Question
A wave $\text{Y} = \text{A}\sin(\omega\text{t} - \text{kx})$ allowed through a string gets reflected from a rigid support and forms stationary wave. Derive the expression for the standing wave.
✓
Answer
Given wave: $\text{Y}_\text{i}=\text{A}\sin(\omega\text{t}-\text{Kx})$
Reflected wave:
$\text{Y}_\text{r}=\text{A}\sin (\omega\text{t}+\text{Kx}+\pi)$
$=-\text{A}\sin(\omega\text{t}+\text{Kx})$
Applying superposition principle,
$\text{y}=\text{Y}_\text{i}+\text{Y}_\text{r}$
$=\text{A}\sin(\omega\text{t}-\text{Kx})-\text{A}\sin(\omega\text{t}+\text{Kx})$
$\text{y}=2\text{A}\sin\text{Kx}\cos\omega\text{t}$
Since amplitude $2\text{A}\sin\text{Kx}$ varies with position, it represents a standing wave.
Reflected wave:
$\text{Y}_\text{r}=\text{A}\sin (\omega\text{t}+\text{Kx}+\pi)$
$=-\text{A}\sin(\omega\text{t}+\text{Kx})$
Applying superposition principle,
$\text{y}=\text{Y}_\text{i}+\text{Y}_\text{r}$
$=\text{A}\sin(\omega\text{t}-\text{Kx})-\text{A}\sin(\omega\text{t}+\text{Kx})$
$\text{y}=2\text{A}\sin\text{Kx}\cos\omega\text{t}$
Since amplitude $2\text{A}\sin\text{Kx}$ varies with position, it represents a standing wave.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
A small block of mass m is kept on a bigger block of mass M which is attached to a vertical spring of spring constant k as shown in the figure. The system oscillates vertically.
- Find the resultant force on the smaller block when it is displaced through a distance x above its equilibrium position.
- Find the normal force on the smaller block at this position. When is this force smallest in magnitude?
- What can be the maximum amplitude with which the two blocks may oscillate together?
A light beam of wavelength $400$ run is incident on a metal plate of work function $2.2eV$.
→- A particular electron absorbs a photon and makes two collisions before coming out of the metal. Assuming that $10\%$ of the extra energy is lost to the metal in each collision, find the kinetic energy of this electron as it comes out of the metal.
- Under the same assumptions, find the maximum number of collisions the electron can suffer before it becomes unable to come out of the metal.
A body is dropped in a hole drilled across a diameter of the earth. Show that it executes S.H.M. Assume the earth to be homogeneous sphere of radius R.
A bullet of mass m moving at a speed v hits a ball of mass M kept at rest. A small part having mass m' breaks from the ball and sticks to the bullet. The remaining ball is found to move at a speed $v_1$ in the direction of the bullet. Find the velocity of the bullet after the collision.
→A laser light bean sent to the moon takes 2.56s to return after reflection at the Moon's surface. Calculate the radius of the lunar orbit around the earth.
If c is r.m.s. speed of molecules in a gas and v is the speed of sound waves in the gas, show that c/ v is constant and independent of temperature for all diatomic gases.
Figure, shows water in a container having $2.0mm$ thick walls made of a material of thermal conductivity $0.50\text{Wm}^{-1}{^{\circ}}\text{C}^{-1}.$ The container is kept in a melting-ice bath at $0^\circ C$. The total surface area in contact with water is $0.05m^2$. A wheel is clamped inside the water and is coupled to a block of mass M as shown in the figure. As the block goes down, the wheel rotates. It is found that after some time a steady state is reached in which the block goes down with a constant speed of $10cms^{-1}$ and the temperature of the water remains constant at $1.0^\circ C$. Find the mass M of the block. Assume that the heat flows out of the water only through the walls in contact. Take $g = 10ms^{-2}$.
→Consider the situation of the previous problem. Suppose each of the blocks is pulled by a constant force F instead of any impulse. Find the maximum elongation that the spring will suffer and the distances moved by the two blocks in the process.
A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of $45Hz$. The mass of the wire is $3.5 \times 10^{–2}kg$ and its linear mass density is $4.0 \times 10^{–2}kg m^{–1}$. What is
→- The speed of a transverse wave on the string,
- The tension in the string?
Two narrow bores of diameters 3.0 mm and 6.0 mm are joined together to form a U-tube open at both ends. If the Utube contains water, what is the difference in its levels in the two limbs of the tube ? Surface tension of water at the temperature of the experiment is $7.3 \times 10^{-2} \mathrm{~N} \mathrm{~m}^{-1}$. Take the angle of contact to be zero and density of water to be $1.0 \times 103 \mathrm{~kg} / \mathrm{m}^{-3}\left(\mathrm{~g}=9.8 \mathrm{~m} / \mathrm{s}^{-2}\right)$.
→