Welcome to Lecture 118 (Master Index Entry: L112) of our comprehensive JEE Physics Master Series! Transitioning into the dynamic study of distributed continuous media in Unit 10: Oscillations and Waves, we present the master foundational lecture on wave dynamics: Wave Motion — Introduction, Classification, and Governing Wave Parameters. While oscillations focus on the periodic localized vibration of a single particle, wave motion examines the collective, phase-coordinated propagation of mechanical disturbances through space. In this comprehensive guide, we dissect the microscopic mechanism of wave transport, contrast mechanical and electromagnetic waves, differentiate transverse and longitudinal modes, derive the fundamental wave equation, relate particle velocity to the spatial wave slope, establish rigorous relationships between phase difference and path difference, and master essential JEE Advanced problem patterns.
Detailed Concept Card: Fundamental Wave Mechanics & Classification
1. The Physical Mechanism of Wave Motion
A wave is a self-sustaining disturbance that propagates through a medium or empty space, transporting energy and linear momentum from one region to another without any bulk transport of matter.
- Dual Motion Character:
- Particle Motion: Individual particles of the material medium execute local oscillations (usually simple harmonic motion) about their respective fixed mean positions. No particle travels with the wave.
- Wave Profile Motion: The collective geometric state or phase pattern of the disturbance moves continuously forward with a characteristic propagation speed $v$.
- Necessary Medium Properties for Mechanical Waves:
- Elasticity: Displaced particles must experience linear restoring forces returning them toward equilibrium.
- Inertia (Mass Density): The medium must possess mass to store kinetic energy and permit overshoot past equilibrium.
- Low Internal Friction (Viscosity): Ensures the propagating disturbance does not dissipate rapidly into thermal energy.
2. Comprehensive Classification of Waves
| Classification Basis | Wave Category | Defining Physical Principle | Typical Examples |
|---|---|---|---|
| Material Medium Requirement | Mechanical Waves | Require an elastic, deformable material medium; governed by Newton’s laws. | Sound waves, seismic waves, vibrating strings, water ripples. |
| Electromagnetic Waves | Do not require a material medium; self-propagating oscillations of $\vec{E}$ and $\vec{B}$ fields travelling at $c \approx 3 \times 10^8\text{ m/s}$. | Light, X-rays, microwaves, radio waves. | |
| Particle Vibration Direction | Transverse Waves | Particle displacement is strictly perpendicular ($\perp$) to the propagation velocity ($\vec{v}$). Consists of crests and troughs. Requires shear elasticity. | Stretched string waves, seismic S-waves, light waves (EM). |
| Longitudinal Waves | Particle displacement is strictly parallel ($||$) to the propagation velocity ($\vec{v}$). Consists of compressions and rarefactions. Requires bulk modulus. | Sound in air/water, seismic P-waves, compression waves in springs. | |
| Energy Flow Character | Progressive (Travelling) Waves | Energy and momentum advance continuously in one direction through space. | Travelling waves on strings, outgoing sound pulses. |
| Standing (Stationary) Waves | Formed by superposition of identical counter-propagating waves; energy remains trapped between fixed nodes. | Organ pipes, plucked guitar strings, Kundt’s tube. |
3. Fundamental Kinematic Parameters
- Amplitude ($A$): The maximum displacement of any medium particle from its mean position (SI unit: $\text{m}$).
- Wavelength ($\lambda$): The spatial period of the wave — the minimum distance along the propagation direction between any two consecutive points oscillating in identical phase (SI unit: $\text{m}$).
- Time Period ($T$): The temporal period — the time required by any single particle to complete one full oscillation (SI unit: $\text{s}$).
- Frequency ($f$ or $\nu$): The number of complete cycles executed per second ($f = 1/T$, SI unit: $\text{Hz}$ or $\text{s}^{-1}$).
Crucial Rule: Frequency is determined strictly by the vibrating source and remains invariant when the wave crosses an interface into another medium! - Angular Frequency ($\omega$): $\omega = 2\pi f = \frac{2\pi}{T}$ (SI unit: $\text{rad/s}$).
