Review and JEE Preparation Quiz

Review and JEE Preparation Quiz — Sound Waves

This closing quiz consolidates the entire sound-waves course into an exam-ready revision architecture. It tests the formula dependencies, sign conventions, and error patterns that decide marks in IIT JEE.

Wave kinematics: ##y(x,t) = A\sin(kx – \omega t)##, ##v = f\lambda = \omega/k##, and the Newton–Laplace speed ##v = \sqrt{B/\rho}##.
Superposition, standing waves, beats, and the Doppler effect with disciplined sign handling.
Intensity and decibels using ##I_0 = 10^{-12}\ \text{W/m}^2## and ##\beta = 10\log_{10}(I/I_0)##.
Applications: ultrasonics, shock waves, acoustic impedance, underwater and medical acoustics, metamaterials, and psychoacoustics.

Work through each item as if under timed examination conditions, then verify your reasoning against the explanation. Standard JEE conventions apply: speed of sound in air at room temperature is approximately ##343\ \text{m/s}##.

Q1. For a travelling sound wave described by ##y(x,t) = A\sin(kx - \omega t)##, which relation correctly links phase velocity, frequency, and wavelength?

##v = f\lambda = \dfrac{\omega}{k}##

##v = \dfrac{f}{\lambda} = \omega k##

##v = f\lambda = \dfrac{k}{\omega}##

##v = \dfrac{\lambda}{f} = \dfrac{\omega}{k}##

Phase velocity is ##v = f\lambda##, and since ##k = 2\pi/\lambda## and ##\omega = 2\pi f##, the ratio ##\omega/k## equals ##f\lambda##. The other options invert or multiply the quantities incorrectly.

Q2. The Newton–Laplace relation for the speed of sound in a fluid is:

##v = \sqrt{\dfrac{\rho}{B}}##

##v = \sqrt{\dfrac{B}{\rho}}##

##v = \dfrac{B}{\rho}##

##v = \sqrt{B\rho}##

The speed of sound in a fluid is governed by bulk modulus and density as ##v = \sqrt{B/\rho}##. This explains why sound travels roughly four times faster in water than in air at comparable temperatures.

Q3. Why is the adiabatic assumption essential when deriving the speed of sound in a gas?

Because compressions and rarefactions occur too rapidly for heat exchange

Because gases cannot conduct heat at all

Because the pressure is always constant during propagation

Because the density of a gas never changes

Compressions and rarefactions occur too rapidly for heat exchange, so the process is adiabatic. Using ##B = \gamma P## gives the Laplace correction, raising the predicted speed from Newton's isothermal value of about ##280\ \text{m/s}## to the observed ##331\ \text{m/s}## at ##0^\circ\text{C}##.

Q4. Two coherent sound waves of equal amplitude and intensity ##I## superpose at a point with a phase difference of ##\pi## radians. The resultant intensity is:

##4I##

##2I##

##I##

Zero

At a phase difference of ##\pi## the waves are completely out of phase, giving fully destructive interference. Equal amplitudes cancel exactly, so the resultant intensity is zero.

Q5. In a pipe closed at one end, the fundamental resonant frequency is ##f_1##. Which set correctly describes the allowed harmonics?

All integer multiples: ##f_1, 2f_1, 3f_1, \ldots##

Only odd multiples: ##f_1, 3f_1, 5f_1, \ldots##

Only even multiples: ##2f_1, 4f_1, 6f_1, \ldots##

Only ##f_1## and ##2f_1##

A closed pipe has a node at the closed end and an antinode at the open end, so only odd harmonics are supported: ##f_1, 3f_1, 5f_1, \ldots##. An open pipe supports all integer multiples.

Q6. Two tuning forks of frequencies ##256\ \text{Hz}## and ##260\ \text{Hz}## are sounded together. The beat frequency heard is:

##2\ \text{Hz}##

##4\ \text{Hz}##

##8\ \text{Hz}##

##516\ \text{Hz}##

The beat frequency equals the difference of the two frequencies: ##260 - 256 = 4\ \text{Hz}##. Beats arise from the periodic reinforcement and cancellation of two nearly equal frequencies.

