The figure shows a snap photograph of a vibrating string at t = 0 .The particle P is observed moving up with velocity 20 3 cm s - 1 . The tangent at P makes an angle 60 ° with x -axis. (a) Find the direction in which the wave is moving. (b) Write the equation of the wave. (c) Find the total energy carried by the wave per cycle of the string. Assume that the mass per unit length of the string is 50 g m - 1 .

(a) From the above figure, slope at a point P is, d y d x = tan 60 ° = 3 . The given velocity of a particle at P is, v p = 20 3 cm s - 1 = 20 3 × 10 - 2 m s - 1 . From relation, v p = - v w d y d x . . . 1 Putting values in the equation 1 , 20 3 × 10 - 2 = - v w 3 v w = - 0 . 20 m s - 1 . From above, we get the velocity of wave as negative which means the wave is travelling in the direction of negative x -axis. (b) General equation of wave travelling in negative x -axis is, y = A sin ω t + k x + ϕ . . . 1 Given: A = 4 × 10 - 3 m λ = 4 × 10 - 2 m k = 2 π λ = 2 π 4 × 10 - 2 = 50 π radian m - 1 ω = v w k = 50 π × 0 . 20 = 10 π radian s - 1 μ = 50 g m - 1 = 0 . 05 kg m - 1 . At t = 0 , x = 0 , y = 2 2 × 10 - 3 m . Putting these values in equation 1 , 2 2 × 10 - 3 = 4 × 10 - 3 sin ϕ sin ϕ = 1 2 ϕ = π 4 Putting above all values in equation 1 , y = 4 × 10 - 3 sin 10 π t + 50 π x + π 4 (c) The energy carried per cycle, E = P T = P f = 2 π 2 A 2 f μ ν . . . 2 Putting values in the above equation 2 , E = 2 × 3 . 14 2 × 4 × 10 - 2 2 × 5 × 0 . 05 × 0 . 20 = 1 . 6 × 10 - 5 J .

A wave pulse is traveling on a string with a speed v towards the positive x -axis. The shape of the string at t = 0 is given by y ( x ) = A sin x a where, A and a are constants. (a) What are the dimensions of A and a ? (b) Write the equation of the wave for a general time t if the wave speed is v .

(a) From the given equation at t = 0 , y = A sin x a . . . 1 From dimension analysis, → The dimension of the left-hand side is equal to right-hand side. → All trigonometric functions are dimensionless. From equation 1 , dimension of A is equal to dimension of y which is M 0 L T 0 . dimension of a is equal to dimension of x which is M 0 L T 0 . (b) As the wave is travelling in positive x -direction, there should be a negative sign between the term of x and term of t . The general equation is given by, y = A sin α x - β t . . . 2 α = coefficient of x , β = coefficient of t . From formula, speed of wave = coefficient of t coefficient of x . . . 3 Comparing equations 1 and 2 , α = 1 a . From equation 3 , v = β α = a β β = v a . Putting above values in equation 2 , y = A sin x - v t a .

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In the wave equation, $y = A \sin 2 π (\frac{x}{a} - \frac{t}{b})$ ,

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Important Questions on Wave Motion

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Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 1 - Wave Motion>Exercises>Q 6

Corresponding to the $y - t$ graph of a transverse harmonic wave shown in the figure, choose the correct options at same position.

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Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 1 - Wave Motion>Exercises>Q 1

The figure shows a snap photograph of a vibrating string at $t = 0$ .The particle $P$ is observed moving up with velocity $20 \sqrt{3} cm s^{- 1}$ . The tangent at $P$ makes an angle $60 °$ with $x$ -axis.

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(a) Find the direction in which the wave is moving.
(b) Write the equation of the wave.
(c) Find the total energy carried by the wave per cycle of the string. Assume that the mass per unit length of the string is $50 g m^{- 1}$ .

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Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 1 - Wave Motion>Exercises>Q 2

A long string, having a cross-sectional area $0.80 {mm}^{2}$ and density $12.5 g {cm}^{- 3}$ , is subjected to a tension of $64 N$ along the positive $x$ -axis. One end of this string is attached to a vibrator at $x = 0$ , moving in the transverse direction at a frequency of $20 Hz$ . At $t = 0$ , the source is at maximum displacement, $y = 1.0 cm$ .

(a) Find the speed of the wave traveling on the string.
(b) Write the equation for the wave.
(c) What is the displacement of the particle of the string at $x = 50 cm$ at time, $t = 0.05 s$ ?
(d) What is the velocity of this particle at this instant?

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Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 1 - Wave Motion>Exercises>Q 3

One end of each of two identical springs, each of force constant $0.5 N m^{- 1}$ , are attached on the opposite sides of a wooden block of mass $0.01 kg$ . The other ends of the springs are connected to separate rigid supports such that the springs are unstretched and are collinear in a horizontal plane. To the wooden piece is fixed a pointer which touches a vertically moving plane paper. The wooden piece kept on a smooth horizontal table is now displaced by $0.02 m$ along the line of springs and released. If the speed of paper is $0.1 m s^{- 1}$ , find the equation of the path traced by the pointer on the paper and the distance between two consecutive maxima on this path.

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Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 1 - Wave Motion>Exercises>Q 4

A wave pulse is traveling on a string with a speed

v

towards the positive

x

-axis. The shape of the string at

t = 0

is given by

y (x) = A \sin (\frac{x}{a})

where,

A

and

a

are constants.
(a) What are the dimensions of

A

and

a

?
(b) Write the equation of the wave for a general time

t

if the wave speed is

v

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Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 1 - Wave Motion>Exercises>Q 5

Figure shows a plot of the transverse displacement of the particle of a string at $t = 0$ through which a traveling wave is passing in the positive $x$ -direction. The wave speed is $20 cm s^{- 1}$ . Find (a) the amplitude (b) the wavelength (c) the wave number and (d) the frequency of the wave.

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Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 1 - Wave Motion>Exercises>Q 6

Two wires of different densities but same area of cross-section are soldered together at one end and are stretched to tension

T

. The velocity of a transverse wave in the first wire is double of that in the second wire. Find the ratio of the density of the first wire to that of the second wire.

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JEE Advanced

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Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 1 - Wave Motion>Exercises>Q 7

Two long strings

A

and

B

, each having linear mass density

1.2 \times 10^{- 2} kg m^{- 1}

, are stretched by different tensions

4.8 N

and

7.5 N

, respectively, and are kept parallel to each other with their left ends at

x = 0

. Wave pulses are produced on the strings at the left ends at

t = 0

on string

A

and at

t = 20 ms

on string

B

. When and where will the pulse on

B

overtake that on

A

In the wave equation, y=Asin2πxa-tb,

Important Questions on Wave Motion

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In the wave equation, $y = A \sin 2 π (\frac{x}{a} - \frac{t}{b})$ ,