- Propagation Constant / Angular Wave Number ($k$): The spatial frequency of the wave: $$\mathbf{k = \frac{2\pi}{\lambda}} \quad (\text{SI unit: rad/m})$$
- Wave Speed ($v$): The rate at which the wave phase advances through space: $$\mathbf{v = \frac{\lambda}{T} = f \lambda = \frac{\omega}{k}}$$ Crucial Insight: Wave speed $v$ is an intrinsic property of the medium (determined by elasticity and inertia, e.g., $v = \sqrt{T/\mu}$ on a string), completely independent of source frequency $f$ or amplitude $A$.
4. Particle Velocity ($v_p$) vs Wave Velocity ($v$)
For a general one-dimensional progressive wave travelling along the $+x$-direction, the displacement is expressed as $y(x, t) = f(x – vt)$:
- Particle Velocity ($v_p$): The instantaneous velocity of a specific medium element located at coordinate $x$: $$v_p = \frac{\partial y}{\partial t}$$
- Spatial Slope: The spatial derivative representing the instantaneous gradient of the wave profile: $$\text{Slope} = \frac{\partial y}{\partial x}$$
- The Fundamental Link:
$$v_p = \frac{\partial y}{\partial t} = f'(x – vt) \cdot (-v) = -v \left[f'(x – vt)\right] = \mathbf{-v \left(\frac{\partial y}{\partial x}\right)}$$
$$\mathbf{v_p = -v \times (\text{Slope of the wave profile})}$$
- If slope is negative ($\frac{\partial y}{\partial x} < 0$), particle moves upwards ($v_p > 0$).
- If slope is positive ($\frac{\partial y}{\partial x} > 0$), particle moves downwards ($v_p < 0$).
- Particle Acceleration ($a_p$): $$a_p = \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}$$
5. The Classical Linear Differential Wave Equation
Equating the second partial derivatives yields the master governing equation of all non-dispersive classical waves:
$$\mathbf{\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}}$$
Any physical function $y(x, t) = f(kx \mp \omega t)$ that is finite, single-valued, and twice-differentiable represents a physically realizable travelling wave.
6. Phase, Path Difference & Time Difference Relations
For a harmonic wave $y(x, t) = A \sin(\omega t – kx + \phi_0)$, the total phase is $\Phi = \omega t – kx + \phi_0$:
| Comparison Scenario | Phase Difference ($\Delta\Phi$) | Governing Formula |
|---|---|---|
| Between two points separated by path difference $\Delta x$ at the same instant | $$\Delta\Phi = k \Delta x$$ | $$\mathbf{\Delta\Phi = \frac{2\pi}{\lambda} \Delta x}$$ |
| For a single particle over time interval $\Delta t$ at a fixed location | $$\Delta\Phi = \omega \Delta t$$ | $$\mathbf{\Delta\Phi = \frac{2\pi}{T} \Delta t = 2\pi f \Delta t}$$ |
Four Solved Examples (Graded Basic to Advanced)
Example 1: Basic Application — Extracting Wave and Particle Kinematics
Problem Statement: A transverse harmonic wave propagating along a stretched horizontal string is described by: $$y(x, t) = 0.05 \sin(4\pi t – 2\pi x)$$ where $x$ and $y$ are in meters and $t$ is in seconds.
(a) Identify the amplitude $A$, angular frequency $\omega$, wave number $k$, frequency $f$, wavelength $\lambda$, and wave speed $v$.
(b) Determine the direction of wave propagation.
(c) Calculate the maximum transverse speed $v_{p,\text{max}}$ and maximum transverse acceleration $a_{p,\text{max}}$ of any particle on the string.
(d) Find the transverse displacement and velocity of the particle at $x = 0.25\text{ m}$ at time $t = 0.50\text{ s}$.
Solution:
(a) Wave Parameters:
Comparing with the canonical form $y(x, t) = A \sin(\omega t – kx)$:
$$\mathbf{A = 0.05\text{ m} = 5.0\text{ cm}}$$
$$\mathbf{\omega = 4\pi\text{ rad/s}}$$
$$\mathbf{k = 2\pi\text{ rad/m}}$$
$$f = \frac{\omega}{2\pi} = \frac{4\pi}{2\pi} = \mathbf{2.0\text{ Hz}}$$
$$\lambda = \frac{2\pi}{k} = \frac{2\pi}{2\pi} = \mathbf{1.0\text{ m}}$$
$$v = \frac{\omega}{k} = \frac{4\pi}{2\pi} = \mathbf{2.0\text{ m/s}}$$
(b) Propagation Direction:
The argument has the form $(\omega t – kx)$. As $t$ increases, a constant phase requires $x$ to increase. Hence, the wave propagates along the positive $x$-direction ($+x$).
