14.2 Sound Intensity and Sound Level

Sound Intensity and Perception

Sound intensity quantifies how much energy a sound wave delivers per unit area, while the decibel scale gives us a practical way to express the enormous range of intensities the human ear can detect. Together, these concepts connect the physics of wave energy to how we actually experience loudness.

Amplitude, Energy, and Loudness Relationships

Sound waves are longitudinal pressure waves, meaning particles oscillate parallel to the direction the wave travels. The amplitude of a sound wave is the maximum displacement of those particles from their equilibrium position. Waves with higher amplitude carry more energy.

Sound intensity measures the power delivered by a sound wave per unit area, in units of $W/m^2$. The relationship between intensity and amplitude is:

$I \propto A^2$

This square relationship matters: if you double the amplitude, the intensity doesn't just double, it quadruples ($2^2 = 4$).

Perceived loudness is subjective. It depends on both intensity and frequency. The human ear is most sensitive to frequencies between about 2 kHz and 5 kHz, which overlaps with the speech range. A sound at 3 kHz will seem louder than a sound of equal intensity at 100 Hz. Also, perceived loudness doesn't scale linearly with intensity. Roughly speaking, a sound must be about 10 times more intense to sound twice as loud to a listener.

Decibel Scale for Sound Intensity

The human ear can detect intensities spanning about 12 orders of magnitude. Working with raw $W/m^2$ values across that range is impractical, so we use the decibel (dB) scale, which compresses it logarithmically:

$\beta = 10 \log_{10} \left(\frac{I}{I_0}\right)$

Here, $I_0 = 10^{-12} \, W/m^2$ is the reference intensity, chosen to approximate the threshold of human hearing.

Key patterns to internalize:

• Every +10 dB corresponds to a 10× increase in intensity
• Every +20 dB corresponds to a 100× increase in intensity
• Every +3 dB roughly doubles the intensity (since $\log_{10}(2) \approx 0.3$)

Typical sound levels for reference:

Notice that a normal conversation (60 dB) is $10^6$ times more intense than the threshold of hearing, not just "60 times." That's the whole point of the logarithmic scale.

Calculations of Sound Wave Intensity

Inverse-square law for a point source:

For a source radiating sound uniformly in all directions with power $P$, the intensity at distance $r$ is:

$I = \frac{P}{4\pi r^2}$

The $4\pi r^2$ in the denominator is the surface area of a sphere. As the wavefront expands, the same power spreads over a larger area, so intensity drops with the square of the distance. Double your distance from the source, and intensity falls to one-quarter ($\frac{1}{2^2} = \frac{1}{4}$).

Converting between dB and intensity:

To find the dB level from a known intensity:

$\beta = 10 \log_{10} \left(\frac{I}{I_0}\right)$

To find the intensity from a known dB level, rearrange:

$I = I_0 \cdot 10^{\beta/10}$

Example calculation: A speaker outputs 0.5 W of acoustic power. What is the intensity and sound level at 4 m away?

1. Apply the inverse-square law: $I = \frac{0.5}{4\pi (4)^2} = \frac{0.5}{201.1} \approx 2.49 \times 10^{-3} \, W/m^2$
2. Convert to decibels: $\beta = 10 \log_{10}\left(\frac{2.49 \times 10^{-3}}{10^{-12}}\right) = 10 \log_{10}(2.49 \times 10^{9}) \approx 10(9.40) \approx 94 \, \text{dB}$

Sound pressure and intensity:

Intensity can also be expressed in terms of the pressure variation $p$ in the wave:

$I = \frac{p^2}{2\rho v}$

where $\rho$ is the density of the medium (for air at sea level, $\rho \approx 1.225 \, kg/m^3$) and $v$ is the speed of sound (in air at 20°C, $v \approx 343 \, m/s$). The product $\rho v$ is called the acoustic impedance of the medium. Differences in acoustic impedance between two media determine how much sound is reflected versus transmitted at a boundary.

Physiology of Sound Production and Perception

Sound production in humans:

1. Air from the lungs passes over the vocal cords in the larynx, causing them to vibrate and produce pressure waves.
2. The frequency of vibration sets the pitch (higher frequency = higher pitch).
3. The vocal tract (throat, mouth, nasal cavity) acts as a resonance chamber, amplifying certain frequencies and shaping the character of the sound.

Sound perception involves three regions of the ear:

1. Outer ear: The pinna (visible part) funnels sound waves into the ear canal toward the eardrum.
2. Middle ear: The eardrum (tympanic membrane) vibrates in response to incoming pressure waves. Three tiny bones called ossicles (malleus, incus, stapes) transmit and amplify these vibrations. This mechanical amplification helps compensate for the energy loss that occurs when sound transitions from air to the fluid-filled inner ear.
3. Inner ear: The stapes pushes against the oval window of the cochlea, a fluid-filled, spiral-shaped structure. Inside, the basilar membrane vibrates at different positions depending on frequency: high frequencies near the base, low frequencies near the tip. Hair cells sitting on the basilar membrane convert these mechanical vibrations into electrical signals, which the auditory nerve carries to the brain.

Wave Properties and Sound Characteristics

• Resonance occurs when an object is driven at its natural frequency, causing a large increase in amplitude. This is why certain notes can make a wine glass vibrate noticeably.
• Wave interference happens when two sound waves overlap. Constructive interference (waves in phase) increases amplitude; destructive interference (waves out of phase) reduces it. This principle is used in noise-canceling headphones.
• The frequency spectrum of a sound, meaning which frequencies are present and their relative amplitudes, determines its timbre. This is why a piano and a violin playing the same note still sound different.
• Sound absorption by materials (carpets, acoustic panels, foam) converts sound energy into heat, reducing reflections and lowering noise levels in a space.