Wave Equation and Speed of Sound
This quiz tests Lesson 2 of the sound-waves course: the derivation of the wave equation for sound and the factors that control the speed of propagation in gases, liquids, and solids.
You will be examined on:
- Newton‘s isothermal formula versus the Laplace adiabatic correction
- Dependence of speed on temperature, pressure, humidity, and molecular mass
- Elastic moduli and the relation ##v = \sqrt{B/\rho}##
- Wave number, angular frequency, and the general wave solution
- Particle velocity versus wave velocity, intensity, decibels, Mach number, and the Doppler effect
Difficulty: Medium. Attempt all 15 questions and read each explanation carefully — the reasoning matters as much as the answer.
Q1. In the derivation of the sound wave equation, the pressure excess is related to the displacement gradient by ##\Delta P = -B\,\partial y/\partial x##. This relation expresses which property of the medium?
The constant ##B## is the bulk modulus, the elastic restoring property that links pressure excess to fractional volume change (strain). Density supplies the inertial part of the wave equation.
Q2. The general one-dimensional wave equation for sound takes the form ##\dfrac{\partial^2 y}{\partial t^2} = v^2 \dfrac{\partial^2 y}{\partial x^2}##. What does the constant ##v## represent in this equation?
In the wave equation the constant ##v## is the wave (phase) speed, the speed at which the disturbance pattern travels, not the velocity of individual particles.
Q3. Newton assumed that sound propagation in a gas is an isothermal process. What speed of sound in air at STP does his formula predict, compared with the measured value?
Newton's isothermal formula gives roughly 280 m/s in air at STP, well below the measured 331 m/s. Laplace later corrected this by using an adiabatic process.
Q4. Laplace corrected Newton's formula by treating the compression and rarefaction in a sound wave as:
Laplace argued that compressions and rarefactions occur so rapidly that heat cannot flow between adjacent layers, so the process is adiabatic. This introduces the factor ##\gamma##.
Q5. The Newton–Laplace expression for the speed of sound in an ideal gas is:
The adiabatic bulk modulus of a gas is ##B = \gamma P##, so ##v = \sqrt{B/\rho} = \sqrt{\gamma P/\rho}##. The isothermal version without ##\gamma## is Newton's original result.
Q6. Using ##v = \sqrt{\gamma R T / M}##, how does the speed of sound in an ideal gas depend on absolute temperature?
Since ##v \propto \sqrt{T}##, the speed rises as the square root of the absolute temperature. This is why the practical formula ##v = 331 + 0.6t## (t in °C) is only an approximation over a modest range.
Q7. For an ideal gas at constant temperature, how does the speed of sound change when the pressure is doubled?
For an ideal gas ##P/\rho = RT/M## depends only on temperature, so at constant temperature doubling the pressure also doubles the density and the speed of sound is unchanged.
Q8. On a humid day, the speed of sound in air increases slightly. The main reason is that:
Water vapour (molar mass about 18 g/mol) is lighter than the average air molecule (about 29 g/mol). Replacing air molecules with vapour lowers the mean molar mass ##M##, and since ##v \propto 1/\sqrt{M}##, the speed rises.
Q9. The speed of sound in a fluid medium is given by ##v = \sqrt{B/\rho}##. Which comparison of measured speeds is consistent with this relation?
Steel has a very large bulk modulus and moderate density, giving the highest speed; water is intermediate; air, with a small bulk modulus, is slowest. Density alone does not decide the speed.
Q10. A sound wave is described by ##y = A\sin(kx - \omega t)##. The wave number ##k## is related to the wavelength ##\lambda## by:
By definition the wave number is ##k = 2\pi/\lambda##, while the angular frequency is ##\omega = 2\pi f##. The wave speed follows from ##v = \omega/k##.
Q11. For a sinusoidal sound wave, which statement correctly distinguishes particle velocity from wave velocity?
Individual particles oscillate about their mean positions with a velocity that varies sinusoidally, whereas the wave pattern travels at the constant phase speed ##v##.
Q12. The intensity of a sound wave is proportional to:
Intensity depends on the square of the amplitude and the square of the frequency (through the particle velocity), which is why high-frequency sound carries more energy for the same amplitude.
Q13. The sound intensity level in decibels is defined as ##\beta = 10\log_{10}(I/I_0)##. If the intensity of a sound increases by a factor of 100, the intensity level increases by:
Since ##\log_{10}(100) = 2##, the increase is ##10 \times 2 = 20## dB. Each factor of 10 in intensity corresponds to 10 dB.
Q14. An aircraft flying at a speed greater than the speed of sound in air produces a sonic boom. This is best explained as:
When the source moves faster than sound, the wavefronts pile up into a cone-shaped shock wave. The sudden pressure change of this shock front is heard as the sonic boom.
Q15. The Mach number of an object is defined as the ratio of:
Mach number is the object's speed divided by the local speed of sound. A Mach number greater than 1 means supersonic motion and the possibility of a shock wave.