(c) Maximum Particle Kinematics:
$$v_{p,\text{max}} = \omega A = (4\pi\text{ rad/s})(0.05\text{ m}) = \mathbf{0.20\pi\text{ m/s} \approx 0.628\text{ m/s}}$$
$$a_{p,\text{max}} = \omega^2 A = (4\pi)^2 (0.05) = 16\pi^2 (0.05) = \mathbf{0.80\pi^2\text{ m/s}^2 \approx 7.90\text{ m/s}^2}$$
(d) Particle State at $x = 0.25\text{ m}, t = 0.50\text{ s}$:
Phase: $\Phi = 4\pi(0.50) – 2\pi(0.25) = 2\pi – 0.5\pi = 1.5\pi = \frac{3\pi}{2}\text{ rad}$.
$$y = 0.05 \sin\left(\frac{3\pi}{2}\right) = 0.05(-1) = \mathbf{-0.05\text{ m} = -5.0\text{ cm}}$$
$$v_p = \frac{\partial y}{\partial t} = 0.05(4\pi) \cos\left(\frac{3\pi}{2}\right) = 0.20\pi (0) = \mathbf{0\text{ m/s}}$$
The particle is instantaneously at rest at its negative extreme position.
Example 2: Intermediate Application — Particle Velocity via Wave Slope & Waveform Inversion
Problem Statement: A bell-shaped wave pulse propagates along the $+x$-direction with speed $v = 10\text{ m/s}$. At time $t = 0$, its spatial profile is given by: $$y(x, 0) = \frac{0.10}{1 + x^2}$$ where $x$ and $y$ are in meters.
(a) Write the mathematical function describing the wave profile $y(x, t)$ at any general time $t$.
(b) Calculate the spatial slope $\frac{\partial y}{\partial x}$ of the string at position $x = 1.0\text{ m}$ at $t = 0$.
(c) Using the fundamental relationship $v_p = -v \left(\frac{\partial y}{\partial x}\right)$, calculate the instantaneous transverse velocity of the particle at $x = 1.0\text{ m}$ at $t = 0$.
Solution:
(a) General Wave Function:
Since the pulse propagates without distortion in the $+x$-direction with speed $v = 10\text{ m/s}$, we replace $x$ by $(x – vt) = (x – 10t)$:
$$\mathbf{y(x, t) = \frac{0.10}{1 + (x – 10t)^2}\text{ m}}$$
(b) Spatial Slope at $x = 1.0\text{ m}, t = 0$:
Differentiating $y(x, 0)$ with respect to $x$:
$$\frac{\partial y}{\partial x} = \frac{d}{dx}\left[0.10(1 + x^2)^{-1}\right] = 0.10(-1)(1 + x^2)^{-2}(2x) = -\frac{0.20 x}{(1 + x^2)^2}$$
Evaluating at $x = 1.0\text{ m}$:
$$\mathbf{\left.\frac{\partial y}{\partial x}\right|_{x=1.0} = -\frac{0.20(1.0)}{(1 + 1^2)^2} = -\frac{0.20}{4} = -0.050}$$
(c) Transverse Particle Velocity:
Applying the slope-velocity relation:
$$v_p = -v \left(\frac{\partial y}{\partial x}\right) = -(10\text{ m/s})(-0.050) = \mathbf{+0.50\text{ m/s}}$$
The particle at $x = 1.0\text{ m}$ is moving upwards in the $+y$ direction at $0.50\text{ m/s}$.
Direct Verification:
$$\frac{\partial y}{\partial t} = 0.10(-1)[1 + (x – 10t)^2]^{-2} \cdot [-20(x – 10t)] = \frac{2.0(x – 10t)}{[1 + (x – 10t)^2]^2}$$
At $x = 1.0, t = 0$: $v_p = \frac{2.0(1.0)}{(1 + 1)^2} = \frac{2.0}{4} = +0.50\text{ m/s}$. The two methods match identically.
Example 3: Advanced Application — Phase, Path, and Temporal Delays in Sound Waves
Problem Statement: A loudspeaker emits a pure harmonic sound wave into air at frequency $f = 500\text{ Hz}$. The speed of sound in the room is $v = 340\text{ m/s}$. Two listening microphones $A$ and $B$ are positioned along the propagation line at distances $x_A$ and $x_B$ such that their separation distance is $\Delta x = 0.85\text{ m}$.
