The equation of a one dimensional transverse wave is given by $\psi =\left({10}^{-3}\right)\sin \pi \left(10t-3x-2y+z\right)$ where $x,y$ and $\psi$ are expressed in meter and $t$ in second Choose the correct option(s)

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

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 $\left(d\right)$ and stress $\left(\sigma \right).$ 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 simple harmonic motion is represented by:
$y=5\left(\sin 3\pi t+\sqrt{3}\cos 3\pi t\right) \mathrm{cm}$
The amplitude and time period of the motion are:

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 $16 \mathrm{s}$. After $2 \mathrm{s}$ its velocity is $\pi \mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}.$ Amplitude of oscillation is $\left(\cos {45}^o=\frac{1}{\surd 2}\right)$

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 $\mathrm{y}=0.03 \mathrm{s}\mathrm{i}\mathrm{n}\pi \left(2t-0.01x\right) \mathrm{m}$. At a given instant of time, the phase difference between two particles $25 \mathrm{m}$ 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.

A sound wave of frequency $245 \mathrm{Hz}$ travels with the speed of $300{ \mathrm{m} \mathrm{s}}^{-1}$ along the positive x-axis. Each point of the wave moves to and fro through a total distance of $6 \mathrm{cm}$. What will be the mathematical expression of this travelling wave?

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 : $\text{y}=\frac{10}{\pi }sin\left(\frac{2\pi }{\text{T}}\text{t}-\frac{2\pi }{\lambda }\text{x}\right)$ For what value of the wavelength the wave velocity is twice the maximum particle velocity?

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 $\overrightarrow{A}$ and $\overrightarrow{B}$ are defined as $\overrightarrow{A}=a\widehat{i}$ and $\overrightarrow{B}=a\left(\cos \omega t\widehat{i}+\sin \omega t\widehat{j}\right)$, where a is a constant and $\omega =\frac{\pi }{6} \mathrm{rad}\mathrm{ }{\mathrm{s}}^{-1}.$ If $\left|\overrightarrow{A}+\overrightarrow{B}\right|=\sqrt{3}\left|\overrightarrow{A}-\overrightarrow{B}\right|$ at time $t=\tau$ for the first time, the value of $\tau$ 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.

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

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 equation of a wave is given by $y=10\sin \left(\left(\frac{2\pi }{45}\right)t+\alpha \right).$ If the displacement at $t=0 \mathrm{s}$ is $5 \mathrm{cm},$ then the total phase at $t=7.5 \mathrm{s}$ 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 $x-\alpha \sin \left(\omega t-\pi /6\right)$and $x=\alpha \cos \omega t$, then the phase difference between the two waves 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 travelling wave in a medium is given by the equation $\mathrm{y}=\mathrm{asin} (\mathrm{\omega t}-\mathrm{kx})$. 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 $y_1=\frac{5}{{\left(3x-4t\right)}^2+2}$ and another wave pulse in the same string is described by $y_2=\frac{-5}{{\left(3x+4t-6\right)}^2+2}$ . 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.

For a transvers wave travelling, along a straight line, the distance between two peaks (crests) is $5\text{ m}$, while the distance between one crest and one trough is $1.5\text{ m}$. The possible wavelengths (in $\text{m}$) 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 $y=0.6\times {10}^{-3}\sin \left(500t-0.05x\right),$ where all the quantities are in their proper units. The maximum particle velocity $\left(\mathrm{i}\mathrm{n}\mathrm{ }m{s}^{-1}\right)$ 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.

The phase difference between the following two waves $y_2$ and $y_1$ is:

$y_1=a\sin \left(\omega t-kx\right)$

$y_2=b\cos \left(\omega t-kx+\frac{\pi }{3}\right)$

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 $360 \mathrm{m} {\mathrm{s}}^{-1}$ and frequency $500 \mathrm{Hz}$ 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 of mass $\frac{2}{3} \mathrm{kg}$ with velocity $v=-15 \mathrm{m} {\mathrm{s}}^{-1}$ at $t=-2 \mathrm{s}$ is acted upon by a force $F=k-\beta t^2$. Here, $k=8 \mathrm{N}$ and $\beta =2 \mathrm{N} {\mathrm{s}}^{-2}$. 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 harmonic wave on a string is given by $y\left(x, t\right)=5 \sin \left(6t+0.003x\right)$ where $x$ and $y$ are in $\mathrm{cm}$ and $t$ in $\sec$. The wave velocity is _________ $\mathrm{m} {\mathrm{s}}^{-1}$.

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 $y\left(x,\mathrm{t}\right)=\mathrm{A}\mathrm{ }\mathrm{s}\mathrm{i}\mathrm{n}(\mathrm{k}x-\mathrm{\omega }\mathrm{t}+\mathrm{\varphi })$ . Its snapshot at $\mathrm{t}=0$ is given in the figure.

For this wave, the phase $\mathrm{\varphi }$ 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 maximum transverse velocity and maximum transverse acceleration of a harmonic wave in a $\mathrm{o}\mathrm{n}\mathrm{e}-\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}$ string are $\mathrm{1 }m{s}^{-1}$ and $1 m{s}^{-2}$ respectively. The phase velocity of the wave is $\mathrm{1 }m{s}^{-1}$ . The waveform is