Intensity, Decibels, Pitch & Quality

We describe a sound by how loud it is, how high it is, and what it sounds like. Physics ties each of these to something measurable: intensity, frequency and waveform.

Intensity#

A sound wave carries energy. The intensity is the average energy crossing unit area (perpendicular to the direction of travel) per unit time. Its unit is W/m².

As in a wave on a string, the energy carried depends on the square of the amplitude and of the frequency. In terms of the displacement amplitude:

Using p₀ = Bks₀ and v² = B/ρ, the same thing in terms of the pressure amplitude:

The decibel scale#

The faintest sound an average ear detects at 1 kHz has an intensity of about $I_0 = 10^{-12}\ \text{W/m}^2$; sound becomes painful near 1 W/m². That is a range of a trillion. Our sense of loudness grows roughly with the logarithm of the intensity, so we use a logarithmic scale:

• Multiplying the intensity by 10 adds 10 dB; by 100 adds 20 dB.
• Doubling the intensity adds 10 log 2 ≈ 3 dB.
• A whisper is about 10 dB, a conversation 60 dB, a rock concert 110 dB; 120 dB is the threshold of pain.

Pitch, loudness and quality#

Pitch

Pitch rises with frequency. A sound containing several frequencies is heard with the pitch of its dominant one. Women’s and children’s voices usually have higher pitch than men’s because their vocal cords are shorter and lighter.

Loudness

Loudness is our sensation of intensity. It isn’t just intensity: the ear is most sensitive between about 1 and 4 kHz, so a 50 Hz hum and a 2 kHz tone of the same intensity do not sound equally loud.

Quality

A violin and a flute playing the same note have the same dominant frequency, yet we tell them apart instantly. Each instrument produces the fundamental together with a set of harmonics (whole-number multiples of it) in its own proportions. That mix gives the waveform its shape, and the shape is what we call the quality or timbre.