Q7. A source of sound moves towards a stationary observer. According to the Doppler formula, the observed frequency:

Decreases because the wavelength increases

Increases because the effective wavelength decreases

Remains unchanged because the medium is at rest

Increases because the speed of sound increases

When the source approaches, successive wavefronts are emitted from progressively closer positions, so the effective wavelength shortens and the observed frequency rises. The speed of sound in the medium is unchanged.

Q8. The sound intensity level in decibels is defined as:

##\beta = 10\log_{10}\left(\dfrac{I_0}{I}\right)##

##\beta = 10\log_{10}\left(\dfrac{I}{I_0}\right)##

##\beta = \log_{10}\left(\dfrac{I}{I_0}\right)##

##\beta = 20\log_{10}\left(\dfrac{I}{I_0}\right)##

The intensity level is ##\beta = 10\log_{10}(I/I_0)## with the reference intensity ##I_0 = 10^{-12}\ \text{W/m}^2##. The factor 10 converts bel to decibel, and the ratio must be intensity over reference.

Q9. When a sound wave travelling in air strikes a rigid wall, the reflected wave undergoes:

No phase change in displacement

A phase change of ##\pi## in displacement

A phase change of ##\pi/2## in displacement

Complete absorption with no reflection

Reflection at a rigid boundary produces a displacement node at the wall, which corresponds to a phase change of ##\pi## in the displacement wave. Pressure, by contrast, is reflected without phase change.

Q10. Acoustic impedance of a medium is defined as the product of:

Density and bulk modulus

Density and wave speed

Pressure and wavelength

Frequency and wavelength

Acoustic impedance is ##Z = \rho v##, the product of density and wave speed. Mismatch in acoustic impedance at an interface governs how much sound is reflected and how much is transmitted.

Q11. Which statement about ultrasonic waves is correct in the context of JEE applications?

They have frequencies below the audible range and are used in noise control

They have frequencies above the audible range and are used in medical imaging and sonar

They cannot propagate through liquids

They travel slower than audible sound in the same medium

Ultrasonics refers to frequencies above the audible range (above roughly ##20\ \text{kHz}##). They are widely used in medical imaging, sonar, and non-destructive testing, and they propagate through liquids.

Q12. A shock wave is best described as:

A wave of very low amplitude travelling slowly

A steep pressure discontinuity travelling faster than the local speed of sound

A standing wave formed in a closed pipe

A wave that exists only in solids

A shock wave is a steep pressure discontinuity that travels faster than the local speed of sound in the medium. It is associated with supersonic motion and nonlinear steepening of the waveform.

Q13. Which of the following is the most effective revision strategy for the final weeks before the JEE examination?

Passively rereading notes repeatedly

Reconstructing derivations from first principles and attempting timed mixed-topic problem sets

Memorising only the final answers of solved examples

Skipping verification to save time

The lesson emphasises that structured revision succeeds: candidates who reconstruct derivations and test themselves against timed, mixed-topic problem sets retain far more than those who reread notes passively.

Q14. A common mistake in Doppler-effect problems is:

Using the correct sign convention for source and observer motion

Ignoring the sign convention for the direction of source and observer motion

Converting all units to SI before substitution

Checking that the medium is at rest

Precise sign conventions are essential in Doppler problems. Ignoring the direction of source and observer motion is a frequent error that costs marks, along with careless unit handling.

Q15. In psychoacoustics, the perceived loudness of a sound depends primarily on:

Only the physical intensity of the sound

The frequency as well as the intensity of the sound

Only the wavelength of the sound

Only the acoustic impedance of the medium

Psychoacoustics recognises that perceived loudness depends on frequency as well as intensity, because the ear's sensitivity varies with frequency. This is why equal-intensity tones at different frequencies are not heard as equally loud.