Nonlinear Acoustics and Shock Waves Quiz

Nonlinear Acoustics and Shock Waves

Linear acoustics assumes infinitesimal pressure disturbances travelling at a fixed sound speed. Real sound waves are finite in amplitude, and once the pressure swing becomes a meaningful fraction of ambient pressure, the medium stops behaving like a perfect linear spring.

In this quiz you will test your understanding of:

Why compressions travel faster than rarefactions, and how this steepens the waveform
The parameter of nonlinearity ##\beta = 1 + B/(2A)## and its values in air, water and tissue
Harmonic generation, shock formation distance and the sonic boom
The parametric acoustic array and its exploitation of nonlinearity
Acoustic streaming, radiation pressure, cavitation and therapeutic ultrasound

Work carefully with the local sound speed relation ##c(u) = c_0 + \beta u## and the adiabatic relation ##p/\rho^\gamma = \text{constant}##. Every question is answerable from the lesson material alone.

Q1. Linear acoustics treats pressure disturbances as infinitesimal perturbations. Which assumption is central to that treatment?

The sound speed is fixed and independent of amplitude

The wave carries no energy

The medium must be a solid

Superposition fails for all waves

Linear acoustics assumes infinitesimal perturbations travelling at a fixed sound speed, so the medium behaves like a perfect linear spring.

Q2. In a finite-amplitude wave in a gas, why does the local sound speed vary across a single cycle?

Because compressions heat the gas and rarefactions cool it, changing ##c## locally

Because the frequency changes with distance

Because the gas molecules are destroyed in compressions

Because the wavelength becomes infinite

Adiabatic compression raises both pressure and temperature, increasing ##c## locally, while rarefaction lowers it. This amplitude-dependent speed seeds nonlinear acoustics.

Q3. For an ideal gas undergoing adiabatic compression, the local sound speed is given by:

##c = \sqrt{\gamma p/\rho}##

##c = \gamma p \rho##

##c = \sqrt{\rho/(\gamma p)}##

##c = \gamma \rho / p##

Using ##p/\rho^\gamma = \text{constant}##, the local sound speed scales as ##c = \sqrt{\gamma p/\rho}##, so compression raises ##c##.

Q4. The parameter of nonlinearity is defined as:

##\beta = 1 + B/(2A)##

##\beta = 1 + A/(2B)##

##\beta = B/A##

##\beta = 2A/B##

##\beta = 1 + B/(2A)##, where ##B/A## is the ratio of second-order to first-order coefficients in the pressure-density expansion.

Q5. Which set of values of ##\beta## correctly matches air, water and biological tissue?

Air ##\approx 1.2##, water ##\approx 3.5##, tissue ##\approx 4## to ##7##

Air ##\approx 3.5##, water ##\approx 1.2##, tissue ##\approx 1##

Air ##\approx 7##, water ##\approx 4##, tissue ##\approx 1.2##

Air ##\approx 1.2##, water ##\approx 7##, tissue ##\approx 3.5##

The lesson states air ##\beta \approx 1.2##, water ##\beta \approx 3.5##, and biological tissue clusters near 4 to 7.

Q6. The local sound speed in a nonlinear medium is written as ##c(u) = c_0 + \beta u##. What does this imply about the wave profile?

Points of higher forward velocity travel faster, so the profile steepens

Points of higher forward velocity travel slower, so the profile flattens

All points travel at the same speed, so the profile is unchanged

The profile reverses direction

Since ##c## increases with forward particle velocity ##u##, crests outrun troughs and the waveform steepens as it propagates.

Q7. Why does medical ultrasound rely on tissue nonlinearity far more than airborne audio does?

Because tissue has a much larger ##\beta##, giving stronger distortion at the same Mach number

Because tissue has no acoustic impedance

Because ultrasound frequencies are below the audible range

Because tissue is a perfect linear spring

Tissue values of ##\beta## cluster near 4 to 7 compared with about 1.2 for air, so distortion is stronger at the same Mach number.

Q8. In linear acoustics, the condensation ##s = (\rho - \rho_0)/\rho_0## is assumed to satisfy:

##|s| \ll 1##

##|s| \approx 1##

##|s| \gg 1##

##s = 0## exactly

The small-condensation assumption ##|s| \ll 1## lets us discard second-order terms in the continuity and momentum equations.

Q9. When the acoustic Mach number ##M = u/c_0## approaches unity, what happens to the wave equation?

It acquires quadratic corrections that linear superposition cannot absorb

It becomes exactly linear

It loses all pressure dependence

It reduces to a static equation

The discarded second-order terms return with measurable force, and the wave equation acquires quadratic corrections that linear superposition cannot absorb.

Q10. The parametric acoustic array exploits nonlinearity to produce:

Highly directional low-frequency beams from compact transducers

Perfectly linear plane waves

Zero radiation pressure

Infinite shock formation distance

The parametric array uses nonlinearity to create highly directional low-frequency beams from compact transducers, reusing the interference principle.

Q11. Which statement best distinguishes a jet crackle from a pure tone?

The crackle contains generated harmonics and steepened waveform features absent in a pure tone

The crackle has no harmonics because it is linear

The pure tone steepens faster than the crackle

Both are identical in nonlinear behaviour

Nonlinear steepening and harmonic generation give the jet crackle its rich, sharp character, unlike a pure sinusoidal tone.

Q12. The shock relations used in nonlinear acoustics reuse which linear principles?

Continuity and momentum balance

Only the Doppler effect

Only Ohm's law

Only the photoelectric effect

The lesson notes that shock relations reuse continuity and momentum balance, while steepening reuses superposition and the parametric array reuses interference.

Q13. A larger value of ##\beta## at the same acoustic Mach number means:

Stronger waveform distortion

Weaker waveform distortion

No distortion at all

The sound speed becomes zero

Since ##c(u) = c_0 + \beta u##, a larger ##\beta## produces a larger amplitude-dependent speed variation and hence stronger distortion.

Q14. Which of the following is a listed consequence of finite-amplitude sound propagation?

Waveform steepening and shock formation at finite distances

Permanent loss of all wave energy at the source

Instantaneous decay of the wave to zero amplitude

Complete absence of harmonics

The lesson lists waveform steepening, shock formation at finite distances, harmonic generation and the parametric array as concrete consequences.

Q15. Why is nonlinear acoustics described as a bridge rather than an isolated topic in this course?

Because its mathematics reuses superposition, continuity, momentum balance and interference from earlier lessons

Because it replaces all earlier lessons entirely

Because it has no connection to wave speed or energy transport

Because it applies only to solids

Steepening reuses superposition, shock relations reuse continuity and momentum balance, and the parametric array reuses interference, linking it to earlier material.