4.1 Waves in air, fluids and solids — Learn

Prerequisites: Section 3.1, especially speed as distance divided by time; reading a graph; multiplying and dividing decimals; using a formula and rearranging it.

By the end of this section you can: describe a wave in terms of amplitude, wavelength, frequency and period; use the wave equation; use the period equation; describe the difference between a transverse wave and a longitudinal wave; and describe a method to measure the speed of sound in air and the speed of ripples on a water surface.

1. What a wave is

A wave transfers energy from one place to another. The material the wave passes through is called the medium. The wave does not carry the medium with it.

Water ripples spread across a pond. Sound travels from a speaker to your ear. Light travels from a lamp to a book. In every case, energy moves from one place to another, but the water, the air and the light path stay where they are.

A wave made in one short push is called a pulse. A wave that repeats is described by four quantities: amplitude, wavelength, frequency and period.

2. Amplitude, wavelength, frequency and period

The amplitude of a wave is the maximum displacement of a point on the wave from its undisturbed position. A large amplitude means a large vibration.

The wavelength is the distance from a point on one wave to the equivalent point on the next wave. It is measured in metres. The symbol for wavelength is the Greek letter lambda, $\lambda$ .

The frequency is the number of waves that pass a point each second. It is measured in hertz (Hz). One hertz is one wave per second.

The period is the time for one complete wave to pass a point. It is measured in seconds. The period and the frequency are linked:

\text{period} = \frac{1}{\text{frequency}} \qquad T = \frac{1}{f}

Worked example 1. A wave has a frequency of 4.0 Hz. Calculate its period.

\begin{aligned} T &= \frac{1}{f} \\ &= \frac{1}{4.0} \\ &= 0.25\,\text{s} \end{aligned}

The wave equation links the speed of a wave to its frequency and wavelength:

\text{wave speed} = \text{frequency} \times \text{wavelength} \qquad v = f \lambda

where $v$ is the wave speed in metres per second, $f$ is the frequency in hertz, and $\lambda$ is the wavelength in metres. The equation is given on the equations sheet, so you choose it and use it.

Worked example 2. A ripple tank makes water waves with a frequency of 2.5 Hz and a wavelength of 0.20 m. Calculate the wave speed.

\begin{aligned} v &= f \lambda \\ &= 2.5 \times 0.20 \\ &= 0.50\,\text{m/s} \end{aligned}

Worked example 3. A sound wave in air has a frequency of 440 Hz. The speed of sound in air is 330 m/s. Calculate the wavelength.

\begin{aligned} v &= f \lambda \\ \lambda &= \frac{v}{f} \\ &= \frac{330}{440} \\ &= 0.75\,\text{m} \end{aligned}

Worked example 4. A wave travels along a spring at 6.0 m/s. Its wavelength is 0.40 m. Calculate the frequency.

\begin{aligned} v &= f \lambda \\ f &= \frac{v}{\lambda} \\ &= \frac{6.0}{0.40} \\ &= 15\,\text{Hz} \end{aligned}

Worked example 5. In the chapter problem the ripple tank makes waves with a frequency of 5.0 Hz and a wavelength of 0.12 m. Calculate the wave speed.

\begin{aligned} v &= f \lambda \\ &= 5.0 \times 0.12 \\ &= 0.60\,\text{m/s} \end{aligned}

Worked example 6. A sound wave travels through a steel rail at 5000 m/s. Its frequency is 1000 Hz. Calculate the wavelength.

\begin{aligned} v &= f \lambda \\ \lambda &= \frac{v}{f} \\ &= \frac{5000}{1000} \\ &= 5.0\,\text{m} \end{aligned}

3. The wave travels, not the medium

Put a cork on a pond and watch a ripple pass. The cork moves up and down, but it does not travel across the pond with the wave. The cork returns to the same place. This is evidence that the wave carries energy through the water, and the water itself does not travel along with it.

The same is true of sound. When you speak, the air molecules near your mouth vibrate back and forth. They push the neighbouring molecules, and a compression travels to the listener. The air molecules do not travel from your mouth to the listener.

Try it: the wave equation

The figure draws a snapshot of a wave on a rope. You can change the frequency and the wave speed.

Predict first. Do not move the sliders yet. Write down your answers.

If you double the frequency and keep the wave speed the same, what happens to the wavelength?
If you double the wave speed and keep the frequency the same, what happens to the wavelength?

Test. Set the frequency to 2.0 Hz and the wave speed to 4.0 m/s. Read the wavelength. Now double the frequency, then double the wave speed, and read the wavelength each time.

Explain. The wave equation is $v = f\lambda$ , so $\lambda = \dfrac{v}{f}$ . Why does doubling the frequency halve the wavelength, while doubling the wave speed doubles it? Write your answer in two or three sentences.

4. Transverse and longitudinal waves

Waves are grouped by how the vibration compares with the direction of travel.

In a transverse wave, the vibration is at right angles to the direction in which the wave travels. Ripples on a water surface are transverse waves.

In a longitudinal wave, the vibration is along the same line as the direction in which the wave travels. A longitudinal wave has areas of compression, where the particles are close together, and rarefaction, where the particles are spread out. Sound waves travelling through air are longitudinal waves.

You can make both kinds with a spring. Shake a spring up and down, and a transverse wave travels along it. Push and pull the end of the spring along its length, and a longitudinal wave travels along it.

5. Measuring the speed of a wave

To find a wave speed, measure a distance and the time the wave takes to travel it, and use $v = \dfrac{s}{t}$ .

To measure the speed of ripples on a water surface, make ripples in a ripple tank at a steady frequency. Measure the wavelength from a photograph or a stroboscope, and use $v = f\lambda$ .

To measure the speed of sound in air, stand a measured distance from a large wall or building. Make a short, loud sound, such as a clap, and measure the time until you hear the echo. The sound travels to the wall and back, so the distance is twice the measured distance.

Watch out. Frequency and wavelength are not the same thing, and one is not the other rearranged. Frequency is how many waves pass each second, measured in hertz. Wavelength is the length of one wave, measured in metres. When the wave speed is fixed, doubling the frequency halves the wavelength, because $v = f\lambda$ . Also, a wave carries energy and not the medium, so the water in a ripple tank does not travel across the tank with the ripples.

Progress check

Return to the chapter problem. The echo sounder sends sound through sea water at a frequency of 50 kHz. The speed of sound in sea water is 1500 m/s. Work out the wavelength of the sound wave. Give your answer in metres.

You should now be able to

describe a wave as a transfer of energy through a medium;
define amplitude, wavelength, frequency and period, with their units;
use $T = \dfrac{1}{f}$ to link period and frequency;
use $v = f\lambda$ to find a wave speed, a frequency or a wavelength;
describe the difference between a transverse wave and a longitudinal wave, and give an example of each;
explain the evidence that a wave transfers energy without transferring the medium;
describe a method to measure the speed of sound in air with an echo.