(a) Determine the wavelength $\lambda$ of the sound wave.
(b) Calculate the phase difference $\Delta\Phi$ between the signals recorded at $A$ and $B$ at the same instant.
(c) If the particle at microphone $A$ reaches its positive maximum displacement at $t = 0$, find the first subsequent time instant at which the particle at microphone $B$ reaches its positive maximum displacement.
(d) Find the phase change experienced by the air particle at $A$ during a time interval of $\Delta t = 1.5\text{ ms}$.
Solution:
(a) Wavelength:
$$\mathbf{\lambda = \frac{v}{f} = \frac{340\text{ m/s}}{500\text{ Hz}} = 0.68\text{ m} = 68\text{ cm}}$$
(b) Spatial Phase Difference:
$$\Delta\Phi = \frac{2\pi}{\lambda} \Delta x = \frac{2\pi}{0.68\text{ m}}(0.85\text{ m}) = 2\pi \left(\frac{85}{68}\right) = 2\pi \left(\frac{5}{4}\right) = \mathbf{\frac{5\pi}{2}\text{ rad} = 2.5\pi\text{ rad} = 450^\circ}$$
Expressed as an effective principal angle within $[0, 2\pi)$:
$$\mathbf{\Delta\Phi_{\text{eff}} = \frac{5\pi}{2} – 2\pi = \frac{\pi}{2}\text{ rad} \ (90^\circ)}$$
(c) Transit Delay to Maximum Displacement:
Because the wave disturbance travels from $A$ to $B$ at speed $v = 340\text{ m/s}$, the time delay is:
$$\mathbf{\Delta t_{\text{delay}} = \frac{\Delta x}{v} = \frac{0.85\text{ m}}{340\text{ m/s}} = \frac{1}{400}\text{ s} = 2.5\text{ ms} = 2.5 \times 10^{-3}\text{ s}}$$
(d) Temporal Phase Change:
$$\mathbf{\Delta\Phi_t = \omega \Delta t = (2\pi f) \Delta t = (2\pi \times 500)(1.5 \times 10^{-3}\text{ s}) = 1.5\pi\text{ rad} = \frac{3\pi}{2}\text{ rad} = 270^\circ}$$
Example 4: JEE Advanced Scenario — Rigorous Verification of the Wave Equation for a Gaussian Pulse
Problem Statement: Consider a localized Gaussian pulse propagating along the $x$-axis described by: $$y(x, t) = A \exp\left[-\left(\frac{x – vt}{b}\right)^2\right]$$ where $A$ is the amplitude, $v$ is the pulse velocity, and $b$ is a characteristic width parameter.
(a) Prove rigorously that $y(x, t)$ satisfies the one-dimensional classical wave equation $\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$.
(b) Identify the trajectory of the peak of the pulse in the $x-t$ plane.
(c) Given $A = 0.040\text{ m}, b = 0.20\text{ m},$ and $v = 5.0\text{ m/s}$, calculate the transverse velocity of the particle at coordinate $x = 0.20\text{ m}$ at time $t = 0$.
Solution:
(a) Wave Equation Proof:
Define the dimensionless coordinate $u = \frac{x – vt}{b}$. Then $y = A e^{-u^2}$.
Using the chain rule: $\frac{\partial u}{\partial x} = \frac{1}{b}$ and $\frac{\partial u}{\partial t} = -\frac{v}{b}$.
