2.3 Intensity and Loudness

Sound Intensity and Energy Transfer

Sound waves carry energy outward from a source, and intensity is how we quantify that energy flow. It connects the physical power of a sound source to what actually reaches your ear at a given distance.

Defining Sound Intensity

Sound intensity measures the sound energy transmitted through a unit area per unit time, expressed in $\text{W/m}^2$. Think of it as "how much energy is hitting each square meter every second."

A few key properties:

• Intensity is directly proportional to the power of the sound source
• It's inversely proportional to the surface area over which the sound spreads
• It follows the inverse square law: as you double your distance from the source, intensity drops to one-quarter (because the same energy spreads over four times the area)
• It's a scalar quantity (energy flux through a surface), though the energy propagation has a direction perpendicular to the wavefront

The energy itself transfers through compressions and rarefactions of the medium (usually air) as the wave propagates outward.

Relationship to Power and Energy

The power of a sound source is the total energy it emits per unit time, measured in watts. Since a point source radiates equally in all directions, that power spreads over the surface of an expanding sphere. This gives us the intensity formula:

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

• $P$ = power of the sound source (W)
• $r$ = distance from the source (m)
• $4\pi r^2$ = surface area of a sphere at distance $r$

To get a feel for the numbers:

• A whisper has an intensity around $10^{-10} \text{ W/m}^2$
• Normal conversation is roughly $10^{-6} \text{ W/m}^2$

That's a factor of 10,000 between them, which is why we need a compressed scale to talk about sound levels practically.

Calculating Sound Intensity Levels

Decibel Scale Basics

Human hearing spans an enormous range of intensities, from about $10^{-12} \text{ W/m}^2$ (barely audible) to $1 \text{ W/m}^2$ (painfully loud). That's a factor of $10^{12}$. The decibel scale uses logarithms to compress this range into manageable numbers.

The sound intensity level $\beta$ in decibels (dB) is:

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

• $I$ = measured intensity
• $I_0 = 10^{-12} \text{ W/m}^2$, the reference intensity (threshold of human hearing at 1000 Hz)

At the reference intensity, $\beta = 0 \text{ dB}$. Two rules of thumb worth memorizing:

• Every 10 dB increase = intensity multiplied by 10
• Every 3 dB increase ≈ intensity roughly doubled (since $10^{0.3} \approx 2$)

Applying the Decibel Scale

To compare two sounds without knowing their absolute intensities, use the difference formula:

$\Delta\beta = \beta_2 - \beta_1 = 10 \log_{10}\left(\frac{I_2}{I_1}\right)$

To convert a dB level back to an absolute intensity:

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

Some reference points to anchor your intuition:

Notice that a rock concert at 110 dB is $10^{5}$ (100,000) times more intense than a conversation at 60 dB, not just "almost twice as loud." This disconnect between the numbers and your gut feeling is exactly why the logarithmic scale matters.

Quick example: Suppose a speaker outputs 0.5 W of acoustic power. What's the intensity at 4 m away?

1. Use $I = \frac{P}{4\pi r^2} = \frac{0.5}{4\pi(4)^2} = \frac{0.5}{201.1} \approx 2.49 \times 10^{-3} \text{ W/m}^2$
2. Convert to dB: $\beta = 10\log_{10}\left(\frac{2.49 \times 10^{-3}}{10^{-12}}\right) = 10\log_{10}(2.49 \times 10^{9}) \approx 10(9.40) = 94.0 \text{ dB}$

Loudness and Frequency Dependence

Understanding Loudness Perception

Loudness is the subjective perception of sound intensity. Two sounds with identical intensities can sound different in loudness depending on their frequency. Your ears are not equally sensitive across all frequencies.

Equal-loudness contours (also called Fletcher-Munson curves) map this out. Each curve connects frequency-intensity combinations that a listener perceives as equally loud. The key takeaway: human ears are most sensitive to frequencies between about 2000 and 5000 Hz. A 3000 Hz tone doesn't need as much intensity to sound "loud" compared to a 100 Hz tone at the same perceived loudness.

This is why you lose the bass first when you turn music down to low volume. At lower intensities, the equal-loudness contours spread farther apart at low frequencies, meaning those bass notes fall below your perception threshold while midrange frequencies remain audible.

Measuring Loudness

Sound pressure level (SPL) is closely related to intensity and is what most sound meters actually measure. It's defined as:

$\text{SPL} = 20 \log_{10}\left(\frac{p}{p_0}\right)$

where $p_0 = 20 \text{ μPa}$ is the reference pressure. The factor of 20 (instead of 10) appears because intensity is proportional to pressure squared: $I \propto p^2$. In many practical situations, SPL and intensity level in dB give the same numerical value, so they're often used interchangeably.

The phon scale quantifies perceived loudness more directly:

• 1 phon is defined as the perceived loudness level of a 1000 Hz tone at 1 dB SPL
• The equal-loudness contours are lines of constant phon value across different frequencies
• At 1000 Hz, the phon value equals the dB SPL value by definition; at other frequencies, it diverges

For relating perceived loudness to intensity quantitatively, Stevens' power law gives:

$L \propto I^{0.3}$

where $L$ is perceived loudness and $I$ is sound intensity. This means that roughly a tenfold increase in intensity (10 dB) corresponds to a perceived doubling of loudness.

Factors Affecting Loudness Perception

Human Auditory System

Your ear processes sound through three stages: the outer ear funnels sound in, the middle ear amplifies vibrations through the ossicles (the three smallest bones in your body), and the inner ear converts mechanical vibrations to neural signals via hair cells.

Inside the cochlea, the basilar membrane performs frequency analysis. Different positions along the membrane respond to different frequencies, an arrangement called tonotopic organization. High frequencies activate the base (near the oval window), and low frequencies activate the apex (the far end). This spatial mapping is how your brain distinguishes pitch.

Two additional effects shape what you perceive:

• Auditory masking: one sound can make another harder to hear. A loud tone at 1000 Hz can mask a quieter tone at 1100 Hz because both excite overlapping regions of the basilar membrane.
• Temporal integration: longer-duration sounds are generally perceived as louder than shorter sounds of the same intensity. Your auditory system effectively "adds up" energy over time windows of roughly 200 ms.

Psychoacoustic Factors

• Critical band theory describes how the auditory system groups nearby frequencies together when assessing loudness. If two tones fall within the same critical band (~1/3 octave wide), they don't add to perceived loudness the way two widely spaced tones would. Two tones separated by more than a critical bandwidth are processed independently, so their loudnesses sum.
• Binaural loudness summation: a sound presented to both ears is perceived as louder than the same sound presented to just one ear (by about 3 to 6 dB in perceived level).
• Loudness adaptation and fatigue: prolonged exposure to a sound reduces its perceived loudness over time. Extended exposure to high-intensity sound can cause temporary threshold shifts or, with repeated exposure, permanent hearing damage.

A couple of well-known psychoacoustic phenomena:

• The cocktail party effect is your ability to focus on a single conversation in a noisy room, selectively filtering competing sounds.
• Shepard tones create the illusion of a pitch that seems to rise (or fall) endlessly, demonstrating that pitch perception involves more than just frequency.