How waves carry energy

TL;DR:

• A wave moves energy from place to place while the material only moves back and forth around where it was.

• Speed, frequency and wavelength are tied together by v = fλ. A 440 Hz sound in air has a wavelength of about 0.78 m.

• Wave energy grows with the square of the amplitude. Doubling the amplitude gives four times the energy.

Drop a pebble in a pond and a ring spreads outward. Wait a moment and a leaf floating near the edge bobs up and down when the ring reaches it. The leaf did not travel across the pond. The disturbance did, and it carried energy with it.

Energy moves, matter stays

A wave is a pattern of disturbance that travels through a medium, or in the case of light, through empty space. Each bit of the medium moves a little and returns. The energy is handed along from one bit to the next.

That is why you can hear a person across a room without any air travelling from their mouth to your ear. The air molecules near them push on the ones next to them, and so on. Each molecule shuffles back and forth over a tiny distance.

Transverse and longitudinal waves

In a transverse wave the medium moves at right angles to the direction the wave travels. A wave on a rope, flicked up and down, is the classic case. In a longitudinal wave the medium moves back and forth along the direction of travel, forming squeezed regions (compressions) and stretched regions (rarefactions). Sound in air is longitudinal. Light is transverse, though it needs no medium.

The wave equation, v = fλ

Three numbers describe a repeating wave. The frequency f is how many cycles pass a point each second, in hertz (Hz). The wavelength λ is the distance from one crest to the next, in metres. The speed v is how fast the pattern moves.

In one second, f crests go past, and each takes up a length λ, so the pattern has moved a distance f × λ. That is the speed: v = fλ. The period, the time for one cycle, is T = 1/f.

The speed is set by the medium, not the source. In a given material at a given temperature, a higher frequency simply means a shorter wavelength.

Worked example: the wavelength of a 440 Hz note

Sound travels in air at about 343 m/s at 20 °C. Find the wavelength and period of a 440 Hz tone, the standard tuning note A.

1. Rearrange v = fλ to λ = v / f.

2. Substitute: λ = 343 / 440.

3. Compute: 343 ÷ 440 = 0.7795, so λ ≈ 0.78 m.

4. Period: T = 1 / 440 = 0.00227 s, about 2.3 ms.

Now check by multiplying back: f × λ = 440 × 0.78 = 343.2 m/s, which matches the speed we started with. Also check units: Hz × m = (1/s) × m = m/s.

What if the same note moves into water, where sound travels at about 1480 m/s? The frequency stays at 440 Hz, because the source sets it. The wavelength becomes 1480 / 440 ≈ 3.4 m, over four times longer. The speed changed and the frequency did not, so the wavelength had to change.

How amplitude relates to energy

The amplitude is the biggest displacement from rest. For sound it relates to loudness, and for a rope wave to how far the rope swings. The energy a wave carries goes with the square of its amplitude.

Take a wave and double the amplitude. Each bit of the medium moves twice as far in the same time, so it moves twice as fast, and kinetic energy goes with speed squared: 2² = 4. The wave carries four times the energy. Triple the amplitude and it is 3² = 9 times.

Do not confuse amplitude with frequency. A higher-pitched sound is not necessarily louder, and a louder sound is not higher in pitch.

How to check your answer

• Round trip. Multiply your wavelength back by the frequency and see if you get the speed.

• Direction check. Higher frequency in the same medium must give a shorter wavelength.

• Sensible size. Audible sound has wavelengths from centimetres to a few metres, so 0.78 m is reasonable.

• Units. Convert kilohertz to hertz and centimetres to metres before you start.

Loud sound can hurt: More amplitude means more energy reaching your ear. Loud sound over time can damage hearing for good, so keep headphone volumes down and wear ear protection around loud machinery.

Mistakes that cost marks

• Thinking the medium travels along with the wave.

• Changing the frequency when the wave enters a new medium. The frequency stays the same, and the speed and wavelength change.

• Calling sound transverse. In air it is longitudinal.

• Using centimetres in v = fλ with a speed in m/s.

• Saying energy doubles when amplitude doubles. It is four times.

Where you will use it next

The same equation holds for light, radio and water waves, and it explains the bending in refraction, where speed changes at a boundary. Light itself comes from the changing fields in electric and magnetic fields in plain words.

Frequently asked questions

Why can you hear around a corner but not see around it?

Sound wavelengths are about the size of doorways, so the waves spread out (diffract) as they pass. Light has far shorter wavelengths, so it bends much less.

Does a wave with a higher frequency always travel faster?

No. In a uniform medium the speed is fixed by the medium, and a higher frequency just packs in shorter wavelengths.