Sinusoidal Wave

Q. For a given wave , maximum particle velocity is $5$ times the wave velocity corresponding to the equation,
$y=A\sin (5t-0.05x)$ in which $A$ , $x$ are in metres and $t$ is in seconds. The value of $A$ is

Q. The equation for a wave travelling in $x$ direction on a string is given as
$y=10\sin (\pi t-0.5x)$, where $y$ and $x$ are in $\text{m}$ and time$\left(t\right)$ in $\text{sec}$. Find the acceleration of a particle at $x=\pi \text{m}$ at time $t=1\text{sec}$.

1. $-10\pi {\text{m/s}}^2$
2. $5\pi {\text{m/s}}^2$
3. $-10{\pi }^2{\text{m/s}}^2$
4. $-5\pi {\text{m/s}}^2$

Q. When a sound wave of wavelength $\lambda$ is propagating in a medium, the maximum velocity of the particle is equal to the wave velocity. The amplitude of wave is:

1. $\frac{\lambda }{2}$
2. $\lambda$
3. $\frac{\lambda }{4\pi }$
4. $\frac{\lambda }{2\pi }$

Q.

The equation of a transverse wave travelling on a rope is given by $y=10sin\pi (0.001x-2.00t)$ where y and x are in centimeters and t in seconds. The maximum transverse speed of a particle in the rope is about.

1. 121 cm/s

2. 63 cm/s

3. 100 cm/s

4. 75 cm/s

Q.

The equation of a transverse wave travelling on a rope is given by $y=10sin\pi (0.001x-2.00t)$ where y and x are in centimeters and t in seconds. The maximum transverse speed of a particle in the rope is about.

1. 63 cm/s

2. 75 cm/s

3. 100 cm/s

4. 121 cm/s

Q.

Why a transverse wave can be either mechanical or non-mechanical wave ?

Q. Equation of a plane progressive wave is given by $y=0$.$6\sin 2\pi (t-\frac{x}{2})$. On reflection from a denser medium its amplitude becomes $2/3$ of the amplitude of the incident wave. The equation of the reflected wave is

1. $y=-0.4\sin 2\pi (t+\frac{x}{2})$
2. $y=0.6\sin 2\pi (t+\frac{x}{2})$
3. $y=-0.4\sin 2\pi (t-\frac{x}{2})$
4. $y=0.4\sin 2\pi (t+\frac{x}{2})$

Q. A long wire $PQR$ is made by joining two wires $PQ$ and $QR$ of equal radii. $PQ$ has length $4.8m$ and mass $0.06kg$. $QR$ has length $2.56m$ and mass $0.2kg$. The wire $PQR$ is under a tension of $80N$. A sinusoidal wave-pulse of amplitude $3.5cm$ is sent along the wire $PQ$ from the end $P$. No power is dissipated during the propagation of the wave-pulse. Calculate the time taken (in $s$) by the wave pulse to reach the other end $R$ of the wire.

Q. Poynting vectors $\overrightarrow{S}$ is defined as a vector whose magnitude is equal to the wave intensity and whose direction is along the direction of wave propagation.
Mathematically, it is given by $\overrightarrow{S}=\frac{1}{{\mu }_0}(\overrightarrow{E}\times \overrightarrow{B})$.
Show the nature of $Svst$ graph.

Q. Which of the following function correctly represent the travelling wave equation for finite values of $x$ and $t$?

1. $y=x^2-t^2$
2. $y=log(x^2-t^2)-log(x-t)$
3. $y=e^{2x}\sin t$
4. $y=\cos x^2\sin t$

Q. A wave pulse, travelling on a two-piece string, gets partially reflected and partially transmitted at the junction. The reflected wave is inverted in shape as compared to the incident one. If the incident wave has wavelength $\lambda$ and transmitted wave $\lambda \text{'}$. Then

1. $\lambda \text{'}<\lambda$
2. $\lambda \text{'}=\lambda$
3. $\lambda \text{'}>\lambda$
4. None

Q.

When a wave travels in a medium, the particle displacement is given by
$y(x,t)=0.03sin\pi (2t-0.01x)$
where y and x are expressed in metres and t in seconds. At a given instant of time, the phase difference between two particles 25 m apart is

1. $\frac{\pi }{8}$

2. $\frac{\pi }{4}$

3. $\frac{\pi }{2}$

4. $\pi$

Q. The phenomena involved in the reflection of radio waves by ionosphere is similar to or not as the total internal reflection of light in air during a mirage.

Q. A transverse wave on a string has an amplitude of 0.2 m and a frequency of 175 Hz. Consider a particle of the string at x = 0. It begins with a displacement y = 0, at t = 0, according to equation $y=0.2sin(kx\pm \omega t)$. How much time passes between the first two instant when this particle has a displacement of y = 0.1 m?

1. 0.5 ms
2. 2.4 ms
3. 1.9 ms
4. 3.9 ms

Q.

Figure (15-E2) shows a plot of the transverse displacements of the particles of a string at t = 0 through which a travelling wave is passing in the positive x-direction. The wave speed is $20cm{s}^{-1}$. Find (a) the amplitude, (b) the wavelength, (c) the wave number and (d) the frequency of the wave.

Q.

As the wavelength of a wave in a uniform medium increases, then what will happen with its speed?

Q.