First spatial derivative:
$$\frac{\partial y}{\partial x} = \frac{dy}{du}\frac{\partial u}{\partial x} = A(-2u e^{-u^2})\left(\frac{1}{b}\right) = -\frac{2Au}{b} e^{-u^2}$$
Second spatial derivative:
$$\frac{\partial^2 y}{\partial x^2} = \frac{\partial}{\partial x}\left[\frac{\partial y}{\partial x}\right] = \frac{1}{b}\frac{d}{du}\left[-\frac{2Au}{b} e^{-u^2}\right] = -\frac{2A}{b^2} \left[e^{-u^2} – 2u^2 e^{-u^2}\right] = \mathbf{\frac{2A}{b^2} (2u^2 – 1) e^{-u^2}}$$
First temporal derivative:
$$\frac{\partial y}{\partial t} = \frac{dy}{du}\frac{\partial u}{\partial t} = A(-2u e^{-u^2})\left(-\frac{v}{b}\right) = \frac{2Avu}{b} e^{-u^2}$$
Second temporal derivative:
$$\frac{\partial^2 y}{\partial t^2} = \left(-\frac{v}{b}\right)\frac{d}{du}\left[\frac{2Avu}{b} e^{-u^2}\right] = \left(-\frac{v}{b}\right)\left(\frac{2Av}{b}\right)\left[e^{-u^2} – 2u^2 e^{-u^2}\right] = \mathbf{\frac{2Av^2}{b^2} (2u^2 – 1) e^{-u^2}}$$
Comparing the two results:
$$\frac{1}{v^2} \frac{\partial^2 y}{\partial t^2} = \frac{1}{v^2} \left[\frac{2Av^2}{b^2} (2u^2 – 1) e^{-u^2}\right] = \frac{2A}{b^2} (2u^2 – 1) e^{-u^2} = \mathbf{\frac{\partial^2 y}{\partial x^2}}$$
Hence, the Gaussian pulse strictly satisfies the classical wave equation.
(b) Peak Trajectory:
The function $e^{-u^2}$ attains its maximum value of $1$ when $u = 0$:
$$u = 0 \implies x – vt = 0 \implies \mathbf{x = vt}$$
The peak moves along a straight line in the $x-t$ plane with slope $dx/dt = v$.
(c) Transverse Velocity at $x = 0.20\text{ m}, t = 0$:
At $t = 0$ and $x = 0.20\text{ m}$:
$$u = \frac{0.20 – 0}{0.20} = 1.0$$
Using the first temporal derivative expression:
$$v_p = \frac{\partial y}{\partial t} = \frac{2Avu}{b} e^{-u^2} = \frac{2(0.040\text{ m})(5.0\text{ m/s})(1.0)}{0.20\text{ m}} e^{-(1.0)^2} = \frac{0.40}{0.20} e^{-1} = \mathbf{2.0 e^{-1}\text{ m/s} = \frac{2}{e}\text{ m/s} \approx 0.736\text{ m/s}}$$
Practice Worksheet: 10 Target JEE Problems
Solidify your understanding of wave definitions, propagation constraints, and particle kinematics. Attempt all 10 questions before checking the answer key and derivations below.
Section I: Single Correct Choice Questions (Q1–Q5)
Q1. Which of the following physical quantities is/are transported by a travelling progressive mechanical wave across a medium?
(A) Matter only
(B) Energy and linear momentum, but no net matter
(C) Energy only, without linear momentum
(D) Both matter and energy simultaneously
Q2. Transverse mechanical waves cannot propagate through which of the following media?
(A) A solid steel girder
(B) A taut guitar string
(C) An ideal gas (air)
(D) A quartz crystal bar
Q3. A progressive harmonic wave of frequency $f = 250\text{ Hz}$ propagates with a phase speed $v = 350\text{ m/s}$. The phase difference between two points separated by a distance $\Delta x = 0.70\text{ m}$ along the line of propagation is:
(A) $\pi\text{ rad}$
(B) $\pi/2\text{ rad}$
(C) $2\pi\text{ rad}$
(D) $\pi/4\text{ rad}$
Q4. The displacement equation of a travelling transverse wave is given by $y = A \sin(\omega t – kx)$. The ratio of the maximum particle speed to the wave propagation speed is:
(A) $\omega A$
(B) $\lambda / A$
(C) $A / \lambda$
(D) $k A$
Q5. When a sound wave travels from air into water, which of the following wave parameters remains strictly unchanged?
(A) Wave propagation speed
(B) Wavelength
(C) Frequency
(D) Wave amplitude
Section II: One or More Than One Correct Questions (Q6–Q8)
Q6. A transverse progressive harmonic wave is represented by $y(x, t) = A \sin(kx – \omega t)$. Which of the following statements is/are correct?
(A) The wave propagates along the positive $x$-direction with speed $v = \omega/k$.
(B) The particle velocity is given by $v_p = -\omega A \cos(kx – \omega t)$.
(C) The particle velocity is related to the spatial slope of the wave profile by $v_p = -v \left(\frac{\partial y}{\partial x}\right)$.
(D) Any two particles separated by an integer multiple of wavelength ($n\lambda$) oscillate in identical phase.
Q7. Which of the following mathematical functions can represent a physically acceptable travelling wave along the $x$-axis?
