A wave travels out in all directions from a point source. Justify the expression y = a 0 r sin k ( r - v t ) , at a distance r from the source. Find the intensity of the wave.

Let, P be the power of the source. Then the intensity in terms of power, I = P 4 π r 2 ⇒ I ∝ 1 r 2 But we know that the relation between intensity and amplitude, I = k a 2 , here k is the constant, ⇒ I ∝ a 2 ⇒ a ∝ 1 r The equation in the standard form is y = a sin k ( r - v t ) Therefore, y = a 0 r sin k ( r - v t ) , simply, replacing a by a 0 r , where a 0 is a constant. Compared with the usual form, y = a sin ( k r - ω t ) ω = k v ⇒ f = k v 2 π and, k = 2 π λ ⇒ λ = 2 π k ⇒ c = f λ = k v 2 π × 2 π k = v , and time period, T = 1 f = 2 π k v So the intensity of the wave, I = 1 2 ρ a 2 ω 2 v I = 1 2 ρ a 0 2 r 2 k 2 v 2 v ⇒ I = 1 2 ρ a 0 2 k 2 v 3 r 2

For plane waves in the air of frequency, 1000 Hz and the displacement amplitude, 2 × 10 - 8 m , deduce the velocity amplitude. (Take ρ = 1 . 3 kg m - 3 , c = 340 m s - 1 )

First, recall the relation between linear velocity and angular velocity in terms of radius say displacement amplitude, v max ( velocity amplitude ) = a ω ⇒ v max = 2 × 10 - 8 × ( 2 π × 1000 ) ⇒ v max = 1 . 3 × 10 - 4 m s - 1 Now recall the formula of the intensity of the travelling wave in terms of amplitude, I = 1 2 ρ a 2 ω 2 c = 1 2 ρ v max 2 c ⇒ I = 1 2 × 1 . 3 1 . 3 × 10 - 4 2 × 340 = 3 . 7 × 10 - 6 W m - 2

For plane waves in the air of frequency, 1000 Hz and the displacement amplitude 2 × 10 - 8 m , deduce the intensity. (Take ρ = 1 . 3 kg m - 3 , c = 340 m s - 1 )

First, recall the relation between linear velocity and angular velocity in terms of radius say displacement amplitude, v max ( velocity amplitude ) = a ω ⇒ v max = 2 × 10 - 8 × ( 2 π × 1000 ) ⇒ v max = 1 . 3 × 10 - 4 m s - 1 Now recall the formula of the intensity of the travelling wave in terms of amplitude, I = 1 2 ρ a 2 ω 2 c = 1 2 ρ v max 2 c ⇒ I = 1 2 × 1 . 3 1 . 3 × 10 - 4 2 × 340 = 3 . 7 × 10 - 6 W m - 2

MEDIUM

Physics

IMPORTANT

Earn 100

Write down the wave equation for a plane progressive wave of amplitude $5 cm$ , wavelength $2 m$ and speed $330 {ms}^{- 1}$ . If the wave is moving in positive $x$ -direction and at $t = 0$ and $x = 0 : y = 0$ and $d y / d t > 0$ .

Important Questions on Travelling Waves

HARD

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 6 - Travelling Waves>CONCEPT APPLICATION EXERCISE>Q 10

A simple harmonic wave travelling along the positive

x

-axis has an amplitude of,

7 cm

, velocity

80 cm s^{- 1}

and frequency,

50

vibration per second. The wave travelling along the positive

x

-axis and at,

t = 0

and

x = 0 : y = 0

and

\frac{d y}{d t} > 0

. Calculate the displacement, velocity, and acceleration of a medium particle at a position,

x = 399 cm

at the instant,

t = 5 s

HARD

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 6 - Travelling Waves>CONCEPT APPLICATION EXERCISE>Q 1

A wave travels out in all directions from a point source. Justify the expression

y = (\frac{a_{0}}{r}) \sin k (r - v t)

, at a distance

r

from the source. Find the intensity of the wave.

MEDIUM

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 6 - Travelling Waves>CONCEPT APPLICATION EXERCISE>Q 2

For plane waves in the air of frequency, $1000 Hz$ and the displacement amplitude, $2 \times 10^{- 8} m$ , deduce the velocity amplitude.

(Take $ρ = 1.3 kg m^{- 3}, c = 340 m s^{- 1})$

HARD

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 6 - Travelling Waves>CONCEPT APPLICATION EXERCISE>Q 2

For plane waves in the air of frequency, $1000 Hz$ and the displacement amplitude $2 \times 10^{- 8} m$ , deduce the intensity.

(Take $ρ = 1.3 kg m^{- 3}, c = 340 m s^{- 1})$

MEDIUM

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 6 - Travelling Waves>CONCEPT APPLICATION EXERCISE>Q 3

A piano wire with mass $3.00 g$ and length $80.0 cm$ i stretched with a tension of $25.0 N$ . A wave with frequency $120.0 Hz$ and amplitude $1.6 mm$ travels along the wire

Calculate the average power carried by the wave

MEDIUM

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 6 - Travelling Waves>CONCEPT APPLICATION EXERCISE>Q 3

A piano wire with mass $3.00 g$ and length $80.0 cm$ is stretched with a tension of $25.0 N$ . A wave with frequency $120.0 Hz$ and amplitude $1.6 mm$ travels along the wire

What happens to the average power if the wave amplitude is halved?

MEDIUM

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 6 - Travelling Waves>CONCEPT APPLICATION EXERCISE>Q 4

A steel wire has a mass of, $50 g m^{- 1}$ and is under tension $450 N$ . Find the maximum average power that can be carried by the transverse wave in the wire if the amplitude is not to exceed, $20 %$ of the wavelength.

MEDIUM

Physics

IMPORTANT

PHYSICS FOR JOINT ENTRANCE EXAMINATION WAVES AND THERMODYNAMICS>Chapter 6 - Travelling Waves>CONCEPT APPLICATION EXERCISE>Q 4

A steel wire has a mass of $50 g m^{- 1}$ and is under tension, $450 N$ . The change in maximum average power if the mass per unit length of the wire is doubled. (Use $π^{2} ≅ 10$ )

Write down the wave equation for a plane progressive wave of amplitude 5 cm, wavelength 2 m and speed 330 ms-1. If the wave is moving in positive x-direction and at t=0 and x=0: y=0 and dy / dt>0.

Important Questions on Travelling Waves

Learn and Prepare for any exam you want !

Write down the wave equation for a plane progressive wave of amplitude $5 cm$ , wavelength $2 m$ and speed $330 {ms}^{- 1}$ . If the wave is moving in positive $x$ -direction and at $t = 0$ and $x = 0 : y = 0$ and $d y / d t > 0$ .