Simplify the expression $1-{\sin }^{2}x$

Q. A sound signal is sent through a composite tube as shown in the figure. The radius of the semicircle is $R$ and speed of sound in air is $v$. If '$n$' is an integer, then the frequencies of the signal for obtaining maximum value of intensity are given by

1. $\frac{nv}{(R-2)\pi }$
2. $\frac{nv}{\pi R}$
3. $\frac{nv}{R}$
4. $\frac{nv}{R(\pi -2)}$

Q. In the equation representing wave function,
$y=A\sin (kx-\omega t+{\varphi }_0)$
The term phase is defined as:

1. ${\varphi }_0$
2. ${\varphi }_0-\omega t$
3. $kx+{\varphi }_0$
4. $kx-\omega t+{\varphi }_0$

Q. Graph shows three waves that are separately sent along a string that is stretched under a certain tension along x-axis. If ${\omega }_1,{\omega }_{2}and{\omega }_3$ are their angular frequencies, respectively, then:

1. 
2. 
3. 
4. 

Q. Angular frequency is defined as the rate of change of angular displacement.

1. True
2. False

Q. For the equation, $y=A\sin 2\pi (ax+bt+\pi /4)$, match the following columns.

$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{(A) Frequency of wave(Hz)}& \text{(p) a}\\ \text{(B) wavelength of wave(m)}& \text{(q) b}\\ \text{(C) Phase difference between two points}\frac{1}{4a}\text{m}\text{distance apart}& \text{(r)}\pi \\ \text{(D) Phase difference of a particle after a time interval of}\frac{1}{8b}\text{s}& \text{(s)}\frac{\pi }{2}\\ & \text{(t) None of above}\end{array}$

1. $\begin{array}{cccc}\text{A}& \text{B}& \text{C}& \text{D}\\ \text{q}& \text{t}& \text{s}& \text{t}\end{array}$
2. $\begin{array}{cccc}\text{A}& \text{B}& \text{C}& \text{D}\\ \text{t}& \text{q}& \text{r}& \text{s}\end{array}$
3. $\begin{array}{cccc}\text{A}& \text{B}& \text{C}& \text{D}\\ \text{p}& \text{q}& \text{r}& \text{s}\end{array}$
4. $\begin{array}{cccc}\text{A}& \text{B}& \text{C}& \text{D}\\ \text{t}& \text{s}& \text{p}& \text{r}\end{array}$

Q. The maximum acceleration of a particle in a sinusoidal wave is (symbols have their usual meaning)

1. A$\omega$
2. A${\omega }^2$
3. A${\omega }^3$
4. A${\omega }^4$

Q. The equation of a wave is $y(x,t)=0.1\sin [\frac{\pi }{3}(10x-30t)-\frac{\pi }{6}]\text{m}$.
Find the acceleration of the particle at $x=0.85\text{m}$ and $t=0.25\text{s}$
[Assume, ${\pi }^2=10$ ; $\uparrow$ denotes positive $y-$direction and $\downarrow$ denotes negative $y-$direction ]

1. $40{\text{m/s}}^2\uparrow$
2. $50{\text{m/s}}^2\downarrow$
3. $40{\text{m/s}}^2\downarrow$
4. $50{\text{m/s}}^2\uparrow$

Q.

The equation for a standing wave generated by a string, with both ends fixed, is$y=(0.4cm)\sin (0.314c{m}^{-1})x]\cos [\left(600\pi s^{-1}\right)t].$. Figure out the smallest length of the string.

Q. If vibration frequency $f$ of the star is given by the following equation: $f=K{R}^a{P}^b{G}^c$ where $R$ is the radius of star, $\rho$ is the density of star, $G$ is the universal gravitational constant and $k$is a constant, then $(b+c)$is equal to

Q. In interference of two waves, two individual amplitudes are $A_0$ each and the intensity is $I_0$ each. The resultant amplitude at a point where phase difference between two waves is ${60}^{\circ }$ is

1. $\sqrt{2}{A}_0$
2. $\sqrt{3}{A}_0$
3. $A_0$
4. $2A_0$

Q.

Figure (16-E12) shows a source of sound moving along the X-axis at a speed of $22m{s}^{-1}$ continuously emitting a sound of frequency 2.0 kHz which travels in air at a speed of $330m{s}^{-1}$. A listener Q stands on the Y-axis at a distance of 330 m from the origin. At t = 0, the source crosses the origin P. (a) When does the sound emitted from the source at P reach the listener Q ? (b) What will be the frequency heard by the listener at this instant? (c) Where will the source be at this instant ?

Q. Write the equation of the wave shown in figure, if its position is shown at $t=0.6\text{sec}$. The wavelength is $15\text{cm}$ and amplitude is $4\text{cm}$ and the crest $X$ was at $x=0$ at $t=0$.

1. $y(x,t)=\left(2\text{cm}\right)\sin \left[\right(1.68\text{rad/s})t-(1{\text{cm}}^{-1}\left)x\right]$
2. $y(x,t)=\sin \left[\right(3.2\text{rad/s})t-(0.42{\text{cm}}^{-1}\left)x\right]$
3. $y(x,t)=\left(2\text{cm}\right)\cos \left[\right(3.2\text{rad/s})t-(0.42{\text{cm}}^{-1}\left)x\right]$
4. $y(x,t)=\left(4\text{cm}\right)\cos \left[\right(1.68\text{rad/s})t-(0.42{\text{cm}}^{-1}\left)x\right]$

Q. Two sinusoidal waves are defined by the functions :
$y_1=\left(2.00\text{cm}\right)\sin (20x-32t)$ and $y_2=\left(2.00\text{cm}\right)\sin (25x-40t)$ where $y_1,y_2$ and $x$ are in centimeters and $t$ is in seconds. What is the phase difference between these two waves at the point $x=5.00\text{cm}$ at $t=2\text{s}$?

1. ${516}^{\circ }$
2. ${258}^{\circ }$
3. ${333}^{\circ }$
4. ${412}^{\circ }$