Mention the important properties which a medium must possess for the propagation of mechanical waves through it?

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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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

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

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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 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

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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.

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

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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.

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

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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.

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

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

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

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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 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:

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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 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?

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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.

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

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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 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?

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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 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

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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.

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_______

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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.

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

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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.

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:

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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.

It $a, b$ be two fixed positive integers such that $f\left(a+x\right)=b+{\left[b^3+1-3b^{2}f\left(x\right)+3b{\left\{f\left(x\right)\right\}}^2-{\left\{f\left(x\right)\right\}}^3\right]}^{1/3}$ for all real $x$ then $f\left(x\right)$ is a periodic function with period

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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.

If $f:R\to R$ is a function satisfying $f\left(2x+3\right)+f\left(2x+7\right)=2 \forall x\in R$, then the period of $f\left(x\right)$ 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 described by $y=\left(0.02 \mathrm{m}\right)\sin \left[\left(1.0{ \mathrm{m}}^{-1}\right)x+\left(30{ \mathrm{s}}^{-1}\right)t\right]$ propagates on a stretched string having a linear mass density of $1.2\times {10}^{-4} \mathrm{kg} {\mathrm{m}}^{-1}$. Tension in the string 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.

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

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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 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 $\left(\cos {45}^o=\frac{1}{\surd 2}\right)$

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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 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: