Waves - Class 11 Part 2

In a Young's Double Slit Experiment, the slits $S_1$ and $S_2$ are separated by a distance $d$ and the screen is placed at a distance $D$ ($D \gg d$). The space between the slits and the screen is filled with a non-homogeneous transparent medium whose refractive index varies with the transverse coordinate $y$ as $\mu(y) = \mu_0 + \alpha y$, where $\mu_0$ and $\alpha$ are positive constants. The $y$-axis is parallel to the plane of the slits, with $y=0$ passing through the midpoint of $S_1$ and $S_2$. Slit $S_1$ is located at $y = d/2$ and $S_2$ is at $y = -d/2$. Assuming $\alpha D \ll \mu_0$ such that light rays travel in approximately straight lines, what is the $y$-coordinate of the central maximum on the screen?

A damped harmonic oscillator consists of a mass $m$, a spring of constant $k$, and a damping medium providing a damping force $-b\dot{x}$. It is driven by a time-varying force $F(t) = F_0 \cos(\theta(t))$, where the instantaneous driving frequency $\omega(t) = \frac{d\theta}{dt}$ is slowly swept linearly as $\omega(t) = \omega_0 + \beta t$. Here $\omega_0 = \sqrt{k/m}$ is the undamped natural frequency, and $\beta$ is a small positive constant. Assuming the system continuously adapts to a quasi-steady state, its displacement is $x(t) = A(t) \cos(\theta(t) - \delta(t))$, where $\delta(t)$ is the instantaneous phase lag. At $t=0$, the driving frequency passes through resonance ($\omega = \omega_0$). What is the difference between the instantaneous frequency of the driving force and the instantaneous frequency of the displacement at $t=0$?

A continuous harmonic transverse wave of wavelength $\lambda$ is generated at $x=0$ on a taut string of linear mass density $\mu_1$. A detector is fixed at $x=L$. The string is joined smoothly at $x=L+d$ to a second semi-infinite string of linear mass density $\mu_2$ under the same tension. The detector measures the time-averaged intensity of the superposition of the incident wave and the first reflected wave. Let $d_1$ be the minimum non-zero distance $d$ to observe a local maximum in intensity at the detector when $\mu_2 = 4\mu_1$. Let $d_2$ be the minimum non-zero distance $d$ to observe a local maximum in intensity when $\mu_2 = \frac{\mu_1}{4}$. Which of the following represents the correct pair $(d_1, d_2)$?

A 2D anisotropic lattice of coupled pendulums lies in the $x$-$y$ plane. Each pendulum has length $L = 0.625 \text{ m}$ and oscillates under gravity ($g = 10 \text{ m/s}^2$). For small angular displacements $\theta(x,y,t)$, the continuum limit wave equation is: $\frac{\partial^2 \theta}{\partial t^2} = c_x^2 \frac{\partial^2 \theta}{\partial x^2} + c_y^2 \frac{\partial^2 \theta}{\partial y^2} - \omega_0^2 \theta$ where $c_x = 3 \text{ m/s}$ and $c_y = 1.5 \text{ m/s}$ are the coupling velocities, and $\omega_0$ is the natural angular frequency of an uncoupled pendulum. A continuous harmonic driving force at the origin $(0,0)$ maintains the central pendulum's oscillation at an angular frequency $\omega = 5 \text{ rad/s}$. Assuming steady-state outward wave propagation without reflections, what is the magnitude of the phase difference between the oscillations of the pendulums at point $A(6 \text{ m}, 0)$ and point $B(0, 1.5 \text{ m})$?

A long string under constant tension $T$ has a variable mass per unit length $\mu(x) = \mu_0 (1 + \alpha x)^4$ for $x \ge 0$. A single positive transverse wave pulse (displacement $y \ge 0$) is generated at $x=0$ with an initial amplitude $A_0$ and propagates in the $+x$ direction. The string rests on a surface that exerts a constant transverse kinetic friction force of magnitude $f_k$ per unit length, which always opposes the transverse velocity of the string particles. Assuming $f_k$ is sufficiently small such that the spatial variation of the pulse amplitude is predominantly governed by the non-uniform density (WKB approximation), what is the magnitude of the total work done by friction on the string segment from $x=0$ to $x=L$ during the complete passage of the pulse?

A plane electromagnetic wave propagates in vacuum along the direction given by the unit vector $\hat{n} = \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}}$. A particle of mass $m$ and charge $q$ is released from rest at the origin. Neglecting radiation loss and gravity, and assuming the motion remains non-relativistic ($v \ll c$), let $\vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k}$ be the instantaneous velocity of the particle at any time $t$. Which of the following relations is always true?

Two transverse waves propagate on a taut string of linear mass density $\mu = 0.01 \text{ kg/m}$ under a tension $T = 100 \text{ N}$. The individual wave displacements are given by $y_1(x,t) = A \sin(kx - \omega t)$ and $y_2(x,t) = A \sin(kx + \omega t + \phi)$. The resultant displacement of the string is $y = y_1 + y_2$. At $x = 0$ and $t = 0$, the instantaneous spatial slope of the string is $\frac{\partial y}{\partial x} = 0.03$, and the phase constant is $\phi = \frac{\pi}{3}$. What is the magnitude of the transverse particle velocity of the string at $x = 0$ and $t = 0$?

An electromagnetic wave is incident from Medium 1 ($x < 0$, refractive index $n_1 = 5$) onto a planar interface at $x = 0$ with Medium 2 ($x > 0$, refractive index $n_2 = 4$). The incident electric field vector is given by $\vec{E}_i = E_0 (3\hat{i} - 4\hat{j}) \cos(\vec{k}_i \cdot \vec{r} - \omega t)$, where $E_0 > 0 \text{ V/m}$. Given that the incident wave vector $\vec{k}_i$ lies entirely in the $x-y$ plane, which of the following represents the unit vector in the direction of the transmitted electric field amplitude $\vec{E}_t$?

A plane electromagnetic wave propagates in a vacuum. At $x=0$, the electric field is $\mathbf{E}(0,t) = \alpha q(t) \hat{j}$ and the magnetic field is $\mathbf{B}(0,t) = \beta \frac{di}{dt} \hat{k}$, where $q(t)$ is the instantaneous charge on a capacitor in an ideal LC circuit, $i(t) = \frac{dq}{dt}$ is the current, and $\alpha, \beta$ are positive constants. The capacitor charge varies as $q(t) = q_0 \cos(\omega t)$. Let $E_0 = \alpha q_0$ and $S_0 = \frac{\alpha \beta q_0^2 \omega^2}{2\mu_0}$. Which of the following correctly describes the electric field $\mathbf{E}(x,t)$ and the time-averaged Poynting vector $\langle \mathbf{S} \rangle$ of the wave?

Consider a cylindrical plasma column of radius $R$ with its axis aligned along the $x$-axis. The equilibrium electron number density varies radially as $n_0(r) = \eta r$, where $r$ is the distance from the $x$-axis and $\eta$ is a positive constant. An electron plasma wave propagates in the positive $x$-direction. Assume the electron number density remains approximately equal to its equilibrium value $n_0(r)$ for calculating the current density. At $t=0$, the spatial profile of the electron drift velocity is given by $\vec{v}_d(x,0) = v_0 \sin(kx) \hat{i}$ (where $v_0 > 0$ and $k > 0$). The wave has a temporal angular frequency $\omega$. What is the net charge $\Delta Q$ that flows through the cross-section at $x=0$ in the positive $x$-direction during the time interval $t=0$ to $t=\frac{\pi}{\omega}$? (Here $e$ represents the elementary positive charge).

A long taut string of linear mass density $\mu$ and tension $T$ is immersed in a viscous fluid. The fluid exerts a damping force per unit length equal to $-b \frac{\partial y}{\partial t}$, where $y(x,t)$ is the transverse displacement and $b$ is a positive constant. A harmonic wave of angular frequency $\omega$ is generated at $x=0$ and propagates in the $+x$ direction. Assuming the damping is small ($b \ll \mu \omega$), the wave amplitude decays exponentially as $e^{-\alpha x}$. Which of the following expressions correctly represents the spatial attenuation constant $\alpha$ and the phase velocity $v_p$ of the wave, up to the second order in the dimensionless parameter $(b/\mu\omega)$?

A plane electromagnetic wave is normally incident from vacuum ($z < 0$) onto a non-uniform non-conducting medium ($z > 0$). The incident electric field is given by $\vec{E}_i = E_0 \cos(kz - \omega t) \hat{y}$. The medium in the region $z > 0$ has a uniform permittivity $\epsilon = \epsilon_0$ but a spatially varying magnetic permeability $\mu(x) = \mu_0 (1 + \alpha x)^2$, where $\alpha > 0$ is a constant. Determine the reflected magnetic field $\vec{B}_r$ at the interface $z = 0$, at the position $x = \frac{2}{\alpha}$ and time $t = 0$.

A high-frequency periodic thermal source generates a temperature wave in a semi-infinite rod (Rod 1). The wave travels towards a junction at $x=0$ where Rod 1 is perfectly joined to another semi-infinite rod (Rod 2). The amplitude of the incident temperature wave at the junction is $T_0 = 60 \text{ K}$. At the junction, both the temperature and the total heat current are continuous. The properties of the rods are given as:

Rod 1: Cross-sectional area $A_1 = A$, thermal conductivity $\kappa_1 = 2\kappa$, volumetric heat capacity $(\rho c)_1 = 5C$. Rod 2: Cross-sectional area $A_2 = 2A$, thermal conductivity $\kappa_2 = 5\kappa$, volumetric heat capacity $(\rho c)_2 = 8C$.

Assuming the thermal wave propagates as $T(x,t) \propto e^{-\gamma x} \cos(\omega t - \gamma x)$ where $\gamma = \sqrt{\omega \rho c / 2\kappa}$, what is the amplitude of the transmitted temperature wave in Rod 2 at the junction?

A series LCR circuit consists of a resistor with resistance $R = 4 \, \Omega$, an inductor with inductance $L = 10 \, \text{mH}$, and a capacitor with capacitance $C = 2.5 \, \text{mF}$. The circuit is driven by a time-varying voltage source given by $V(t) = 15\sqrt{2} \sin(100t) + 12\sqrt{2} \sin(200t)$ where $V$ is in volts and $t$ is in seconds. A hot-wire ammeter of negligible resistance is connected in series with the circuit. What is the reading of the ammeter?

Two coherent plane electromagnetic waves, Wave 1 and Wave 2, of wavelength $\lambda$ propagate in the $+z$ direction. Wave 1 has a uniform intensity $I_1 = 9I_0$ and travels through vacuum. Wave 2 has a uniform intensity $I_2 = 4I_0$ and passes through a transparent slab of uniform thickness $d$ placed just before the $z=0$ plane (occupying $-d \le z \le 0$). Due to a spatially varying medium density, the refractive index of the slab varies along the $x$-axis as $n(x) = 1 + \frac{\lambda}{4ad}x$, where $a$ is a positive constant. An opaque screen with a rectangular aperture is placed at the $z=0$ plane. The aperture is bounded by $0 \le x \le a$ and $0 \le y \le b$. Assuming both waves have identical phase at $z = -d$, what is the net time-averaged power (intensity flux) passing through the aperture?

A massless disk of radius $R = 0.5 \text{ m}$ is pivoted at its center and can rotate freely in a horizontal plane. Two identical long strings, each of linear mass density $\mu = 0.1 \text{ kg/m}$ and under tension $T = 40 \text{ N}$, are attached to a pin on the rim of the disk. String 1 extends along the negative x-axis ($x < 0$) and String 2 extends along the positive x-axis ($x > 0$). The transverse displacement of the strings at $x=0$ is constrained to $y(0,t) = R\theta(t)$, where $\theta(t)$ is the angular displacement of the disk. The disk is subjected to a viscous rotational damping torque $\tau_d = -b \omega$, where $b = 1.0 \text{ N m s}$ and $\omega = \dot{\theta}$. Two rectangular wave pulses are sent towards the disk from opposite sides. The incident pulses are characterized by their constant transverse velocities $v_{y1} = \frac{\partial y_{i1}}{\partial t} = 0.4 \text{ m/s}$ (for Pulse 1 on String 1) and $v_{y2} = \frac{\partial y_{i2}}{\partial t} = 0.2 \text{ m/s}$ (for Pulse 2 on String 2). Both pulses last for a duration $t_0 = 0.5 \text{ s}$ and are timed such that they arrive at the disk simultaneously, overlapping completely. Calculate the total energy dissipated by the damping torque during this interaction.

Two coherent electromagnetic waves, having vacuum intensities $I_1 = 4 \text{ W/m}^2$ and $I_2 = 9 \text{ W/m}^2$, are superposed in phase at $x = 0$ to form a single resultant wave. This resultant wave propagates in the $+x$ direction through a semiconductor medium of length $L = \ln(2) \text{ m}$. Inside the medium, local energy dissipation alters the band gap, creating a feedback loop between the wave intensity and the medium's properties. The local steady-state refractive index $n(x)$ depends on the local intensity $I(x)$ as $n(x) = n_0 + \mu I(x)$. Consequently, the local absorption coefficient $\alpha(x)$ of the medium becomes intensity-dependent and is given by $\alpha(x) = \alpha_0 + \sigma (n(x) - n_0)$, where $\alpha_0 = 1 \text{ m}^{-1}$ is the linear absorption coefficient. Given that the product $\sigma \mu = 0.2 \text{ m/W}$, and the intensity decays according to $\frac{dI}{dx} = -\alpha(x) I(x)$, determine the intensity of the wave as it exits the medium at $x = L$.

A rigid platform is moving along the $+x$ direction through still air. The speed of sound in still air is $c$. A sound detector $D$ is mounted at the front of the platform. Two point sound sources, $S_1$ and $S_2$, are mounted on the platform at distances $L_1$ and $L_2$ respectively behind the detector ($L_1 > L_2$). Both sources emit sound isotropically at a constant frequency $f_0$ in their instantaneous rest frame. The platform starts from rest at $t=0$ and moves with a constant acceleration $a$. At a later time $t$, the detector has a velocity $v$ ($v < c$). Assume the sound received by the detector at this instant was emitted after $t=0$, and that $\frac{aL_1}{(c-v)^2} \ll 1$. Which of the following expressions best represents the beat frequency recorded by the detector at this instant?

A transverse wave propagates along a long stretched string of linear mass density $\mu$ and tension $T$. Due to a viscous surrounding medium, the wave amplitude attenuates exponentially. The transverse displacement of the string is given by: $y(x,t) = A_0 e^{-\gamma x} \sin(\omega t - kx)$ where $A_0$, $\gamma$, $\omega$, and $k$ are positive constants. Which of the following expressions correctly represents the time-averaged total mechanical energy density (energy per unit length) $\langle u \rangle$ of the wave at a position $x$? Assume all quantities are expressed in coherent SI units.

A uniform wire of unstretched length $L_0$, cross-sectional area $A_0$, mass $m$, and coefficient of linear expansion $\alpha$ is suspended vertically from a rigid support. A block of mass $M$ ($M \gg m$) is attached to its lower end. A vibrator at the top end drives the wire at a constant frequency $f$, forming a standing wave. The $n$-th node from the top is at a distance $x_n$ (Assume the top end acts as a node). If the temperature of the wire is increased by $\Delta T$, the fractional change in the position of the $n$-th node, $\frac{\Delta x_n}{x_n}$, is given by: (Assume $\alpha \Delta T \ll 1$, initial stress $\sigma = \frac{Mg}{A_0} \ll Y$, where $Y$ is the Young's modulus of the wire)

A perfectly flexible, insulating string of length $L$, uniform mass per unit length $\mu$, and uniform linear charge density $\lambda$ is stretched with tension $T$ along the central axis of a long, grounded, conducting cylindrical shell of inner radius $R$ ($R \ll L$). The ends of the string are fixed at $x=0$ and $x=L$ on the axis. Assuming the transverse displacement $y(x,t)$ of the string is very small ($y \ll R$), find the fundamental angular frequency $\omega_1$ of the transverse vibrations of the string. (Ignore gravity and edge effects of the cylinder).

An electromagnetic standing wave is established inside a one-dimensional cavity of length $L$ bounded by perfectly reflecting mirrors at $x = 0$ and $x = L$. The cavity is filled with a non-magnetic dielectric medium whose refractive index varies spatially as $n(x) = n_0 \left(1 + \frac{x}{L}\right)$, where $n_0$ is a positive dimensionless constant. The standing wave is excited in a resonant mode such that exactly two intermediate nodes exist between the mirrors (excluding the boundaries). What is the position $x_1$ of the first intermediate node from the mirror at $x = 0$?

A semi-infinite string of linear mass density $\mu$ and tension $T$ lies along the negative x-axis ($x \le 0$). Its end at $x=0$ is attached to a massless conducting rod of length $l$ oriented parallel to the z-axis. The rod can slide without friction along two parallel conducting rails that are parallel to the y-axis. A uniform magnetic field $B$ points in the positive x-direction. The rails are connected to a DC voltage source $V_0$, a resistor $R$, and an inductor $L_0$ to form a closed series circuit. At $t=0$, the circuit is closed. As the rod moves, it generates transverse waves in the string. Assuming the string only supports transverse waves and remains taut, what is the steady-state energy stored in the inductor after a long time? (All given quantities are in standard SI units)

An electron of mass $m$ is confined in a one-dimensional infinite potential well of width $L$ between $x = 0$ and $x = L$. The electron is in its first excited state ($n=2$). Let $\Delta x$, $\Delta p$, and $\Delta E$ denote the standard deviations (uncertainties) in the position, momentum, and total energy of the electron in this state, respectively, and let $h$ be Planck's constant. A flawed argument suggests that because the probability density is exactly zero at the node $x = L/2$, the electron is perfectly excluded from this point, implying a local spatial uncertainty $\Delta x \to 0$ and thus $\Delta E \to \infty$. By correctly applying the principles of quantum mechanics to the entire stationary state, determine the true values of $\Delta x$, $\Delta p$, and $\Delta E$.

A uniform heavy wire of length $L$, cross-sectional area $A$, volume mass density $\rho$, Young's modulus $Y$, and coefficient of linear thermal expansion $\alpha$ is clamped vertically between two perfectly rigid horizontal supports. Initially, at a temperature $\theta_0$, the wire is straight and taut such that the tension at its lowest point is exactly zero. The temperature of the wire is then uniformly decreased by $\Delta \theta$. Assuming the cross-sectional area remains approximately constant and $g$ is the acceleration due to gravity, which of the following expressions gives the fundamental frequency of transverse vibrations of the wire at the new temperature?

A uniform flexible wire of length $L$ and mass per unit length $\mu$ is suspended from a rigid support at its upper end, while its lower end is free. The wire carries a steady current $I$. A large fixed vertical sheet is placed nearby, carrying a uniform vertical surface current density $K$. This creates a uniform horizontal magnetic field $\vec{B}$ across the region of the wire. The wire reaches a steady equilibrium state. A small transverse wave pulse is generated at the free lower end. What is the time $\tau$ taken by the pulse to travel along the wire and reach the rigid support? (Assume the magnetic field of the flexible wire does not affect the sheet, and $g$ is the acceleration due to gravity).

A string of length $L = 1 \text{ m}$ is kept under a constant tension $T = 100 \text{ N}$. The linear mass density of the string is non-uniform and varies with the distance $x$ from one end as $\lambda(x) = \lambda_0(1 + \alpha x)$, where $\lambda_0 = 0.01 \text{ kg/m}$ and $\alpha = 3 \text{ m}^{-1}$. Using the approximation that the fundamental period of vibration is twice the time taken by a transverse wave pulse to travel the entire length of the string, determine the fundamental frequency of the string.

A block of mass $M$ is constrained to move on a frictionless horizontal track. A string of uniform linear mass density $\mu$ is attached to the block, passes through a small smooth ring fixed at a height $h$ directly above the track, and supports a hanging mass $m_0$. An external mechanism maintains the block's velocity at a constant value $v$ directed away from the point directly below the ring. At the instant the block is at a horizontal distance $x$ from the point directly below the ring, what is the instantaneous fundamental frequency of the transverse vibrations of the string segment between the ring and the block? (Define the frequency as the inverse of the round-trip time of a wave pulse, and assume $v$ is sufficiently small).

A cylindrical string of length $L$, cross-sectional area $A$, and intrinsic linear mass density $\mu_0$ is stretched with a constant tension $T$ between two rigid supports inside a closed chamber. The chamber is filled with an ideal monoatomic gas ($\gamma = \frac{5}{3}$). Due to the fluid's added mass effect, the effective linear mass density of the string during transverse vibrations is given by $\mu_{\text{eff}} = \mu_0 + \rho_g A$, where $\rho_g$ is the density of the gas. Initially, the gas has an energy density (internal energy per unit volume) of $u_0$ and a mass density of $\rho_0$. In this initial state, it is given that $\mu_0 = \rho_0 A$, and the string's fundamental frequency is $f_0$. The gas is then rapidly and adiabatically compressed until its energy density becomes $32 u_0$. Assuming $L$, $A$, and $T$ remain unchanged, what is the new fundamental frequency $f_1$ of the string?

An observer is stationary at the origin $(0,0)$ of a Cartesian coordinate system. At time $t = 0$, two identical sound sources, $S_1$ and $S_2$, are located at a height $H = 110 \text{ m}$ on the y-axis (initial position $(0, 110 \text{ m})$). Source $S_1$ is projected horizontally with a velocity $\vec{u}_1 = 10 \hat{i} \text{ m/s}$. Source $S_2$ is simultaneously dropped from rest. Both sources continuously emit sound of proper frequency $f_0 = 1215 \text{ Hz}$. The speed of sound in air is $v_s = 340 \text{ m/s}$, and the acceleration due to gravity is $g = 10 \text{ m/s}^2$ (acting in the $-\hat{j}$ direction). Calculate the beat frequency detected by the observer corresponding to the sound waves emitted by the sources at the exact instant $t = 4 \text{ s}$.

Two identical satellites, $S_1$ and $S_2$, are located in deep space at $0\text{ K}$. Each contains an identical open organ pipe filled with an ideal diatomic gas. The pipes are continuously sounded at their fundamental frequencies. The walls of the pipes have a zero coefficient of thermal expansion, and the heat capacity of the gas is negligible. Satellite $S_1$ is maintained at a constant temperature $T_0$, emitting sound of frequency $f_0$ and constant acoustic intensity $I_0$. Satellite $S_2$ is initially at $T_0$. At $t=0$, its heating system fails, and it cools by radiating as a perfect black body. $S_2$ has mass $M$, specific heat capacity $c_s$, and surface area $A$. As $S_2$ cools, the local speed of sound drops, modulating the frequency $f(t)$ of its pipe. Furthermore, the acoustic intensity $I(t)$ of the pipe in $S_2$ is directly proportional to the absolute temperature of the gas ($I(t) \propto T$). An observer counts the acoustic beats between the two pipes. However, the beat detection system automatically shuts off when the intensity $I(t)$ from $S_2$ drops to $\frac{I_0}{4}$. Find the total number of beats registered by the observer from $t=0$ until the detection system shuts off. (Assume $\sigma$ is the Stefan-Boltzmann constant).

Two harmonic signals of frequencies $f_1$ and $f_2$ ($f_1 > f_2$) are fed into a phase detector. At $t = 0$, the signals are perfectly in phase. The phase detector outputs a voltage $V(t) = \alpha \Delta \phi(t) + V_0$, where $\Delta \phi(t)$ is the instantaneous phase difference between the signals, $\alpha = \frac{5}{\pi} \text{ mV/rad}$ is the calibration constant, and $V_0 = 16 \text{ mV}$ is a systematic zero-error (DC offset). A student wishes to determine the beat frequency $f_b = f_1 - f_2$. They calculate the ideal target voltage $V_{ideal}$ that corresponds to exactly $N = 8$ beats, assuming no zero-error. The student then measures the time $T$ it takes for the actual output voltage $V(t)$ to reach this calculated $V_{ideal}$ from $t=0$. Using the measured time $T$, the student calculates the apparent beat frequency as $f_b^* = \frac{N}{T}$ and finds it to be $5.0 \text{ Hz}$. What is the true beat frequency $f_b$?

A sound source $S_1$ and a stationary observer $O$ are situated in a horizontal plane. $S_1$ moves on a circular track of radius $R$ centered at the origin $(0,0)$. The observer $O$ is located at a distance $L$ from the origin along the $+x$-axis, where $L \gg R$. At $t=0$, $S_1$ starts from rest at $(R,0)$ and moves counterclockwise with a constant angular acceleration $\alpha$. $S_1$ continuously emits sound of frequency $f_0$. A stationary source $S_2$ located at the origin also emits sound of frequency $f_0$. The speed of sound in air is $c$ (assume $c \gg R\sqrt{\alpha}$). Find the average beat frequency recorded by the observer $O$ during the time interval corresponding to the first complete revolution of $S_1$.

In a modified Young's Double Slit Experiment, a monochromatic light of vacuum wavelength $\lambda$ is incident on two narrow slits $S_1$ and $S_2$ separated by a distance $d$. A screen is placed at a distance $D$ ($D \gg d$) from the slits. The region between the slits and the screen is filled with a transparent medium whose refractive index varies linearly along the transverse direction $y$ (parallel to the screen and perpendicular to the slits) as $n(y) = n_0 + \alpha y$, where $n_0$ and $\alpha$ are positive constants. The origin $y=0$ is on the central axis equidistant from the slits, with $S_1$ at $y = d/2$ and $S_2$ at $y = -d/2$. Assuming that light rays travel in straight lines from the slits to the screen and $d \ll y \ll D$, which of the following expressions gives the local fringe width $\beta(y)$ at a coordinate $y$ on the screen?

A small block of mass $m = 1 \text{ kg}$ is placed on a long horizontal plank of mass $M = 3 \text{ kg}$. The plank is attached to a fixed wall by a horizontal ideal spring of force constant $k = 400 \text{ N/m}$. The coefficient of static friction between the block and the plank is $\mu_s = \frac{\sqrt{3}}{2}$. The plank rests on a frictionless horizontal floor. Let $x=0$ be the equilibrium position of the plank. At $t = 0$, the plank is at $x = -0.05 \text{ m}$ and is moving with a velocity $v = -\frac{\sqrt{3}}{2} \text{ m/s}$. Assuming the block does not slip on the plank initially, find the exact time $t$ at which the block first begins to slip. (Take $g = 10 \text{ m/s}^2$)

A long, taut string of linear mass density $\mu$ and tension $T$ lies along the $x$-axis. A small bead of mass $M$ is fixed to the string at the origin ($x = 0$). Two harmonic waves are incident from the negative $x$-direction, given by the superposition $y_i(x,t) = A \cos(\omega_1 t - k_1 x) + A \cos(\omega_2 t - k_2 x)$, where $k_1$ and $k_2$ are the respective wave numbers. The angular frequencies are $\omega_1 = \frac{2T}{Mv}$ and $\omega_2 = \frac{2\sqrt{3}T}{Mv}$, where $v = \sqrt{\frac{T}{\mu}}$ is the wave speed. The waves are partially transmitted into the region $x > 0$. The transmitted wave can be written as $y_t(x,t) = A_1 \cos(\omega_1 t - k_1 x + \phi_1) + A_2 \cos(\omega_2 t - k_2 x + \phi_2)$, where $\phi_1$ and $\phi_2$ are the phase constants introduced at the boundary. Assuming the phase constants are chosen in the principal range $(-\pi, \pi]$, what is the magnitude of the relative phase shift $|\phi_1 - \phi_2|$ between the two transmitted components?

