Differential Equation of Wave

Q. The wave function of a triangular wave pulse is defined by the relation below at time $t=0\text{sec}$
$y=\{\begin{array}{cc}mx& \text{for}0\le x\le \frac{a}{2}\\ -m(x-a)& \text{for}\frac{a}{2}\le x\le a\\ 0& \text{every where else}\end{array}$

The wave pulse is moving in the + X direction in a string having tension $T$ and mass per unit lengh $\mu$. The total kinetic energy present with the wave pulse is

1. $\frac{m^{2}Ta}{2}$
2. $\frac{3m^{2}Ta}{2}$
3. $m^{2}Ta$
4. None of these

Q.

A wave pulse passing on a string with a speed of $40cm{s}^{-1}$in the negative $X-direction$ has its maximum at $x=0att=0$ . Where will this maximum be located at $t=5s$?

Q.

A bat emitting an ultrasonic wave. of frequency $4.5\times {10}^4$ Hz flies at a speed of 6 m/s between two parallel walls. Find the two frequencies heard by the bat and the beat frequency between the two. The speed sound is 330 m/s.

Q.

An electromagnetic wave of frequency$5GHz$, is travelling in a medium whose relative electric permittivity and relative magnetic permeability both are $2$. Its velocity in this medium is _______ $\times {10}^{7}m/s$.

Q. Electromagnetic waves are represented in which of the following format?

1. Square waves
2. Triangular waves
3. Sinusiodal waves
4. None of the above

Q.

How does intensity depend on amplitude?

Q. Two particles of same mass are performing simple harmonic oscillation along the $y-$axis whose displacement time graphs are shown. If $E_1$ and $E_2$ are their energies of oscillation, then

1. $E_1=2E_2$
2. $E_1=4E_2$
3. $E_1=E_2$
4. $E_2=4E_1$

Q.

If the speed of a transverse wave on a stretched string of length 1 m is $60m{s}^{-1}$, what is the fundamental. frequency of vibration ?

Q. A transverse wave $y=5\text{sin}(3t-x)$, where $y,x,t$ are in SI units, is propagating on a stretched string having density $\rho =1{\text{kg/m}}^3$. Find the intensity of propagating wave.

1. $325{\text{W/m}}^2$
2. $200{\text{W/m}}^2$
3. $337.5{\text{W/m}}^2$
4. $300{\text{W/m}}^2$

Q.

A plane electromagnetic waves of frequency 25 mega hertz travel in free space along x direction. At a particular point in the space and time E= 6.3j V/m.what is B vector of that point

Q. A tuning fork of frequency $580Hz$ is employed to produce transverse waves on a long rope. The distance between the nearest crests is found to be $20cm$. The velocity of the wave is

1. $58{ms}^{-1}$
2. $580{ms}^{-1}$
3. $20{ms}^{-1}$
4. $116{ms}^{-1}$

Q. A wave is described by the equation $$y=(1-0mm)\sin \pi (\frac{x}{20cm}-\frac{t}{0-01s})$$ (a) Find the time period and the wavelength . (b) Write the equation for the velocity of the particles. Find the speed of the particle at $x=1\cdot 0cm$ at time $t=0\cdot 01\cdot s$ (c) What are the speeds Of the particles at $x=3\cdot 0cm$, $5.0cm$and $7.0cm$ at $t=0.01s$ ? (d) What are the speeds of the particles at $x=1\cdot 0cm$ at $t=0\cdot 011,0\cdot 012$ and $0.013s$ ?

Q. A transverse wave is described by the equation $y=y_{0}sin2\pi (ft-\frac{x}{\lambda })$. The maximum particle velocity is four times the wave velocity if

1. $\lambda =\pi y_0$
2. $\lambda =\frac{\pi y_0}{4}$
3. $\lambda =2\pi y_0$
4. $\lambda =\frac{\pi y_0}{2}$

Q.

What information does Ψ indicate about the wave?

Q. The wave function of a triangular wave pulse is defined by the relation below at time $t=0\text{sec}$
$y=\{\begin{array}{cc}mx& \text{for}0\le x\le \frac{a}{2}\\ -m(x-a)& \text{for}\frac{a}{2}\le x\le a\\ 0& \text{every where else}\end{array}$

The wave pulse is moving in the + X direction in a string having tension $T$ and mass per unit lengh $\mu$. The total kinetic energy present with the wave pulse is

1. $\frac{m^{2}Ta}{2}$
2. $m^{2}Ta$
3. $\frac{3m^{2}Ta}{2}$
4. None of these

Q. A transverse wave is represented by $y=Asin(\omega t-kx)$. For what value of the wavelength is the wave velocity equal to the maximum particle velocity?

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

Q. A beam of protons with velocity $4\times {10}^{5}m/s$ enters a uniform magnetic field of 0.22 T at an angle of ${60}^o$ with magnetic field. Pitch of helical path traced in $cm$ is [Take mass of proton = $1.6\times {10}^{-27}kg$ and $\pi$ = $\frac{22}{7}$] (Write upto two digits after the decimal point.)

Q.

The given figure shows the shape of part of a long string in which transverse waves are produced by attaching one end of the string to tuning fork of frequency 250 Hz. What is the velocity of the waves?

1. $1.0m{s}^{-1}$

2. $2.0m{s}^{-1}$

3. $1.5m{s}^{-1}$

4. $2.5m{s}^{-1}$

Q. For standing waves in string, the minimum gap between two points which are fixed or do not oscillate is: ($\lambda =$ wavelength)

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

Q. A wave is given by $y=3sin2\pi (\frac{t}{0.04}-\frac{x}{0.01})$, where y is in cm. frequency o wave and maximum acceleration of particle will be

1. $100HZ,4.7\times {10}^{3}cm/s^2$
2. $50HZ,7.5\times {10}^{3}cm/s^2$
3. $25HZ,4.7\times {10}^{4}cm/s^2$
4. $25HZ,7.4\times {10}^{4}cm/s^2$

Q. A transverse wave is described by the equation $Y=Y_0\sin 2\pi (ft-\frac{x}{\lambda })$. The maximum particle velocity is equal to four times the wave velocity if:

1. $\lambda =\frac{\pi Y_0}{4}$
2. $\lambda =\frac{\pi Y_0}{2}$
3. $\lambda =\pi Y_0$
4. $\lambda =2\pi Y_0$

Q. A transverse wave of amplitude $0.5$ m and wavelength $1$ m and frequency $2$ Hz is propagating in a string in the negative x direction. The expression for this wave is?

1. $y=0.5\sin (\pi x-2\pi t)$
2. $y=0.5\sin (2\pi x-4\pi t)$
3. $y=0.5\sin (2\pi x+2\pi t)$
4. $y=0.5\sin (2\pi x+4\pi t)$

Q. The particle displacement in a wave is given by
$y=0.2\times {10}^{-5}\cos (500t-0.025x)$
Where the distances are measured in meters and time in seconds. Now

1. Wave velocity is $2\times {10}^4{ms}^{-1}$
2. Particle velocity is $2\times {10}^4{ms}^{-1}$
3. Initial phase difference is $\frac{\pi }{2}$
4. Wavelength of the wave is $\left(80\pi \right)m$

Q. A flood light is covered with a filter that transmits red light. The electric field of the emerging beam is represented by a sinusoidal plane wave $E_x=36\sin (1.20\times {10}^{7}z+6\times {10}^{15}t)V/m$. The average intensity of the beam will be

1. $0.86W/m^2$
2. $3.44W/m^2$
3. $1.72W/m^2$
4. $6.88W/m^2$

Q. The distance between $2^{nd}$ crest and $6^{th}$ trough of a wave shown below is $24cm$. If the wave velocity of the moving crests is $20m{s}^{-1}$, then frequency of rocking of the boat is

1. $24Hz$
2. $20Hz$
3. $375Hz$
4. $240Hz$

Q. You have learnt that a travelling wave in one dimension is represented by a function $y=f(x,t)$ where $x$ and $t$ must appear in the combination $x-v$ t or $x+vt,$ i.e., $y=f(x\pm vt).$ Is the converse true? Examine if the following functions for y can possibly represent a travelling wave:
$\left(a\right)(x-vt{)}^2$
(b) $\log \left[\right(x+\vee y\left)/x_0\right]$
(c) $1/(x+vt)$

Q. A transverse wave is expressed as, $y=y_0\sin 2\pi ft$.
For what value of $\lambda$, maximum particle velocity equals to 4 times the wave velocity?

1. $y_0\frac{\pi }{2}$
2. $y_0\pi$
3. $y_0\frac{\pi }{4}$
4. $2y_0\pi$

Q. The equation of a travelling wave is $y=60cos(1800t-6x)$
where $y$ is in microns, $t$ in second and $x$ in metre. The ratio of maximum particle velocity to the velocity of wave propagation is

1. $3.6\times {10}^{-4}$
2. $3.6\times {10}^{-6}$
3. $3.6\times {10}^{-8}$
4. None of these

Q. A point sound source is situated in a medium of bulk modulus $1.6b\times {10}^{5}N/m^2$ . The equation for the wave emitted it is given by $y=Asin(7.5\pi x-300\pi t)$ . Velocity of wave is $\nu$ and the displacement amplitude of the waves received by the observer standing at a distance $5m$ from the source is A . The density of medium is $\rho$ . The pressure amplitude at the observer ear is $30Pa$ . The intensity of wave received by the observer is I . Then

1. $\rho =1kg/m^3$
2. $\nu =400m/s$
3. $A=\frac{{10}^{-4}}{4\pi }$
4. $I=1W/m^2$

Q. The electric component of an electromagnetic wave (in SI units) are given by
$E_x={10}^{2}sin\left[\pi \right(9\times {10}^{14}t-3\times {10}^{6}z\left)\right]$
$E_y=0,E_z=0$ find the inensity of this radiation.

1. $1.33W{m}^{-2}$
2. $13.3\times {10}^{-12}W{m}^{-2}$
3. $13.3W{m}^{-2}$
4. $0.133W{m}^{-2}$