Describing Waves

Wave Speed

v = wavelength x frequency

Because one wavelength passes every period, the wave travels at wavelength times frequency.

Example

Because one wavelength passes every period, the wave travels at wavelength times frequency. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

One wavelength passes every period

Put the space and time views together. In one period the wave slides forward by exactly one wavelength. So the wave's speed is a wavelength per period — which, since frequency is cycles per second, is the same as wavelength times frequency.

v=λT=λ fv = \frac{\lambda}{T} = \lambda\,f
The pattern slides to the rightThe wave now and, a moment later, the same shape shifted to the right; the crest has moved along, so the whole pattern travels.crest movesa moment later

A worked value

Multiply the wavelength by the frequency: 3 metres times 2 per second is 6 metres per second. The metres-times-per-second gives metres per second, a speed.

v=λ f=3 m ⋅ 2 Hz=6 m/sv = \lambda\,f = 3\ \text{m} \,\cdot\, 2\ \text{Hz} = \hl{6}\ \text{m}/\text{s}

Same wavelength, higher frequency, faster wave

Hold wavelength fixed. More cycles passing each second means more metres of pattern pass each second.

λfv3 m1 Hz3 m/s3 m2 Hz6 m/s3 m3 Hz9 m/s\begin{array}{c|c|c}\lambda & f & v \\ \hline 3\ \text{m} & 1\ \text{Hz} & 3\ \text{m}/\text{s} \\ 3\ \text{m} & 2\ \text{Hz} & 6\ \text{m}/\text{s} \\ 3\ \text{m} & 3\ \text{Hz} & 9\ \text{m}/\text{s}\end{array}

Same frequency, longer wavelength, faster wave

Hold frequency fixed. A longer cycle means each repeated pattern covers more metres, so the wave speed is larger.

λfv1 m2 Hz2 m/s2 m2 Hz4 m/s3 m2 Hz6 m/s\begin{array}{c|c|c}\lambda & f & v \\ \hline 1\ \text{m} & 2\ \text{Hz} & 2\ \text{m}/\text{s} \\ 2\ \text{m} & 2\ \text{Hz} & 4\ \text{m}/\text{s} \\ 3\ \text{m} & 2\ \text{Hz} & 6\ \text{m}/\text{s}\end{array}
waves Clean 3 m wavelength at 2 Hz gives an exact 6 m/s.