A particle of mass $m$ and charge $q > 0$ is initially at rest at the origin $(0,0,0)$ of an inertial reference frame. At $t=0$, a plane electromagnetic wave is turned on. The electric field of the wave is given by $\vec{E} = E_0 \sin(kx - \omega t + \phi) \hat{j}$ and the magnetic field is $\vec{B} = B_0 \sin(kx - \omega t + \phi) \hat{k}$, where $E_0 = c B_0$ and $c$ is the speed of light in vacuum. The phase constant $\phi$ is a given parameter ($0 \le \phi < 2\pi$). Which of the following expressions represents the maximum longitudinal momentum ($p_x$) acquired by the particle during its subsequent motion?

A spherical soap bubble of equilibrium radius $R_0$ and surface tension $T$ pulsates radially in an atmosphere of constant ambient pressure $P_0$, emitting a sound wave. The intensity of the emitted sound wave depends on the effective stiffness $k$ of the bubble. The gas inside undergoes small adiabatic oscillations such that its radius at any instant is $R = R_0 + x$, where $x \ll R_0$. The specific heat ratio of the gas is $\gamma$. Defining the net radial restoring force on the bubble surface as $F = -kx$, which of the following expressions correctly represents the effective spring constant $k$?

A uniform plane wave of initial intensity $I_0 = 2500 \text{ W m}^{-2}$ propagates in the $+x$ direction through a viscous fluid. The power dissipated per unit volume by the wave, $p(x) = -\frac{dI}{dx}$, locally heats the fluid. In steady state, the local temperature rise above the ambient is given by $\Delta T(x) = \frac{p(x)}{h}$, where $h = 50 \text{ W m}^{-3} \text{ K}^{-1}$ is the volumetric cooling coefficient. The attenuation coefficient of the fluid increases with temperature as $\alpha(x) = \alpha_0 + \gamma \Delta T(x)$, where $\alpha_0 = 0.1 \text{ m}^{-1}$ and $\gamma = 0.01 \text{ m}^{-1} \text{ K}^{-1}$. What is the ratio of the power dissipated per unit volume at $x=0$ to the power dissipated per unit volume at a distance $x=L$ where the wave intensity has decreased to $I_L = 1000 \text{ W m}^{-2}$?

A monochromatic laser beam of initial intensity $I_0$ (in $\text{W m}^{-2}$) propagates along the positive $x$-axis into a thick semiconductor crystal starting at $x=0$. As the beam propagates, photon absorption increases the local temperature of the crystal, which in turn decreases the energy band gap. Assume the local steady-state temperature rise is directly proportional to the local beam intensity $I(x)$, such that the effective absorption coefficient becomes $\alpha(I) = \alpha_0 + \lambda I(x)$. Here, $\alpha_0$ (in $\text{m}^{-1}$) is the intrinsic absorption coefficient and $\lambda$ (in $\text{m W}^{-1}$) is a positive thermal-coupling constant. Determine the depth $x_{1/2}$ (in $\text{m}$) at which the beam intensity drops to half of its initial value ($I_0/2$).

A standing acoustic wave of wavelength $\lambda$ and displacement amplitude $A_0$ is established in a closed cylindrical tube filled with an ideal rigid diatomic gas. The space-and-time averaged mechanical energy density of this standing wave is found to be exactly equal to a small fraction $\alpha$ of the thermal energy density associated with \textit{only the translational} degrees of freedom of the gas molecules. Which of the following expressions correctly gives the displacement amplitude $A_0$?

A uniform string of length $L$ (m) and linear mass density $\mu$ (kg/m) is stretched with a constant tension $T$ (N) along the $x$-axis from $x=0$ to $x=L$. The end at $x=0$ is rigidly fixed. The end at $x=L$ is attached to a solid block of mass $M$ (kg) and uniform cross-sectional area $S$ (m$^2$). The block floats in equilibrium in a liquid of density $\rho_L$ (kg/m$^3$), constrained to move only in the vertical $y$-direction. The system is set into small transverse vertical oscillations. The restoring force on the block is purely due to buoyancy and the string's tension. It is observed that the fundamental angular frequency $\omega$ (rad/s) of this coupled system is exactly equal to the natural angular frequency of the isolated block oscillating vertically in the liquid. Which of the following relations must hold true? (Assume $g$ is the acceleration due to gravity in m/s$^2$)

A lossless transmission line of length $d = \lambda/8$ (where $\lambda$ is the operating wavelength) has a real characteristic impedance $Z_0 = \sqrt{L_c/C_c}$, where $L_c$ and $C_c$ are the inductance and capacitance per unit length. The line is located along the $z$-axis from $z = -d$ to $z = 0$. It is terminated at $z = 0$ by a purely inductive load $L_0$ such that its reactance at the operating angular frequency $\omega$ is exactly equal to $Z_0$ (i.e., $\omega L_0 = Z_0$). A sinusoidal generator is connected at $z = -d$, establishing a steady-state standing wave on the line. Let $U_E$ and $U_M$ be the total time-averaged electric and magnetic energies stored in the entire length of the transmission line, respectively. What is the ratio $\frac{U_E}{U_M}$?