(A) $y = A \sin(kx – \omega t)$
(B) $y = \frac{A}{(x + vt)^2 + b^2}$
(C) $y = A \ln(x – vt)$
(D) $y = A \exp\left[-k(x – vt)^2\right]$
Q8. Which of the following statements is/are correct regarding longitudinal progressive waves?
(A) Medium particles oscillate along lines parallel to the direction of wave velocity.
(B) Longitudinal waves can propagate through solids, liquids, and gases.
(C) The wave consists of alternating regions of compression (elevated pressure and density) and rarefaction (reduced pressure and density).
(D) The pressure variation wave leads the particle displacement wave by a phase of $\pi/2$ radians.
Section III: Numerical / Integer Type Questions (Q9–Q10)
Q9. A transverse harmonic wave propagating along a wire is described by the equation $y = 0.020 \sin(50 t – 2.0 x)$, where $x$ and $y$ are in meters and $t$ is in seconds. Determine the propagation speed of the wave in $\text{m/s}$.
Q10. A rhythmic plunger generates circular ripples on the surface of a pond at a steady frequency of $f = 20\text{ Hz}$. The distance measured between two consecutive crests is observed to be $15\text{ cm}$. Calculate the distance (in meters) travelled by the leading edge of the ripple pattern in a time duration of $2.0\text{ seconds}$.
Comprehensive Solutions & Answer Key
Answer Key
| Question No. | Correct Answer | Key Concept / Formula Tested |
|---|---|---|
| Q1 | (B) | Waves transport energy and momentum without bulk matter transport |
| Q2 | (C) | Transverse waves require shear elasticity, which gases lack |
| Q3 | (A) | $\lambda = v/f = 1.4\text{ m} \implies \Delta\Phi = (2\pi/1.4)(0.70) = \pi\text{ rad}$ |
| Q4 | (D) | $v_{p,\text{max}} = \omega A, v = \omega/k \implies v_{p,\text{max}}/v = kA$ |
| Q5 | (C) | Frequency is an invariant characteristic of the source |
| Q6 | (A, B, C, D) | Wave kinematic relations: phase speed, slope relation, periodic coherence |
| Q7 | (A, B, D) | Valid waves must remain finite everywhere; $\ln(x-vt)$ diverges at $x=vt$ |
| Q8 | (A, B, C, D) | Longitudinal wave properties: compression/rarefaction, phase lead of pressure |
| Q9 | 25 | $v = \omega/k = 50 / 2.0 = 25\text{ m/s}$ |
| Q10 | 6 | $v = f\lambda = 20(0.15) = 3.0\text{ m/s} \implies d = vt = 3.0(2.0) = 6\text{ m}$ |
Detailed Step-by-Step Solutions
Solution to Q1:
A wave is an organized propagation of a physical disturbance. As the wave travels, particles execute oscillatory motion about their local equilibrium positions without undergoing net translational displacement. However, the coordinated phase lag allows energy and linear momentum to be continuously transported through space.
Hence, the correct choice is (B).
Solution to Q2:
Propagation of transverse waves requires shear elasticity (a restoring force opposing shape deformation without volume change). Solids and tensioned strings possess shear modulus and support transverse waves.
Fluids (liquids and gases) have zero shear modulus ($\eta = 0$) and cannot sustain transverse mechanical waves in their bulk interior. Gases support only longitudinal compressional waves.
Hence, the correct choice is (C).
Solution to Q3:
The wavelength of the wave is:
$$\lambda = \frac{v}{f} = \frac{350\text{ m/s}}{250\text{ Hz}} = 1.4\text{ m}$$
The phase difference $\Delta\Phi$ corresponding to path difference $\Delta x = 0.70\text{ m}$ is:
$$\mathbf{\Delta\Phi = \frac{2\pi}{\lambda} \Delta x = \frac{2\pi}{1.4\text{ m}} (0.70\text{ m}) = \frac{2\pi}{2} = \pi\text{ rad}}$$
Hence, the correct choice is (A).
Solution to Q4:
Given the displacement wave $y = A \sin(\omega t – kx)$:
The particle velocity is:
$$v_p = \frac{\partial y}{\partial t} = \omega A \cos(\omega t – kx) \implies v_{p,\text{max}} = \omega A$$
The wave propagation speed is:
$$v = \frac{\omega}{k}$$
The ratio is:
$$\mathbf{\frac{v_{p,\text{max}}}{v} = \frac{\omega A}{\omega / k} = k A}$$
Hence, the correct choice is (D).
