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

The function f x = sin π x n ! − cos π x n + 1 ! is

periodic, with period 2 n + 1 !

What is Angular Frequency What is its SI Unit

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.

In the wave equation

y = 0.5 \sin \frac{2 π}{λ} (400 t - x) m

the velocity of the wave will be :

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 propagation constant or angular wave number is

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})

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

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?

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 travelling wave is described by the equation $y (x, t) = [0.05 \sin (8 x - 4 t)] m$ . The velocity of the wave is:

[All the quantities are in SI unit]

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:

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.

If

f : R \to R

is a function satisfying

f (2 x + 3) + f (2 x + 7) = 2 \forall x \in R

, then the period of

f (x)

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 wave is represented by the equation

y = (0.02 m) \sin (5 π x - 20 t) .

The minimum distance between the two particles always having the same speed is. (Assume

x

and

t

are in SI units)

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

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 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

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

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 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?

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 travelling wave in a medium is given by the equation

y = asin (ωt - kx)

. The maximum acceleration of the particle in the medium 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

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.

The function

f (x) = \sin (\frac{π x}{n!}) - \cos (\frac{π x}{(n + 1)!})

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.

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.

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_______

What is Angular Frequency? What is its S.I Unit?

Important Questions on Use mathematical representations to support a claim regarding relationships among the frequency, wavelength,and speed of waves traveling in various media.

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