Physics · Oscillations and waves · Grade 10

Sound: speed, pitch, ultrasound

How can we measure the distance to a cliff from its echo in the mountains?

The situation

In the Chimgan mountains a hiker shouted at the cliff opposite — and his voice came back a few seconds later. That same day, in a hospital, a doctor “saw” a patient's liver with an ultrasound scanner — with no radiation, using only sound. And a bat catches a fly precisely even in a pitch-dark cave. How fast does sound travel in air, in water and in steel? Why does one sound seem thin, another deep, another loud? And how can a sound that the human ear cannot hear draw a picture?

Topic

Sound: speed, pitch and loudness. Echo in the mountains, a sound level meter and an ultrasound scanner

Key ideas

Speed of sound
Decibel
Ultrasound

Formulas

v = \sqrt{\frac{γ R T}{M}} \approx 331 + 0.6 t m/s

Speed of sound in air ( $t$ in °C). In water ≈ 1 480 m/s, in a steel rail ≈ 5 000 m/s.

D = \frac{v Δ t}{2}

Echo (echo sounder, ultrasound scan): the sound travels to the obstacle and back, so we divide by 2.

L = 10 \lg \frac{I}{I_{0}} = 20 \lg \frac{p}{p_{0}}

Sound level, dB: $I_{0} = 10^{- 12}$ W/m², $p_{0} = 20$ µPa. +10 dB — the intensity increases 10 times.

λ = 2 (L_{2} - L_{1}), v = λ f

Resonance tube: the spacing between neighbouring resonances is $λ / 2$ . In Quincke's tube the shift between neighbouring minima is also $λ / 2$ .

R = \frac{Z_{2} - Z_{1}}{Z_{2} + Z_{1}}, Z = ρ v

The fraction of the wave amplitude reflected at the boundary; $Z$ is the acoustic impedance.

Theory

Sound is a longitudinal wave

The loudspeaker cone alternately compresses and rarefies the air; these compressions and rarefactions travel through the air (the particles in “Properties” mode). The particles only oscillate back and forth. Sound cannot travel without a medium — there is no sound in a vacuum. In a gas the speed is $v = \sqrt{γ p / ρ} = \sqrt{γ R T / M}$ : the compression is adiabatic ( $γ = 1.4$ ), which is why Newton's isothermal calculation (≈ 280 m/s) was wrong until Laplace corrected it. The speed depends not on pressure but on temperature: about 0.6 m/s for every 1 °C. Liquids and solids resist compression many times more strongly — so the speed in them is greater too. When the flow of gas in a pipe approaches the speed of sound, its properties change sharply — more about this in the “Aeroplane wing” experiment.

Echo, echo sounder and rail

Sound reflected from a cliff is heard after $Δ t = 2 D / v$ . For the ear to tell the echo apart as a separate sound, the delay must be more than about 0.1 s — so the obstacle must be at least ≈ 17 m away. The echo sounder on ships measures the depth of water in exactly the same way ( $v \approx 1 480$ m/s). Sound produced by a blow on a rail travels 15 times faster in the steel, so a person with an ear to the rail hears the blow much sooner than it arrives through the air.

Pitch, loudness, timbre

Pitch (high or low) depends on the frequency. Loudness depends on the intensity, but the ear perceives it logarithmically, so the decibel is used: $L = 10 \lg (I / I_{0})$ . 0 dB is the threshold of hearing at 1 kHz, 60 dB is conversation, and about 120–130 dB is the threshold of pain. The threshold depends on frequency: the ear is most sensitive at 2–5 kHz, and at low and very high frequencies the sound has to be much stronger (the “Hearing range” graph). Timbre is the set of harmonics: even if a flute and a violin play the same note, their spectra are different (in detail — in the “Fourier: the spectrum of sound” experiment).

Infrasound and ultrasound

A person hears from about 16–20 Hz up to 20 kHz. Below that is infrasound (earthquakes, storms, the low-frequency “talk” of elephants); above it is ultrasound. Bats and dolphins use ultrasound for echolocation: the short wavelength lets them “see” even small objects. Ultrasound is produced by a piezoelectric crystal: when an alternating voltage is applied it vibrates, and when a reflected wave strikes it, it produces a voltage itself. The frequency of ultrasound reflected from moving blood cells changes (the Doppler effect — in the “Waves and the Doppler effect” experiment); Doppler ultrasound uses this to measure the speed of blood flow.

How an ultrasound scan makes a picture

The probe sends a short pulse of 2–15 MHz and listens for the returning echoes. At the boundary between two tissues the acoustic impedance $Z = ρ v$ changes and part of the wave is reflected: $R = (Z_{2} - Z_{1}) / (Z_{2} + Z_{1})$ . The delay gives the depth: $d = v t / 2$ , $v = 1 540$ m/s. The $Z$ of air is very small — almost everything is reflected at the boundary, which is why gel is needed. The higher the frequency, the sharper the image (smaller $λ$ ), but the wave is attenuated faster and does not reach deep. A fluid-filled cyst looks black (no echoes from inside) with a brighter area behind it; a stone is bright white, with an “acoustic shadow” behind it. The model is simplified: the tissues are uniform, the beam is thin, and the graininess (speckle) is random.

Lab work: measuring the speed of sound

Resonance tube: when the natural frequency of an air column closed by water equals the frequency of the source, the sound becomes sharply louder — acoustic resonance (the general phenomenon of resonance is in the “Bridge resonance” experiment). There is an antinode at the open end and a node at the water surface: $L + e = λ / 4, 3 λ / 4, \dots$ ( $e \approx 0.6 r$ is the end correction). In the difference between two neighbouring resonances $e$ cancels out: $λ = 2 (L_{2} - L_{1})$ . Quincke's tube (an interference method): the sound splits into two paths and then recombines; if the sliding section is moved by $x$ , the path difference changes by $2 x$ . When the path difference is $(m + \frac{1}{2}) λ$ there is a minimum, so the spacing between neighbouring minima is $Δ x = λ / 2$ . In both methods $v = λ f$ , and then $t \approx (v - 331) / 0.6$ .

Did you know?

In 1826 Jean-Daniel Colladon and Charles Sturm measured the speed of sound in water on Lake Geneva: on one boat an underwater bell was struck and gunpowder was flashed at the same moment; on the other boat the sound was listened to through a horn lowered into the water. The result was about 1 435 m/s.
Most bats emit ultrasound of roughly 20–120 kHz for echolocation — the human ear cannot hear it.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.