A standing wave set up in a medium, the equation of wave is given by y = 4 cos π x 3 sin 40 π t where x and y are in cm and t in sec . Find the amplitude and the velocity of the two component waves.

The given equation is: y = 4 cos π x 3 sin 40 π t or y = 2 × 2 cos 2 π x 6 sin 2 π ( 120 ) t 6 . . . i When two travelling waves y 1 = A sin ω t - k x and y 2 = A sin ω t + k x interfere, the standard equation of a stationary wave is given by, y = 2 A cos k x sin ω t ⇒ y = 2 Acos 2 πx λ sin 2 πvt λ . . . ii where k = 2 π λ is the propagation constant, ω = k v = 2 π λ v is the frequency of the wave, v is the velocity of the wave. Comparing the Eq. ( i ) and ( ii ) , we get Amplitude, A = 2 cm , Velocity, v = 120 cm / s , and λ = 6 cm The component waves are, y 1 = A sin 2 π λ v t - x = 2 sin π 3 120 t - x and y 2 = A sin 2 π λ v t + x = 2 sin π 3 120 t + x

A standing wave set up in a medium, the equation of wave is given by y = 4 cos π x 3 sin 40 π t where x and y are in cm and t in sec . Find the distance between adjacent nodes.

The standard equation of a stationary wave is given by, y = 2 A cos k x sin ω t ⇒ y = 2 A cos 2 π λ x sin 2 π λ v t ⋯ i where, k = 2 π λ is the propagation constant, ω = k v = 2 π λ v is the frequency of the wave, v is the velocity. The given equation of wave is, y = 4 cos π x 3 sin 40 π t ⇒ y = 2 × 2 cos 2 π x 6 sin 2 π 6 120 t ⋯ ii Comparing Eq. i and ii , we get, Wavelength, λ = 6 cm Distance between two adjacent nodes = λ 2 = 6 2 = 3 cm

A standing wave set up in a medium, the equation of wave is given by y = 4 cos π x 3 sin 40 π t where x and y are in cm and t in sec . Find the the velocity of a medium particle at x = 3 cm at time 1 / 8 s ?

Equation of the wave is given by, y = 4 cos π x 3 sin 40 π t The velocity of medium particle ∂ y ∂ t = 4 cos π x 3 cos 40 πt × 40 π ⇒ v P = 160 π cos π x 3 cos 40 π t At x = 3 cm , t = 1 / 8 s , the particle velocity is, v P | x = 3 cm , t = 1 8 s = 160 π cos π × 3 3 cos 40 π × 1 8 ⇒ v P | x = 3 cm , t = 1 8 s = 160 π cm / s

In an experiment, to measure the speed of sound by a resonating air column, a tuning fork of frequency 500 Hz is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length 50 .7 cm and 83 .9 cm . Which of the following statements is (are) true?

The wavelength of the sound wave is 66 .4 cm .

The resonance at 50 .7 cm corresponds to the fundamental harmonic.

MEDIUM

Physics

IMPORTANT

Earn 100

Two sinusoidal waves combining in a medium are described by the wave functions

$y_{1} = (3.0 c m) \sin π (x + 0.60 t)$
$y_{2} = (3.0 c m) \sin π (x - 0.60 t)$

where $x$ is in centimetres and $t$ is in seconds. Determine the maximum transverse position of an element of the medium at

$x = 1.50 c m$ .

Important Questions on Superposition and Standing Waves

MEDIUM

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 7 - Superposition and Standing Waves>ILLUSTRATIONS>Q 7.24

Two sinusoidal waves combining in a medium are described by the wave functions

$y_{1} = (3.0 cm) sinπ (x + 0.60 t)$
$y_{2} = (3.0 cm) sinπ (x - 0.60 t)$

where $x$ is in centimetres and $t$ is in seconds. Determine the maximum transverse position of an element of the medium at

Find the three smallest values of $x$ corresponding to antinodes.

MEDIUM

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 7 - Superposition and Standing Waves>ILLUSTRATIONS>Q 7.25

A standing wave set up in a medium, the equation of wave is given by

y = 4 \cos \frac{π x}{3} \sin 40 π t

where

x

and

y

are in

cm

and

t

\sec .

Find the amplitude and the velocity of the two component waves.

MEDIUM

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 7 - Superposition and Standing Waves>ILLUSTRATIONS>Q 7.25

A standing wave set up in a medium, the equation of wave is given by

y = 4 \cos \frac{π x}{3} \sin 40 π t

where

x

and

y

are in

cm

and

t

\sec .

Find the distance between adjacent nodes.

MEDIUM

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 7 - Superposition and Standing Waves>ILLUSTRATIONS>Q 7.25

A standing wave set up in a medium, the equation of wave is given by

y = 4 \cos \frac{π x}{3} \sin 40 π t

where

x

and

y

are in

cm

and

t

\sec .

Find the the velocity of a medium particle at

x = 3 cm

at time

1 / 8 s ?

MEDIUM

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 7 - Superposition and Standing Waves>ILLUSTRATIONS>Q 7.26

A string vibrates in its first normal mode with a frequency of

220

vibrations s^{- 1}

. The vibrating segment is

70.0 cm

long and has a mass of

1.20 g

.
(a) Find the tension in the string.
(b) Determine the frequency of vibration, when the string vibrates in three segments.

HARD

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 7 - Superposition and Standing Waves>ILLUSTRATIONS>Q 7.27

A string, $120$ $cm$ in length, sustains a standing wave, with the points of the string, at which the displacement amplitude is equal to $3.5$ $mm$ , being separated by $15.0$ $cm$ . Find the maximum displacement amplitude. To which overtone do these oscillations correspond?

MEDIUM

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 7 - Superposition and Standing Waves>ILLUSTRATIONS>Q 7.28

Three successive frequencies for a string are $75, 125, 175 Hz .$

State whether the string is fixed out at one end or at both ends.

MEDIUM

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 7 - Superposition and Standing Waves>ILLUSTRATIONS>Q 7.28

Three successive frequencies for a string are $75, 125, 175 Hz .$

What is the fundamental frequency?

Two sinusoidal waves combining in a medium are described by the wave functions y1=(3.0 cm)sinπ(x+0.60t) y2=(3.0 cm)sinπ(x-0.60t) where x is in centimetres and t is in seconds. Determine the maximum transverse position of an element of the medium at x=1.50 cm.

Important Questions on Superposition and Standing Waves

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