Physics

Wave and Sound - 2

Session

Session Objectives

Session Objective

1. Nature and propagation of sound waves

2. Displacement and pressure wave

3. Reflection & Transmission of Waves

4. Linear Wave Equation (Differential Form)

5. Superposition & Interference of Sinusoidal

Waves

6. Speed of Sound Waves in a material and

Laplace correction

Longitudinal waves

Produced by a Vibrating Source

Elasticity and Inertia are necessary

Nature and propagation of sound waves

Propagates Through-

• Compression

Rarefaction

Audible Range: 20 to 20000 Hz

Ultrasonic Range: > 20000 Hz

Infrasonic Range: < 20 Hz

Displacement and pressure wave

BA xP cos t

v v

0

B AP BkA

v

The displacement wave is given by

y A sin t kx

The pressure waves is given by

where

Displacement and pressure wave

Pressure wave and displacement wave

have a phase difference of , i.e.

pressure maxima occurs where

displacement is zero and displacement

maxima occurs where the pressure is at

its normal level.

2

Reflection & Transmission of Waves

Sound wave undergoes a phase change of 180o on reflection from a rigid boundary compared to incident wave

Linear Wave Equation (Differential Form)

2 2

2 2 2

d y 1 d y

dx v dt

The above equation represents a progressive wave.

Any function that satisfies the above equation

represents a progressive wave

•Two or more waves having constant phase difference meet in the same medium

y = y1+y2 = f1(x-vt)+f2(x-vt+ )

y1 and y2 must be specific to find f

Interference

Superposition and Interference of sinusoidal Waves

1 1

2 2

Y A sin t kx

Y A sin t kx

1 2Y Y Y A sin t kx

2 21 2 1 2A = A + A +2A A cosφ -1 2

1 2

A sinφδ = tan

A + A cosφ

2m ax 1 22n I ( I I ) maxima

2m in 1 2(2n 1) I ( I I ) [minima]

Constructive

Destructive

Types of interference

1 2 1 2

x n

I I I 2 I I

1 2 1 2

1x (n )

2

I I I 2 I I

Young’s Double Slit Experiment

In general : Elastic Propertyv =

Inertial Property

In a fluid medium

( density) B

v =ρ

(Y Young'smodulus)In a solid rod : Y

v

Speed of sound in gases = v = P

Speed of Sound Waves in a material and Laplace correction

Class Test

Class Exercise - 1

Two waves represented by and

are superimposed. The

resultant waves will have an amplitude

(a) a (b)

(c) 2a (d) 0

y acos kx t y asin kx t

2a

Two waves have phase difference

2

2 2 2 2A a a 2a cos

2 A 2a

Solution :

Hence answer is (b).

Class Exercise - 2

A propagating sound wave encounters

rigid boundary. The reflected wave will

have the phase difference of

(a) (b)

2

(c) 0 (d) none of these

When sound wave encounters rigid

boundary, there is change in phase by .

Solution :

Hence answer is (b).

Class Exercise - 3

If there are two drum-beaters beating

the drums independently, the

sources would be

(a) coherent

(b) Incoherent

(c) Cannot be determined

(d) Data insufficient

In this case, the frequency difference is constant.

Solution :

Hence answer is (a).

Class Exercise - 4

The velocity of sound wave in a solid does

not depend on

(a) Young’s modulus (b) density

(c) conductivity (d) None of these

Velocity does not depend on conductivity.

Hence answer is (c).

Solution :

Class Exercise - 5

Consider the following statements about

sound passing through a gas.

A : The pressure of the gas at a point

oscillates in time.

B : The position of a small layer of the gas

oscillates in time.

(a) Both A and B are correct (b) A is correct but B is wrong

(c) B is correct but A is wrong (d) Both A and B are wrong

Sound wave is longitudinal wave. Hence, both pressure

and position oscillate with time.

Hence answer is (a).

Solution :

Class Exercise - 6

When we clap our hands, the sound

produced is best described by

0 0

0 n n n

(a) P P sin(kx t) (b) P P coskx sin t

(c) P P sina cos t (d) P Po sin k x t

Sound produced is the mixture of

various sound waves.Hence answer is (d).

Solution :

Class Exercise - 7

The equation of a sound wave in air is

given by

. The speed of sound in air is given by

(a) 330 m/s (b) 335 m/s

(c) 325 m/s (d) None of these

2 1 1P 0.01 N/m sin 1000 s t 3m x

3

v

1000

v 330 m/s3 3

Hence answer is (a).

Solution :

Class Exercise - 8

Two waves reach a point with a path

difference of 12 cm. The waves have

wavelength 4 cm. The phase difference

of two waves is

(a) 6 (b) 3

2(c) (d)

3 3

Hence answer is (a).

Solution :

212 6

4

Phase difference = Path difference × 2

Class Exercise - 9

A wave of frequency 4.5 MHz has a speed

of 1.5 km/s in a material. The wavelength

of wave in this material is

(a) 3.3 × 10–4 m (b) 3.3 × 10–3 m

(c) 3.3 × 10–7 m (d) 3.3 × 10–9 m

Solution

Hence answer is (a).

Frequency f = 4.5 × 106 Hz

Velocity v = 1.5 km/s = 1500 m/s

46

v 15007.6 10 m

f 4.5 10Wave length

Class Exercise - 10

Two point sources of sound are kept at

a separation of 10 cm. They vibrate in

phase to produce waves of wavelength

5 cm. What would be the phase

difference between the two waves

arriving at a point 20 cm from one

source on the line joining the sources?

Solution

Hence answer is (a).

Path difference between waves

coming source S1 and S2 = 30 – 20 =

10 cm.

Hence, phase difference

210 4

5

S1

S2

10

20

D

Thank you