$6-$ The wave$-$particle duality may be well understood by considering that the

electron can be represented by a combination of several wave trains having

slightly different frequencies $$ and $+$ and different wave numbers, $k$

and $k+k.$ Assume two waves ${}_1$ and ${}_2$ given by:

${}_1=sin[kx-t]$

${}_2=sin[(k+k)x-(+)t].$

Show that their supersposition yields:

${}_1+{}_2==$ubrace$(2cos(\frac{}{2}t-\frac{k}{2}x)ubrace{)}_{\{\begin{array}{ccc}\text{:}[& \text{M}\text{o}\text{d}\text{u}\text{l}\text{a}\text{t}\text{e}\text{d}& ]\\ & \text{a}\text{m}\text{p}\text{l}\text{i}\text{t}\text{u}\text{d}\text{e}& \end{array}}*ubrace(sin[(k+\frac{k}{2})x-(+\frac{}{2})t]ubrace{)}_{\text{S}\text{i}\text{n}\text{e}\text{w}\text{a}\text{v}\text{e}}.$