Superposition Principle and Interference Quiz

Sound waves rarely travel alone. In any real acoustic environment, multiple pressure disturbances overlap continuously, and the resulting field is a single, physically measurable pressure history. The superposition principle formalises this: at any point, the instantaneous displacement or pressure equals the algebraic sum of the contributions from each individual wave.

This quiz tests Lesson 3 of the IIT JEE sound-waves series. You will be examined on:

The superposition statement and its basis in the linearity of the acoustic wave equation
Constructive and destructive interference conditions
Path difference and phase difference relations
Coherent sources and sustained interference
Resultant amplitude and intensity, and energy conservation
Beats as interference in time, and interference versus diffraction

Work carefully with the relations ##\phi = \dfrac{2\pi}{\lambda}\Delta x## and ##I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi##. Good luck!

Q1. The superposition principle states that at any point in an acoustic field, the resultant displacement or pressure is:

The vector product of the individual wave contributions

The algebraic sum of the contributions from each individual wave

The average of the individual wave contributions

Always zero when two waves meet

Superposition requires the algebraic (signed) sum of the individual pressure or displacement contributions at that point, preserving each wave's amplitude, phase and frequency.

Q2. Superposition in acoustics is valid because the acoustic wave equation is:

Linear in the pressure perturbation for small-amplitude disturbances

Nonlinear for all amplitudes

Valid only for transverse waves

Independent of the medium density

For small-amplitude disturbances the density variations are a tiny fraction of ambient density, nonlinear convective terms are negligible, and the wave equation is linear in the pressure perturbation.

Q3. Two coherent sound waves of equal amplitude ##p_0## arrive at a point with a phase difference of ##\pi## radians. The resultant pressure amplitude is:

##2p_0##

##p_0##

##\sqrt{2}\,p_0##

Zero

A phase difference of ##\pi## corresponds to destructive interference, so the resultant amplitude is ##\sqrt{p_0^2 + p_0^2 + 2p_0^2\cos\pi} = 0##.

Q4. For two waves of the same frequency, constructive interference occurs when the path difference is:

An odd multiple of ##\dfrac{\lambda}{2}##

An integral multiple of ##\lambda##

An odd multiple of ##\dfrac{\lambda}{4}##

Always zero

Constructive interference requires a phase difference of an even multiple of ##\pi##, which corresponds to a path difference of ##n\lambda##.

Q5. The relation between phase difference ##\phi## and path difference ##\Delta x## for a wave of wavelength ##\lambda## is:

##\phi = \dfrac{2\pi}{\lambda}\Delta x##

##\phi = \dfrac{\lambda}{2\pi}\Delta x##

##\phi = 2\pi\lambda\,\Delta x##

##\phi = \dfrac{\Delta x}{\lambda}##

A path difference of one full wavelength corresponds to a phase difference of ##2\pi##, giving the proportionality ##\phi = \dfrac{2\pi}{\lambda}\Delta x##.

Q6. Two sound sources are called coherent when they:

Have equal amplitudes

Emit waves with a constant phase difference

Are placed at equal distances from the listener

Emit waves of different frequencies

Coherence means the phase difference between the sources remains constant in time, which is the essential condition for a sustained, observable interference pattern.

Q7. Two waves of intensities ##\ce{I1}## and ##\ce{I2}## interfere with a phase difference ##\phi##. The resultant intensity is:

##\ce{I1 + I2}##

##I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi##

##\sqrt{I_1^2 + I_2^2}##

##I_1 + I_2 - 2\sqrt{I_1 I_2}\cos\phi##

Since intensity is proportional to the square of amplitude, the resultant intensity is ##I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi##, which reduces to ##\ce{I1 + I2}## when averaged over all phases.

Q8. Two coherent waves of equal intensity ##\ce{I0}## interfere. The maximum resultant intensity is:

##\ce{I0}##

##\ce{2I0}##

##\ce{4I0}##

##\sqrt{2}\,I_0##

For equal intensities, ##I_{\max} = I_0 + I_0 + 2I_0 = 4I_0##, since the amplitudes add fully in constructive interference.

Q9. In an interference pattern produced by two coherent sources, the energy that disappears at a destructive minimum:

Is permanently lost from the field

Reappears at the constructive maxima, so total energy is conserved

Is converted into heat in the medium

Is radiated back to the sources

Interference redistributes energy in space; the energy missing at the minima appears at the maxima, so the total energy over the pattern is conserved.

Q10. Two loudspeakers driven by the same oscillator produce a stable interference pattern in a hall. This is possible because the speakers act as:

Incoherent sources with random phases

Coherent sources with a constant phase difference

Independent sources of different frequencies

Sources of different wavelengths

Driving both speakers from one oscillator locks their phases, making them coherent sources capable of producing a sustained interference pattern.

Q11. A sound wave reflects from a rigid wall. The phase change introduced in the reflected pressure wave is:

Zero

##\dfrac{\pi}{2}##

##\pi##

##2\pi##

Reflection at a rigid (hard) boundary introduces a phase change of ##\pi## in the reflected wave, which is why standing waves form with a pressure antinode at the wall.

Q12. Beats heard when two sound waves of slightly different frequencies overlap are best described as:

Interference in space

Interference in time

Diffraction of sound

Resonance of the air column

Beats arise because the phase difference between two nearby frequencies changes with time, so the resultant amplitude waxes and wanes periodically — interference in time.

Q13. Which of the following is NOT a condition for sustained interference of sound from two sources?

The sources must be coherent

The sources must have comparable frequencies

The sources must have equal amplitudes

The waves must overlap at the observation point

Equal amplitudes are not required for sustained interference; they only make the minima perfectly zero. Coherence, comparable frequency and overlap are the essential conditions.

Q14. Interference differs from diffraction mainly because interference involves:

Superposition of waves from two or more distinct coherent sources

Bending of waves around obstacles

Superposition of wavelets from a single continuous wavefront

Change of frequency at a boundary

Interference is the superposition of waves from separate coherent sources, whereas diffraction is the superposition of secondary wavelets from the same wavefront.

Q15. Two coherent waves of amplitudes ##3## units and ##4## units interfere in phase. The resultant amplitude is:

##1## unit

##5## units

##7## units

##12## units

For in-phase (constructive) interference the amplitudes add algebraically: ##3 + 4 = 7## units.