Waves and Sound · Grade 9

What happens to the wavelength when the frequency doubles and the speed stays the same?

It halves exactly. The speed of a wave equals its frequency times its wavelength, so if the speed is fixed, raising one factor forces the other down in the same proportion. The frequency is set by the source; the speed is set by the medium; the wavelength is simply what follows from the two.

Learning objectives

Four quantities

The wavelength is the distance between two neighbouring identical points on a wave — crest to crest, or trough to trough. It is written with the Greek letter lambda and measured in metres.

The amplitude is the height from the rest line up to a crest, not from trough to crest. It measures how much energy the wave carries, and in sound it is what sets the loudness.

The frequency is the number of complete cycles passing a point in one second, measured in hertz. The period is its opposite: how many seconds one single cycle takes.

The relation that ties it together

The speed of a wave equals its frequency times its wavelength. This holds for every wave there is — sound, light, sea waves and a wave on a rope.

You can see why without memorising it. If five crests pass a point each second, and each crest is two metres from the next, then ten metres of wave went by in that second. Five times two.

One point hides in here and is easily missed: the speed is set by the medium, not by the source. Play a high note and a low note in the same room and both reach your ear at exactly the same speed — otherwise an orchestra could not be listened to at all.

Frequency and period

The period is one divided by the frequency, and the frequency is one divided by the period. A wave at 50 hertz completes a cycle in 0.02 seconds; a wave whose period is 0.25 seconds has a frequency of 4 hertz.

The human ear picks up roughly 20 to 20,000 hertz. An FM radio station broadcasts near 100 megahertz — a hundred million cycles a second — and that is an electromagnetic wave, not a sound wave.

In calculation questions the only real trap is units. Frequency in hertz, wavelength in metres, speed in metres per second. Centimetres or milliseconds left sitting in the formula are the commonest source of error in this whole topic.

Worked examples

  1. A wave has a frequency of 5 hertz and a wavelength of 2 metres. What is its speed?

    1. Speed = frequency times wavelength
    2. 5 times 2

    Answer: 10 metres per second

  2. Sound travels through air at 340 metres per second. What is the wavelength of a 170 hertz note?

    1. Wavelength = speed divided by frequency
    2. 340 divided by 170

    Answer: 2 metres

  3. One cycle of a wave lasts 0.02 seconds. What is the frequency?

    1. Frequency = 1 divided by the period
    2. 1 divided by 0.02

    Answer: 50 hertz

Common mistakes

Measuring amplitude from trough to crest
Amplitude runs from the rest line to a crest only. Trough to crest is twice that, so this mistake doubles every answer that depends on it.
Thinking a higher note travels faster
The speed of sound is set by the medium, not by the frequency. If different notes moved at different speeds, a tune heard from a distance would arrive scrambled.
Confusing frequency with period
They are reciprocals. A high frequency means a short period, and substituting one for the other turns the answer upside down rather than merely shifting it.

What to remember

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