Wave Behavior

Superposition

overlapping waves add

When waves overlap, the displacement at every point is just the sum of the individual waves.

Example

When waves overlap, the displacement at every point is just the sum of the individual waves. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Overlapping waves add

When two waves pass through the same place at the same time, the string's displacement is simply the sum of what each wave alone would give — added point by point, positive and negative together. This adding-up is called superposition.

ytotal=ya+yb(at every point)y_{\text{total}} = y_a + y_b \quad \text{(at every point)}
Two waves add to a combined shapeTwo component waves and, in purple, their point-by-point sum.sum

At a point where both push up

Take a point where both waves are at their crest, each lifting the string up by 1 metre. The total there is 1 plus 1, which is 2 metres.

ytotal=1+1=2 my_{\text{total}} = 1 + 1 = \hl{2}\ \text{m}

Point by point, add the signed displacements

Superposition is local arithmetic. At each position, add the two signed displacements there: up is positive, down is negative, and zero means the point is on the rest line.

yaybytotal1 m1 m2 m1 m0 m1 m1 m−1 m0 m\begin{array}{c|c|c}y_a & y_b & y_{\text{total}} \\ \hline 1\ \text{m} & 1\ \text{m} & 2\ \text{m} \\ 1\ \text{m} & 0\ \text{m} & 1\ \text{m} \\ 1\ \text{m} & -1\ \text{m} & 0\ \text{m}\end{array}
waves Two equal waves in phase add 1 + 1 = 2 at the crests, point by point.