Pitch, loudness and speed

A sound wave has three numbers you can measure.

• Frequency f: how many complete vibrations happen each second. It is measured in hertz (Hz). 440 Hz means 440 vibrations per second. Frequency sets the pitch: high frequency sounds high, low frequency sounds low.
• Amplitude A: how big the pressure swing is. A bigger amplitude means more energy, and the sound is louder (it also takes more energy to make).
• Wavelength λ (the Greek letter lambda): the distance between two neighbouring compressions, in metres.

Pitch and loudness are independent. You can play a high note quietly and a low note loudly. The common phrase "a high sound" really means a high pitch, not a high volume.

The wave equation. A wave travels one wavelength in the time of one vibration, which is 1/f seconds. Speed is distance over time, so

v = f × λ, which is the same as λ = v / f.

In air the speed depends on the temperature (a warmer gas has faster particles that pass the squeeze along sooner). A good formula for dry air is c ≈ 331.3 + 0.606 × T, with T in °C. At 0 °C that gives 331 m/s, at 20 °C it gives 343 m/s, and at 35 °C about 353 m/s. The speed does not depend on the frequency or the loudness: the bass and the treble of a concert reach the back row at the same moment, so the music stays together.

Examples at 20 °C (c = 343 m/s):

• 440 Hz: λ = 343 / 440 ≈ 0.78 m (78 cm).
• 686 Hz: λ = 343 / 686 = 0.5 m.
• 100 Hz (a deep voice or a bass): λ ≈ 3.4 m.
• 10,000 Hz: λ ≈ 3.4 cm.

Low sounds have long waves, so they flow around corners and through doors. High sounds have short waves, so they are blocked by objects and travel more in straight beams.

Crossing between materials. When a sound passes from air into water, the frequency stays the same, because it is set by the source and each vibration must carry on through. The speed changes, so the wavelength must change too. A 440 Hz tone has a wavelength of about 0.78 m in air. In water (1,480 m/s) it is 1,480 / 440 ≈ 3.4 m, more than four times longer.

What can we hear? Healthy young ears hear from about 20 Hz to 20,000 Hz. Below 20 Hz is infrasound (elephants and some whales use it) and above 20,000 Hz is ultrasound (bats use it, and medical scanners use frequencies from about 1 to 20 million Hz to see inside the body). The top of the range falls with age. By adulthood many people cannot hear much above 15,000 Hz, and the limit keeps drifting down. Hearing loss from loud noise, which you will meet in the next lesson, can make it fall sooner.

Loudness also depends on pitch. Human ears are most sensitive between about 2,000 and 5,000 Hz, so a tone there sounds louder than a tone of the same pressure at 50 Hz. We come back to this when we weight decibels.