Solution to Q5:
When a wave crosses the boundary between two media:
– The frequency $f$ is determined solely by the periodic driver (source) and remains strictly unchanged.
– The wave speed $v$ changes due to differing medium properties ($v = \sqrt{B/\rho}$).
– The wavelength adjusts according to $\lambda = v/f$.
– The amplitude changes due to partial reflection and transmission.
Hence, the correct choice is (C).
Solution to Q6:
Let us evaluate each statement:
(A) The phase is $kx – \omega t$. Setting $kx – \omega t = \text{const} \implies \frac{dx}{dt} = +\frac{\omega}{k} > 0$. The wave moves along $+x$. (A) is correct.
(B) $v_p = \frac{\partial y}{\partial t} = \frac{\partial}{\partial t}[A\sin(kx – \omega t)] = -\omega A \cos(kx – \omega t)$. (B) is correct.
(C) $\frac{\partial y}{\partial x} = k A \cos(kx – \omega t)$. Thus $-v \left(\frac{\partial y}{\partial x}\right) = -\left(\frac{\omega}{k}\right) [kA \cos(kx – \omega t)] = -\omega A \cos(kx – \omega t) = v_p$. (C) is correct.
(D) $\Delta\Phi = \frac{2\pi}{\lambda}(n\lambda) = 2n\pi$, which represents identical oscillatory phase. (D) is correct.
Hence, all choices (A, B, C, D) are correct.
Solution to Q7:
A mathematical function represents a physically acceptable wave if it satisfies the wave equation $\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 y}{\partial t^2}$ and remains finite (bounded) for all values of $x$ and $t$:
(A) $A\sin(kx – \omega t)$ is bounded within $[-A, +A]$ and satisfies the wave equation. (A) is valid.
(B) $y = \frac{A}{(x+vt)^2 + b^2}$ is bounded with maximum value $A/b^2$ at $x = -vt$, and satisfies the wave equation. (B) is valid.
(C) $y = A\ln(x – vt)$ approaches $-\infty$ as $x \to vt$ and is undefined for $x < vt$. Thus it is not a physically acceptable wave. (C is invalid).
(D) $y = A e^{-k(x – vt)^2}$ is bounded within $(0, A]$ and satisfies the wave equation. (D) is valid.
Hence, correct choices are (A, B, D).
Solution to Q8:
(A) In longitudinal waves, particle displacements are parallel to the direction of propagation. (A) is correct.
(B) All states of matter possess a non-zero bulk modulus, enabling longitudinal waves to propagate through solids, liquids, and gases. (B) is correct.
(C) Compressions correspond to volume reduction (high density/pressure), while rarefactions correspond to volume expansion (low density/pressure). (C) is correct.
(D) The excess pressure is $\Delta P = -B \frac{\partial s}{\partial x}$. For $s = s_0 \cos(kx – \omega t)$, $\Delta P = B k s_0 \sin(kx – \omega t) = \Delta P_0 \cos(kx – \omega t – \pi/2)$. Pressure and displacement are out of phase by $\pi/2$. (D) is correct.
Hence, all choices (A, B, C, D) are correct.
Solution to Q9:
Comparing $y = 0.020 \sin(50 t – 2.0 x)$ with $y = A \sin(\omega t – kx)$:
$$\omega = 50\text{ rad/s}, \quad k = 2.0\text{ rad/m}$$
The wave propagation speed is:
$$\mathbf{v = \frac{\omega}{k} = \frac{50\text{ rad/s}}{2.0\text{ rad/m}} = 25\text{ m/s}}$$
Answer: 25
Solution to Q10:
The distance between two consecutive crests equals the wavelength:
$$\lambda = 15\text{ cm} = 0.15\text{ m}$$
Given frequency $f = 20\text{ Hz}$, the speed of the water waves is:
$$v = f \lambda = (20\text{ s}^{-1})(0.15\text{ m}) = 3.0\text{ m/s}$$
The distance travelled by the wave in time $t = 2.0\text{ s}$ is:
$$\mathbf{d = v \times t = (3.0\text{ m/s})(2.0\text{ s}) = 6.0\text{ m} = 6\text{ m}}$$
Answer: 6