1.
Differentiate between transverse and longitudinal waves based on particle motion and the direction of wave propagation. Provide one example of each.
2.
The equation of a transverse wave is given by y = 5 sin(2π(t/0.04 - x/40)), where x and y are in cm and t is in seconds. Calculate the velocity of the wave.
3.
Solids generally possess a higher mass density than gases. Despite this, explain conceptually why the speed of sound is significantly greater in solids than in gases.
4.
What will be the phase difference between two particles vibrating in a homogeneous medium if they are separated by a spatial distance of λ / 4?
5.
State the Principle of Superposition of waves. What happens to the phase of a transverse wave when it suffers reflection from a perfectly rigid boundary?
6.
Discuss Newton's formula for the velocity of sound in air. Why was there a discrepancy with the experimental value, and how did Laplace correct it?
7.
A uniform steel wire 0.72 m long has a mass of 5 × 10-3 kg. If the wire is kept under a tension of 60 N, calculate the speed of transverse waves propagating on the wire.
8.
Two harmonic waves travelling in the same medium are represented by y1 = a sin(ωt - kx) and y2 = a cos(ωt - kx). Determine the phase difference between them and find the amplitude of the resultant wave obtained upon their superposition.
9.
Distinguish between a progressive wave and a stationary (standing) wave. Provide at least three distinct points of difference regarding energy transfer, amplitude, and phase of particles.
10.
A wave travelling along a string is described by the equation y(x, t) = 0.005 sin(80.0 x - 3.0 t), where all numerical constants are in standard SI units. Calculate the wavelength, frequency, and time period of this wave.
11.
(a) Explain the analytical formation of standing waves in a string fixed at both ends using the principle of superposition.
(b) Derive the condition for the formation of nodes and antinodes.
(c) Show mathematically that for a string fixed at both ends, the frequencies of the fundamental mode and subsequent harmonics are in the ratio 1 : 2 : 3 : ...
12.
(a) Derive the general expression for a one-dimensional simple harmonic progressive wave travelling in the positive x-direction.
(b) What is the effect of pressure and temperature on the speed of sound in a gas? Prove mathematically that at a constant temperature, the speed of sound in a gas is independent of variations in pressure.
13.
(a) A wave equation is given by y = 10-4 sin(60t + 2x) where x and y are in metres and t is in seconds. In which direction is the wave travelling? Calculate its wave velocity.
(b) [JEE Application] A uniform heavy string of length L and total mass M hangs freely vertically from a rigid ceiling. A small transverse wave pulse is generated at its lower end. Using the concept of variable tension, derive an expression for the total time taken by the wave pulse to travel from the bottom end to the top of the string.