Wave Relation Cross-Scans

Wavefront Distance Three-Row Scan

a crest moves at speed

Three wavefront rows hold speed fixed while elapsed time moves a crest farther down the path.

waves At 3 m/s, travel times 2 s, 4 s, and 6 s move a crest 6 m, 12 m, and 18 m.

Example

With wave speed fixed, a tracked crest travels distance equal to speed times time. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Wavefront distance is speed times time

Track one crest as it moves. With speed fixed, elapsed time controls how far that crest has travelled.

x=vtx=vt

Short travel time

After 2 s, the crest has travelled 6 m.

x=vt=3 m/s ⋅ 2 s=6 mx=vt=3\ \text{m}/\text{s}\,\cdot\,2\ \text{s}=6\ \text{m}
Wavefront travel rowThe crest starts at the marker and travels along the axis.frontstartx

Middle travel time

After 4 s, the crest has travelled 12 m.

x=vt=3 m/s ⋅ 4 s=12 mx=vt=3\ \text{m}/\text{s}\,\cdot\,4\ \text{s}=12\ \text{m}
Wavefront travel rowThe crest starts at the marker and travels along the axis.frontstartx

Long travel time

After 6 s, the crest has travelled 18 m.

x=vt=3 m/s ⋅ 6 s=18 mx=vt=3\ \text{m}/\text{s}\,\cdot\,6\ \text{s}=18\ \text{m}
Wavefront travel rowThe crest starts at the marker and travels along the axis.frontstartx

The crest position advances at constant speed

The speed column is fixed. Each longer time gives a farther crest position on the diagram and in the table.

vtx3 m/s2 s6 m3 m/s4 s12 m3 m/s6 s18 m\begin{array}{c|c|c}v & t & x \\ \hline 3\ \text{m}/\text{s} & 2\ \text{s} & 6\ \text{m} \\ 3\ \text{m}/\text{s} & 4\ \text{s} & 12\ \text{m} \\ 3\ \text{m}/\text{s} & 6\ \text{s} & 18\ \text{m}\end{array}