Doppler's effect, whenever there is a relative motion between a source of sound and listener, the apparent frequency of sound heard by the listener is different from the actual frequency of sound emitted by the source
Apparent frequency,

f^{'} = \frac{v - v _{L}}{v - v _{S}} \times f

here,

f

is the actual frequency,

v

is the velocity of sound

(\approx 340 m / s)

v_{L}

is the velocity of listener and

v_{S}

is the velocity of source.

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