Physics

Wave and Sound - 4

Session

Session Objectives

Session Objective

Beats

Conditions under which beats occur

Standing Waves in air columns

- All Conditions

Standing waves in a string

(with all conditions)

Fundamental Frequency

Standing waves in a string (with all conditions)

Let the waves

1y A sin(kx t)

2y A sin k x t

with same amplitude, same frequency, having a

phase difference ‘’, superimpose to produce a

standing wave given by

y 2A sin k x cos t2 2

String fixed at both ends

A

N N

Lx = Lx = O

Since x = 0 and x = L are nodes, at these points y = 0 for all ‘t’.

Hence, for x = 0,

y 2A sin cos t

2 2

Considering = 0,

y 2A sin(0)cos t y 0

String fixed at both ends

Similarly, for x = L

2

sin L 0 sinn

where n = 0, 1, 2, …

2L2 L

n orn

String fixed at both ends

If the length of the string is an integral

multiple of , standing waves are

produced.

2

Hence, the other natural frequencies with which

standing waves can be formed on string are

v nv n T

or2L 2L

v

Now

String fixed at both ends

1 0

2 T2

2L

2 0

3 T3

2L

3 0

4 T4

2L

[First overtone or second harmonic]

[Second overtone or third harmonic]

[Third overtone or fourth harmonic]

String fixed at both ends

N N

A AN

N N

A AN N Second overtone

Third harmonic

First overtone

Second harmonic

String fixed at one end

N A

A

NAN

N

A AN N A

Fundamental

First overtone

Second overtone

String fixed at one end

The boundary condition that x = 0

is automatically satisfied by

y = 2A sinkx cost.

For x = L to be an antinode,

sinkL 1

1kL n

2

2 L 1n

2

2L 1n

v 2

12

n1 v Tor n

2 2L 2L

String fixed at one end

Putting n = 0,

0v

4L [Fundamental frequency]

Similarly,

1 03v

34L

2 05v

54L

[First overtone]

[Second overtone]

Fundamental Frequency

The lowest frequency with which a

standing wave can be set up in a

string is called the fundamental

frequency.

Standing Waves in Air Columns

v nvff or n 1,3,5,7,.....[closed organ pipe]

4L

L

L4

Fundamental Frequencyor

First Harmonic

L

3L

4

First Overtoneor

Third Harmonic

L Second Overtoneor

Fifth Harmonic

5L

4

Standing Waves in Air Columns

L2

LFundamental Frequency

orFirst Harmonic

v nvff or n 1,2,3, 4.....[open organ pipe]

2L

2L

2

LFirst Overtone

orSecond Harmonic

3L

2

LSecond Overtone

orThird Harmonic

Beats

The difference between two combining frequencies is called beats.

Beats are produced due to superposition of two or more waves having nearly equal frequencies.

[Beat frequency] beat 1 2f f f

Class Test

Class Exercise - 1

An open organ pipe of length L vibrates

in its fundamental mode. The pressure

variation is maximum

(a) at the two ends

(b) at the middle of the pipe

(c) at distance inside the ends

(d) at distance inside the ends

L

4

L

8

Solution

Hence answer is (a).

Pressure variation is maximum

at the ends.

Class Exercise - 2

An open pipe is suddenly closed at one

end with the result that the frequency of

third harmonic of the closed pipe is

found to be higher by 100 Hz than

fundamental frequency of the open

pipe. The fundamental frequency of the

open pipe is

(a) 200 Hz (b) 300 Hz

(c) 240 Hz (d) 480 Hz

Solution

Hence answer is (a).

3v v v

100 Hz 200 H z4L 2L 2L

Class Exercise - 3

A pipe, open at both ends, gives

frequencies which are

(a) only even multiple of fundamental

frequency

(b) only odd multiple of fundamental

frequency

(c) all integral multiple of fundamental

frequency

(d) all fractional multiple of fundamental

frequency

Solution

Hence answer is (c).

Open pipe produces all integral multiple

of fundamental frequency.

Class Exercise - 4

In a closed organ pipe, the fundamental

frequency is ‘’. What will be the ratio of

the frequencies of the next three

overtones?

(a) 2 : 3 : 4 (b) 3 : 4 : 5

(c) 3 : 7 : 11 (d) 3 : 5 : 7

Solution

Hence answer is (d).

Closed pipe produces only odd

multiples of fundamental frequency.

Class Exercise - 5

Two open organ pipes of length 25

cm and 25.5 cm produce 0.1 beats

per second. The velocity of sound is

(a) 350 m/s (b) 325.5 m/s (c)

303 m/s (d) 255 m/s

Solution

Hence answer is (d).

v v

0.12(25) 2(25.5)

v = 255 m/s

Class Exercise - 6

Velocity of sound in air is 320 m/s. A

pipe closed at one end has a length of 1

m. Neglecting end correction, the air

column in the pipe can resonate for

sound of frequency which is equal to

(More than one options may be correct)

(a) 80 Hz (b) 240 Hz

(c) 320 Hz (d) 400 Hz

Solution

Hence answer is (a, b , d).

Fundamental frequency

0

v 32080 Hz

4L 4 1

Hence, possible frequencies are odd multiples of

fundamental frequency.

Class Exercise - 7

Two waves of wavelengths 2 m and 2.02

m respectively, moving with the same

velocity superimpose to produce 2 beats

per second. The velocity of the wave is

equal to

(a) 400 m/s (b) 402 m/s

(c) 404 m/s (d) 406 m/s

Solution

Hence answer is (c).

v v

22 2.02

v = 404 m/s

Class Exercise - 8

A sonometer wire, 65 cm long, is in

resonance with a tuning fork of frequency

N. If the length of the wire is decreased by

1 cm and it is vibrated with the same

tuning fork, 8 beats are heard per second.

What is the value of N?

(a) 256 Hz (b) 384 Hz

(c) 480 Hz (d) 512 Hz

Solution

Hence answer is (a).

1 TN'

2(65 1) m

1 T

N2 65 m

Given

Also or N' N 8 N' N 8

Hence, N is calculated.

Class Exercise - 9

Beats are result of

(a) diffraction

(b) destructive interference

(c) constructive interference

(d) superposition of two waves of nearly

equal frequencies

Solution

Hence answer is (d).

Definition of beats.

Class Exercise - 10

An organ pipe vibrates in fundamental

resonance with the medium as air, nitrogen

and oxygen. Which is the correct statement?

(a) The wavelength changes with medium

change

(b) Both wavelength and frequency change

(c) Both and remain unaltered with the

medium change

(d) The frequency changes with the medium

change

Solution

Hence answer is (a).

Only wavelength depends

on the medium.

Thank you