4.2 Sound waves and waves for detection and exploration — Practise with help

Work each problem first. Open a hint only if you are stuck, and the full working only to check.

Warm-up: remember

A wave has a frequency of 2.0 Hz and a wavelength of 1.5 m. Calculate the wave speed.
A sound wave has a frequency of 500 Hz. Calculate its period.

Answers

$v = f\lambda = 2.0 \times 1.5 = 3.0\,\text{m/s}$ .
$T = \dfrac{1}{f} = \dfrac{1}{500} = 0.0020\,\text{s}$ .

If question 1 went wrong, look again at the wave equation in Section 4.1. If question 2 went wrong, remember that the period is the reciprocal of the frequency.

Practise with help

1. A pulse of ultrasound travels through tissue at 1500 m/s. The reflection returns after 0.00020 s. Calculate the distance to the boundary.

Hint

The pulse travels to the boundary and back, so the time covers twice the distance you want.

Next step

Write $s = \dfrac{v t}{2}$ , then work out $\dfrac{1500 \times 0.00020}{2}$ .

Full working

\begin{aligned} s &= \frac{v t}{2} \\ &= \frac{1500 \times 0.00020}{2} \\ &= \frac{0.30}{2} \\ &= 0.15\,\text{m} \end{aligned}

2. A ship sends a pulse of sound down through water at 1500 m/s. The reflection returns after 0.040 s. Calculate the depth of the water.

Hint

This is the same shape of problem as question 1. The echo time is for the trip down and back.

Next step

Use $s = \dfrac{v t}{2}$ , and put in $v = 1500$ and $t = 0.040$ .

Full working

\begin{aligned} s &= \frac{v t}{2} \\ &= \frac{1500 \times 0.040}{2} \\ &= \frac{60}{2} \\ &= 30\,\text{m} \end{aligned}

3. An earthquake P-wave travels 45 km in 15 s. Calculate the speed of the P-wave.

Hint

The speed is distance divided by time, but the distance is given in kilometres.

Next step

First write $45\,\text{km} = 45000\,\text{m}$ . Then work out $\dfrac{45000}{15}$ .

Full working

\begin{aligned} 45\,\text{km} &= 45000\,\text{m} \\ v &= \frac{s}{t} \\ &= \frac{45000}{15} \\ &= 3000\,\text{m/s} \end{aligned}

4. An ultrasound scanner works at a frequency of 2.5 MHz. The speed of sound in tissue is 1500 m/s. Calculate the wavelength of the ultrasound.

Hint

The wavelength comes from the wave equation, but the frequency must be in hertz first.

Next step

Write $2.5\,\text{MHz} = 2.5 \times 10^6\,\text{Hz}$ , then use $\lambda = \dfrac{v}{f}$ .

Full working

\begin{aligned} 2.5\,\text{MHz} &= 2.5 \times 10^6\,\text{Hz} \\ \lambda &= \frac{v}{f} \\ &= \frac{1500}{2.5 \times 10^6} \\ &= 6.0 \times 10^{-4}\,\text{m} \end{aligned}

This is 0.60 mm.

5. Explain why a sound wave cannot travel through a vacuum.

Hint

Think about what a longitudinal wave is made of, and what a vacuum contains.

Next step

Say that a sound wave is a series of compressions and rarefactions, then say whether a vacuum has particles that can vibrate.

Full working

A sound wave is a longitudinal wave, so it travels as compressions and rarefactions of the particles in the medium. The particles vibrate back and forth, and pass the wave along. A vacuum contains no particles, so there is nothing to vibrate and the wave cannot travel. This answer is a model answer for you to compare with your own wording.

6. Explain why the ear can detect sound only over a limited range of frequencies.

Hint

Follow the sound from the air to the ear drum, and think about what the ear drum has to do with the wave.

Next step

Say that sound waves make the ear drum vibrate, then say why that vibration cannot happen at every frequency.

Full working

Sound waves arriving at the ear make the ear drum and other parts of the ear vibrate. This converts the sound wave into a vibration of a solid, which causes the sensation of sound. The parts of the ear can only vibrate well over a limited range of frequencies, so the conversion works only over that range. This sets the limits of human hearing, which is about 20 Hz to 20 kHz. This answer is a model answer for you to compare with your own wording.

7. Explain how an ultrasound pulse can be used to find the distance to a boundary inside the body.

Hint

Think about what happens when the wave reaches the boundary, and what the detector measures.

Next step

Say that part of the wave is reflected back, that the detector times the return, and that the time must be halved.

Full working

When the ultrasound wave meets the boundary between two different tissues, part of the wave is reflected back towards the detector. The detector measures the time taken for the reflection to return. The pulse travelled to the boundary and back, so the distance is half of the speed multiplied by the time. This answer is a model answer for you to compare with your own wording.