A rigid cylindrical pipe of length $L$, closed at both ends, contains a pure monoatomic gas A of molar mass $M_0$ at a pressure $P_0$ and temperature $T$. The fundamental frequency of longitudinal acoustic resonances in the pipe is $f_0$. A pure diatomic gas B of molar mass $4M_0$ is then slowly injected into the pipe, maintaining the temperature at $T$, until the total pressure inside the pipe reaches $2P_0$. Assuming ideal gas behavior, what is the frequency of the first overtone of the gas mixture in the pipe?

A cylindrical organ pipe of length $L$ and radius $r$ is closed at one end. The pipe is made of a solid material with a coefficient of linear expansion $\alpha$. The temperature of the pipe is increased by $\Delta T$. To isolate the effect of the pipe's expansion, assume the air inside the pipe is maintained at a constant temperature so that the speed of sound $v$ remains constant. The end correction for the pipe is given by $e = 0.6r$. A student calculates the magnitude of the fractional change in the fundamental frequency, $\left| \frac{\Delta f}{f} \right|$, but incorrectly assumes that the end correction does not change with temperature. What is the absolute difference between the true magnitude of the fractional change and the student's calculated magnitude?

A horizontal rectangular pipe of width $w$, height $d$, and initial length $L_0$ is closed at one end by a rigid wall. The other end is sealed by a movable, frictionless dielectric plunger of mass $M$ and dielectric constant $K$. The plunger perfectly fits the pipe and acts as a rigid node for acoustic waves. A section of the pipe's top and bottom walls forms a parallel-plate capacitor connected to a constant DC voltage source $V$. The plunger is partially inserted into this capacitor region such that an inward displacement of the plunger (compressing the gas) increases the length of the dielectric inside the capacitor. The pipe contains an ideal gas at initial pressure $P_0$. When the voltage is applied, the electrostatic force draws the plunger inward. The plunger moves slowly, compressing the gas isothermally until a new mechanical equilibrium is reached. If $f_0$ is the initial fundamental frequency of the gas column, what is the new fundamental frequency $f'$ at equilibrium? (Ignore fringing fields).

An organ pipe of length $L$, closed at one end, resonates in its fundamental acoustic mode. The open end of the pipe is fitted with a very light, flexible conducting membrane of area $S$. The membrane oscillates with a displacement $y(t) = A \sin(\omega_0 t)$, where $\omega_0$ is the fundamental angular frequency of the pipe and $A$ is the displacement amplitude.

A fixed rigid conducting plate is placed at a distance $d$ ($d \gg A$) from the equilibrium position of the membrane, forming a parallel-plate capacitor $C(t)$. This capacitor is connected in series with an ideal DC voltage source $V_0$, an inductor $L_0$, and a resistor $R$. The electrical circuit is tuned such that its natural resonant frequency matches the acoustic frequency $\omega_0$.

Assuming ideal conditions and neglecting fringing fields, what is the amplitude $I_0$ of the steady-state AC current induced in the circuit, and what is its phase relationship with the membrane's displacement $y(t)$?

A rigid pipe of length $L$, closed at both ends, contains an ideal gas of adiabatic index $\gamma$. The pipe is accelerated uniformly with a constant acceleration $a$ along its longitudinal axis. The gas is maintained at a constant uniform temperature $T$ such that the local speed of sound in the unaccelerated gas is $c = \sqrt{\gamma R T / M}$. Assuming the gas is in equilibrium in the accelerating frame before any acoustic waves are excited, what is the fundamental frequency $f_1$ of the longitudinal acoustic standing waves in the pipe? (Assume the amplitude of oscillations is small and neglect any dissipative effects).

A metallic rod of length $L$, uniform density $\rho$, Young's modulus $Y$, and coefficient of linear thermal expansion $\alpha$ is rigidly clamped between two immovable walls at $x=0$ and $x=L$. The cross-sectional area of the rod varies linearly from $A_0$ at $x=0$ to $2A_0$ at $x=L$. Initially, the rod is in an unstrained state at temperature $\theta_0$. The temperature of the rod is then uniformly decreased by $\Delta\theta$. A small transverse wave pulse is generated at the end $x=0$. Assuming the dimensional changes due to thermal contraction are negligible for calculating the wave speed and density, what is the time $t$ taken by the pulse to reach the other end $x=L$?

A uniform thick string of mass $M$ and length $L$ is suspended from the ceiling of an elevator. A block of mass $3M$ is attached to the lower end of the string. The elevator is accelerating upwards with a constant acceleration $a = \frac{g}{3}$, where $g$ is the acceleration due to gravity. A small transverse wave pulse is generated at the lower end of the string. What is the time taken by the wave pulse to reach the ceiling of the elevator?