Intensity and Loudness of Sound
Sound intensity is a precisely measurable physical quantity — the acoustic power crossing unit area perpendicular to the direction of propagation, expressed in ##\text{W m}^{-2}##. Loudness, by contrast, is a psychological perception created by the auditory system, and it does not scale linearly with intensity.
This quiz tests the core machinery of Lesson 8:
- The intensity–amplitude and intensity–pressure amplitude relations
- The inverse square law for spherical waves
- The decibel scale and the threshold-of-hearing reference
- Phon and sone units, equal loudness contours, and hearing damage limits
- Intensity addition, reflection and transmission, and practical decibel levels
Expect JEE-style numerical work: intensity ratios, distance scaling, and decibel arithmetic. Keep the quadratic dependence ##I \propto p_0^2## firmly in mind — it is the most common source of error.
Q1. Sound intensity is defined as the average acoustic power crossing unit area oriented perpendicular to the direction of propagation. Its SI unit is:
Intensity is power per unit area, so its SI unit is the watt per square metre, ##\text{W m}^{-2}##. It is a scalar quantity despite describing a directed energy flow.
Q2. For a plane progressive wave in a fluid, the intensity is related to the pressure amplitude ##p_0## by ##I = \dfrac{p_0^2}{2\rho v}##. If the pressure amplitude is doubled while the medium is unchanged, the intensity becomes:
Intensity depends on the square of the pressure amplitude, so doubling ##p_0## multiplies ##I## by ##2^2 = 4##. This quadratic dependence is a recurring source of error in examinations.
Q3. A small source radiates sound uniformly into free space. At a distance of ##2\ \text{m}## the intensity is ##I##. At a distance of ##8\ \text{m}## the intensity will be:
For a spherical source radiating uniformly, intensity obeys the inverse square law ##I \propto \dfrac{1}{r^2}##. The distance increases by a factor of 4, so the intensity falls by ##4^2 = 16##.
Q4. The sound intensity level in decibels is defined as:
The intensity level is ##\beta = 10\log_{10}\left(\dfrac{I}{I_0}\right)##, where ##\ce{I0}## is the threshold-of-hearing reference intensity. The factor 20 appears only when the ratio is expressed in pressure amplitudes.
Q5. The reference intensity ##\ce{I0}## that anchors the decibel scale is the threshold of hearing, equal to:
The threshold of hearing is taken as ##I_0 = 10^{-12}\ \text{W m}^{-2}##, and every decibel value on the scale is measured relative to this anchor.
Q6. If the intensity of a sound is increased by a factor of 100, the intensity level increases by:
Using ##\beta = 10\log_{10}\left(\dfrac{I}{I_0}\right)##, a factor of 100 gives an increase of ##10\log_{10}(100) = 10 \times 2 = 20\ \text{dB}##.
Q7. Two independent, incoherent sources each produce an intensity ##I## at a point. The total intensity at that point is:
For incoherent sources the intensities add directly, giving ##\ce{I + I = 2I}##. Amplitudes add only for coherent sources, and even then the result depends on phase.
Q8. Loudness differs from intensity in that loudness is:
Intensity is an objective, instrument-measurable quantity, whereas loudness is the subjective sensation produced by the auditory system. Loudness does not scale linearly with intensity.
Q9. A sound carrying ten times the physical power of another does not sound ten times louder because:
The auditory system compresses a vast range of intensities into a manageable perceptual span, which is precisely why the logarithmic decibel scale is used.
Q10. The loudness level in phons of a sound is numerically equal to:
The phon scale defines the loudness level of any sound as the decibel level of an equally loud pure tone at 1 kHz. Equal loudness contours connect points of equal phon value.
Q11. A sound of loudness level 40 phons is judged to have a loudness of:
By definition, 1 sone corresponds to a loudness level of 40 phons. The sone scale is the linear loudness scale, doubling whenever the loudness is judged twice as great.
Q12. The threshold of pain corresponds to a sound intensity level of approximately:
The threshold of pain lies near 120 dB, roughly twelve orders of magnitude above the threshold of hearing. Prolonged exposure well below this level can still damage hearing.
Q13. For a plane wave, the intensity can be written as ##I = u\,v##, where ##u## is the average energy per unit volume. This relation shows that intensity quantifies:
Intensity quantifies energy flow rather than energy storage, which distinguishes it from quantities such as pressure amplitude alone. The product of energy density and wave speed gives the rate of energy transport per unit area.
Q14. The product ##\rho v## appearing in ##I = \dfrac{p_0^2}{2\rho v}## is called the:
The product ##\rho v## is the characteristic acoustic impedance of the medium. It governs how much power a given pressure amplitude actually transports, so denser, faster media carry more intensity for the same pressure swing.
Q15. When sound travelling in one medium strikes a boundary with a second medium, the incident intensity is:
At a boundary the incident intensity divides into a reflected part and a transmitted part. The partition depends on the acoustic impedances of the two media, and the two parts together account for the incident intensity.