A wave pulse in a string is described by the equation y 1 = 5 3 x - 4 t 2 + 2 and another wave pulse in the same string is described by y 2 = - 5 3 x + 4 t - 6 2 + 2 . The values of y 1 , y 2 and x are in metres and t is in seconds. Which of the following statement is correct?

At x = 1 m , y 1 and y 2 always cancel

y 1 travels along - x -direction and y 2 along + x -direction

Both y 1 and y 2 travel along + x -direction

At time t = 1 s , y 1 and y 2 exactly cancel everywhere

At time t = 3 s , y 1 and y 2 exactly cancel everywhere

If the frequency ofa wave is doubled its wavelength

HARD

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

A particle performing S.H.M. starts from the equilibrium position and its time period is

16 s

. After

2 s

its velocity is

π m s^{- 1} .

Amplitude of oscillation is

(\cos 45^{o} = \frac{1}{\sqrt 2})

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Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

A longitudinal wave is represented by

y = 10 \sin 2 π (n t - \frac{x}{λ}) cm

. The maximum particle velocity will be four times the wave velocity if the determined value of wavelength is equal to

HARD

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

Two vectors

\vec{A}

and

\vec{B}

are defined as

\vec{A} = a \hat{i}

and

\vec{B} = a (\cos ω t \hat{i} + \sin ω t \hat{j})

, where a is a constant and

ω = \frac{π}{6} rad s^{- 1} .

If

|\vec{A} + \vec{B}| = \sqrt{3} |\vec{A} - \vec{B}|

at time

t = τ

for the first time, the value of

τ

in seconds, is_______

EASY

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

A progressive wave travelling along the positive

x -

direction is represented by

y (x, t) = A s i n (k x - ω t + ϕ)

. Its snapshot at

t = 0

is given in the figure.
Question Image

For this wave, the phase

ϕ

is:

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Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

A travelling wave in a medium is given by the equation

y = asin (ωt - kx)

. The maximum acceleration of the particle in the medium is

MEDIUM

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

A wave pulse in a string is described by the equation

y_{1} = \frac{5}{{(3 x - 4 t)}^{2} + 2}

and another wave pulse in the same string is described by

y_{2} = \frac{- 5}{{(3 x + 4 t - 6)}^{2} + 2}

. The values of

y_{1}, y_{2}

and

x

are in metres and

t

is in seconds. Which of the following statement is correct?

MEDIUM

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

A simple harmonic motion is represented by:

y = 5 (\sin 3 π t + \sqrt{3} \cos 3 π t) cm

The amplitude and time period of the motion are:

EASY

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

Let $f (x) = \sin x + \cos a x$ be a periodic function. Then,

HARD

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

A particle of mass

\frac{2}{3} kg

with velocity

v = - 15 m s^{- 1}

at

t = - 2 s

is acted upon by a force

F = k - β t^{2}

. Here,

k = 8 N

and

β = 2 N s^{- 2}

. The motion is one-dimensional. Then, the speed at which the particle acceleration is zero again, is

EASY

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

A transverse wave described by

y = (0.02 m) \sin [(1.0 m^{- 1}) x + (30 s^{- 1}) t]

propagates on a stretched string having a linear mass density of

1.2 \times 10^{- 4} kg m^{- 1}

. Tension in the string is

EASY

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

A simple harmonic progressive wave is represented as

y = 0.03 s i n π (2 t - 0.01 x) m

. At a given instant of time, the phase difference between two particles

25 m

apart is

EASY

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

The displacement of a wave is represented by

y = 0.6 \times 10^{- 3} \sin (500 t - 0.05 x),

where all the quantities are in their proper units. The maximum particle velocity

(i n m s^{- 1})

of the medium is

MEDIUM

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

For a transvers wave travelling, along a straight line, the distance between two peaks (crests) is

5 m

, while the distance between one crest and one trough is

1.5 m

. The possible wavelengths (in

m

) of the waves are:

EASY

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

The maximum transverse velocity and maximum transverse acceleration of a harmonic wave in a

o n e - d i m e n s i o n a l

string are

1 m s^{- 1}

and

1 m s^{- 2}

respectively. The phase velocity of the wave is

1 m s^{- 1}

. The waveform is

EASY

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

The phase difference between the following two waves $y_{2}$ and $y_{1}$ is:

$y_{1} = a \sin (ω t - k x)$

$y_{2} = b \cos (ω t - k x + \frac{π}{3})$

HARD

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

It

a, b

be two fixed positive integers such that

f (a + x) = b + {[b^{3} + 1 - 3 b^{2} f (x) + 3 b {\{f (x)\}}^{2} - {\{f (x)\}}^{3}]}^{1 / 3}

for all real

x

then

f (x)

is a periodic function with period

EASY

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

The distance between two points differing in phase by

60 °

on a wave having wave velocity

360 m s^{- 1}

and frequency

500 Hz

is

EASY

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

Two simple harmonic motion, are represented by the equations $y_{1} = 10 \sin (3 π t + \frac{π}{3}); y_{2} = 5 (\sin 3 π t + \sqrt{3} \cos 3 π t)$ . Ratio of amplitude of $y_{1}$ to $y_{2} = x : 1$ . The value of $x$ is

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Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

A transverse wave is represented by :

y = \frac{10}{π} sin (\frac{2 π}{T} t - \frac{2 π}{λ} x)

For what value of the wavelength the wave velocity is twice the maximum particle velocity?

EASY

Physical Sciences>Waves and Their Applications in Technologies for Information Transfer>Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.>Wave Properties - The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing.

Consider a wire with density

(d)

and stress

(σ) .

For the same density, if the stress increases

2

times, the speed of the transverse waves along the wire changes by

If the frequency of a wave is doubled, its wavelength

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