Section 5.5 · Home exercises · 2 of 4
Home exercise 5.5.2: A pulse released at rest
A pulse released at rest
#At a field is at rest, , in a lossless medium with wave speed . Using d'Alembert's solution and the two initial conditions, find and . Sketch at , and .
d'Alembert's solution has two unknown functions and there are two initial conditions, one on the value and one on the time derivative. Write both at as equations between , and their derivatives, and solve.
At , with and the derivatives with respect to the argument,
The second equation gives , so with a constant . Then and
and the constant cancels in the sum.
At the two half-height copies are centred at and still overlap: , a broad hump with two shoulders. At they are centred at with peaks (plus a negligible from the other copy), and at only is left (Figure 5.5.19).
At the formula returns , and is zero at . Each term is of the form , so the wave equation holds.
A field released at rest has no preferred direction, so it splits into two equal halves running apart at . To launch a single pulse in the direction one also has to give it the right initial rate of change, . For an electromagnetic field this means the right magnetic field from the start, , as Section 6.1 will show. Exercise 5.5.4 gives the field an initial kick instead.