4.2 Sound waves and waves for detection and exploration — Learn

Prerequisites: Section 4.1, especially the wave equation and longitudinal waves; Section 3.1 for speed as distance divided by time; powers of ten and prefixes.

By the end of this section you can: describe how sound travels through solids and how the ear detects it; explain why hearing covers only a limited range of frequencies; calculate a distance from an echo; describe how ultrasound is used for medical and industrial imaging; and describe how seismic waves give evidence about the structure of the Earth.

1. Sound needs a medium

A sound wave is a longitudinal wave. It travels as a series of compressions and rarefactions, so it needs particles to vibrate. A bell inside a sealed jar rings loudly at first. When the air is pumped out of the jar, the ringing fades away, even though the hammer still strikes the bell. The sound has no particles to travel through.

Sound travels at different speeds in different media. The speed depends on the medium, not on the sound. In air at room temperature the speed of sound is about 330 m/s. In water it is about 1500 m/s. In steel it is about 5000 m/s. Sound travels fastest in solids, because the particles are closer together.

2. Sound waves, solids and hearing

Sound waves can travel through a solid and make the solid vibrate. When you tap a table, you can sometimes feel the vibration through the wood.

Inside the ear, sound waves make the ear drum and other parts of the ear vibrate. This vibration causes the sensation of sound. The conversion of a sound wave into a vibration of a solid works only over a limited range of frequencies, because the parts of the ear cannot vibrate at every frequency. That limited range sets the limits of normal human hearing.

Most people can hear frequencies from about 20 Hz to 20 kHz. Below 20 Hz is infrasound. Above 20 kHz is ultrasound. Both are real waves, but you cannot hear them.

Worked example 1. The lowest frequency a healthy young person can hear is 20 Hz. Calculate the period of this sound wave.

\begin{aligned} T &= \frac{1}{f} \\ &= \frac{1}{20} \\ &= 0.050\,\text{s} \end{aligned}

3. Using an echo to find a distance

When a wave meets the boundary between two different media, part of the wave is reflected. The reflection is called an echo. The time taken for the echo to return to a detector can tell you how far away the boundary is.

The pulse travels to the boundary and back, so the distance it covers is twice the distance to the boundary:

\text{distance to boundary} = \frac{\text{speed} \times \text{time}}{2}

Worked example 2. A pulse of ultrasound travels through tissue at 1500 m/s. The reflection returns to the detector after 0.00012 s. Calculate the distance to the boundary.

\begin{aligned} s &= \frac{v t}{2} \\ &= \frac{1500 \times 0.00012}{2} \\ &= \frac{0.18}{2} \\ &= 0.090\,\text{m} \end{aligned}

Worked example 3. In the chapter problem the echo sounder sends a pulse through sea water at 1500 m/s. The reflection returns after 0.020 s. Calculate the depth of the sea floor.

\begin{aligned} s &= \frac{v t}{2} \\ &= \frac{1500 \times 0.020}{2} \\ &= \frac{30}{2} \\ &= 15\,\text{m} \end{aligned}

Try it: an echo and the depth

The figure shows a boat sending a pulse to a boundary. You can change the depth and the speed of the wave.

Predict first. Do not move the sliders yet. Write down your prediction. If you double the depth, what happens to the echo time?

Test. Set the depth to 30 m and the speed of the wave to 1500 m/s. Read the echo time. Now set the depth to 60 m and read the echo time again.

Explain. The pulse has to travel to the boundary and back, so the distance travelled is twice the depth. Why does doubling the depth double the echo time, while doubling the speed halves it? Write your answer in two or three sentences.

4. Ultrasound and echo sounding

Ultrasound is a sound wave with a frequency higher than 20 kHz. Ultrasound waves are partially reflected when they meet a boundary between two different media. The time taken for the reflections to reach a detector shows how far away the boundary is. This is how ultrasound is used for medical and industrial imaging.

Echo sounding uses high-frequency sound to detect objects in deep water and to measure the depth of the water. The survey boat in the chapter problem uses echo sounding.

Worked example 4. A medical scanner uses ultrasound at a frequency of 2.0 MHz. The speed of sound in tissue is 1500 m/s. Calculate the wavelength.

\begin{aligned} 2.0\,\text{MHz} &= 2.0 \times 10^6\,\text{Hz} \\ v &= f \lambda \\ \lambda &= \frac{v}{f} \\ &= \frac{1500}{2.0 \times 10^6} \\ &= 7.5 \times 10^{-4}\,\text{m} \end{aligned}

This is 0.75 mm. A shorter wavelength can show smaller details.

5. Seismic waves

An earthquake produces seismic waves that travel through the Earth. There are two important types.

P-waves are longitudinal. They travel at different speeds through solids and liquids.
S-waves are transverse. They cannot travel through a liquid.

The two types travel at different speeds, so they arrive at a distant detector at different times. This gives evidence about the inside of the Earth.

S-waves do not pass through the liquid outer core, so a region on the far side of the Earth receives no S-waves. P-waves do pass through, but they are bent at the boundary. From this evidence, scientists worked out the size and structure of the Earth's core. The core cannot be seen directly, so the waves are the evidence.

Worked example 5. A seismic P-wave travels 60 km in 12 s. Calculate its speed.

\begin{aligned} 60\,\text{km} &= 60000\,\text{m} \\ v &= \frac{s}{t} \\ &= \frac{60000}{12} \\ &= 5000\,\text{m/s} \end{aligned}

6. Comparing waves in solids and liquids

Different types of wave travel at different speeds, are absorbed differently and are reflected differently in solids and liquids. These differences can be used to detect and to explore structures that cannot be seen directly, in the body, in the Earth's core and in deep water.

An ultrasound pulse sends a reflection back from a boundary inside the body, so a doctor can build a picture of an organ.
A seismic wave changes speed and direction as it crosses the boundary of the core, so a scientist can work out where that boundary is.
An echo sounder detects the sea floor or a shoal of fish from the reflection of a high-frequency sound pulse.

Watch out. The echo time is the time for the pulse to travel to the boundary and back. A common error is to use the full time as if it were a one-way trip, which makes the distance twice as large. Always halve the time before you multiply by the speed. Also keep every quantity in SI units: convert kilometres to metres and megahertz to hertz before you calculate.

Progress check

Return to the chapter problem. The echo sounder sends a pulse through sea water at 1500 m/s, and the reflection returns after 0.020 s. Work out the depth of the sea floor. Then decide whether the channel is deep enough for the ferry, which needs at least 12 m.

You should now be able to

explain why sound needs a medium, and why it cannot travel through a vacuum;
describe how sound waves make solids vibrate, using the ear as an example;
explain why human hearing is limited to about 20 Hz to 20 kHz;
calculate the distance to a boundary from an echo, using $s = \dfrac{v t}{2}$ ;
describe how ultrasound is used for medical and industrial imaging;
describe echo sounding and how it measures the depth of water;
compare P-waves and S-waves and explain how they give evidence about the Earth's core;
explain how differences in velocity, absorption and reflection are used to explore hidden structures.