Physics

Wave and Sound - 1

Session

Session Objectives

Session Objective

1. Introduction to wave motion

(Terminologies)

2. Types of waves

3. Sinusoidal waves

4. Characteristics of sine waves

5. Speed of mechanical waves in one-

dimensional translatory motion

6. Velocity of transverse mechanical waves

in strings

7. Phase and path difference

Introduction to wave motion (terminologies)

‘A wave is a disturbance which

propagates energy (and momentum)

from one place to another without

the transport of matter.’

Amplitude:- Maximum displacement of the elements from

their equilibrium position

Time period:- Time any

wave takes to complete one

oscillation.

T

x, t

A

Introduction to wave motion (terminologies)

Frequency :- It is defined as the number

of oscillations per unit time.

Wavelength :- It is the distance (parallel to

the direction of wave propagation) between

the consecutive repetitions of the shape of

the wave. It is the distance between two

consecutive troughs or crests

Propagation constant :- The quantity is called the

propagation constant,

2

Types of Waves

Mechanical waves: The waves which

require medium for their propagation are

called mechanical waves.e.g. sound waves

Non-mechanical waves: The waves which do

not require medium for their propagation are

called non-mechanical waves, e.g. light

Types of Waves

Transverse waves: If the particles of the

medium vibrate at right angle to the

direction of wave motion or energy

propagation, the wave is called

transverse wave e.g. waves on strings.

Wave motion

Vibration C C C

T T

Types of Waves

Longitudinal waves: If the particles of a

medium vibrate in the direction of wave

motion, the wave is called longitudinal wave.

e.g. sound waves

R R R

CCCC

Sinusoidal waves

At any time t, the displacement y

of the element located at a

position x is given by

y(x, t) A sin(kx t )

Sinusoidal wave

y

x

Characteristics of Sine Waves

The sinusoidal wave represented by

above equation is periodic in

position and time.

The equation of the wave traveling along

positive x-axis is given by

and moving along negative x-axis is given by

In general, we can write y A sin(kx t )

y A sin(kx t )

y A sin(kx t )

Characteristics of Sine Waves

This equation can be represented as

t xy A sin2

T

y A sink(vt x)

xy A sin t

v

xy A sin2 f t

Speed of mechanical waves in one-dimensional translatory motion

d

(kx t) 0dt

d x

k 0d t

d xv

d t k

2 2k

T

v f

T

The relation is valid for all types of progressive waves.

kx t cons tant

v f

Velocity of Transverse mechanical waves on strings

TT

F = 2T sin

sin as is very small and 2

r

2v

ar

m m

Now F = ma

2vT m

r r

T

vm

Phase and Path Difference

If the shape of the wave does not

change as the wave propagates in a

medium, with increase in t, x will also

increase in such a way that

2

x

t kx cons tant

1 1 2 2 = t - kx and = t -kx

2 1 2 1 = k(x - x )

x

x

y

Class Test

Class Exercise - 1

The equation of a transverse wave is

given by y = 10 sin(0.01x – 2t) where

x and y are in centimeters and t is in

seconds, its frequency is

(a) 10 Hz (b) 2 Hz

(c) 1 Hz (d) 0.01 Hz

Solution

Comparing with equation

t xy A sin2

T

y = 10 sin(0.01x – 2t)

0.01y 10sin2 x t

2

We get,

i.e. f = 1 Hz

1

1T

Hence answer is (c).

Class Exercise - 2

0

xy y sin2 f t

h

A transverse wave is described by the

equation . The

maximum particle velocity is equal to

four times the wave velocity if

0 0

0 0

y y(a) (b)

4 2

(c) y (d) 2 y

Solution

We know that the maximum particle

velocity

Hence answer is (b). 0y

2

maxV A

From the given equation, we get

0A y , 2 f

Wave velocity v = f

Given condition 0y × 2 f = 4f

Class Exercise - 3

A source of frequency 500 Hz emits waves

of wavelength 0.2 m. How long does it

take for the wave to travel 300 m?

(a) 70 s (b) 60 s

(c) 12 s (d) 3 s

Solution

Hence answer is (d).

Using the relation v f

we get, v = 500 × 0.2

v = 100 m/s

Dis tance

Time takenVelocity

300

3 s100

Class Exercise - 4

The equation of a plane wave is given

by where y is in

centimeters and t is in seconds. The

phase difference at any instant between

the points separated by 150 cm is

xy 2sin 200t

150

(a) 2 (b)

(c) (d)2 4

Solution

We know that,

2

x

x 150 cm

2150

300

= 300 cm

Hence answer is (b).

Class Exercise - 5

A stone is dropped into a well. If the

depth of water below the top be h and

velocity of sound is v, the splash in

water is heard after T second, then

2h h 2h(a) T (b) T

g v v

2h 2h h(c) T (d) T

g g v

Solution

Time taken by the stone to fall to the

surface of water is given by

1 11

s ut at2

12h

tg

2h

tv

t2 — time taken by sound Total time T = t1 + t2

2h h

Tg v

Hence answer is (a).

Class Exercise - 6

A man standing symmetrically between

two cliffs claps his hands and starts

hearing a series of echoes at intervals

of 1 s. The speed of sound in air is 340

m/s, the distance between parallel cliffs

must be

(a) 340 m (b) 680 m

(c) 1,020 m (d) 170 m

Solution

Let the distance of each cliff from

the man be x.

2x

Then 1v

v

or x2

340

x 170 m2

Distance between cliffs = 2x

= 2 × 170 = 340 mHence answer is (a).

Class Exercise - 7

The relation between the particle

velocity and wave velocity in a wave is

d y dy dy dy1 1(a) (b)

d t v dx dx v dt

dy dy dy dy1(c) (d) v

dx v dt dx dt

Solution

Hence answer is (c).

2

Let y asin (vt x)

dy 2 v 2a cos (vt x)

dt

dy 2 2a cos (vt x)

dx

dy dy

vdt dx

dy 1 dy

dx v dt

Class Exercise - 8

A 5.5 m length of string has a mass of

0.035 kg. If the tension in the string is 77

N, the speed of the wave on the string is

(a) 110 ms–1 (b) 164 ms–1

(c) 77 ms–1 (d) 102 ms–1

Solution

Mass per unit length = 0.035 kg/m

5.5

7

kg/m1100

T

vm

77 1100v

7

v = 110 m/s Hence answer is (a).

Class Exercise - 9

An observer standing at sea coast

observes 54 waves reaching the coast

per minute, if the wavelength of the

wave is 10 m, its velocity is

(a) 3 m/s (b) 6 m/s

(c) 9 m/s (d) 12 m/s

Solution

As 54 waves reach the coast per minute

54 9

f Hz60 10

9

v 1010

v = 9 m/s

Hence answer is (c).

v f

Class Exercise - 10

A progressive wave of frequency 500 Hz

is traveling with a velocity of 360 m/s.

How far apart are the two points 60° out

of phase?

(a) 0.12 m (b) 0.06 m

(c) 0.24 m (d) 0.36 m

Solution

v 360

0.72 mf 500

60 rad

180 3

x

2

0.72

x2 3

x 0.12 m Hence answer is (a).

Thank you