13 WAVE MOTION 13.4 Wave-pulses
vx −>Figure 110: A wave-pulse propagating down the x-axis. The solid, dotted, and dashed curves show
the wave displacement at three successive and equally spaced times.
a y 1 (x, t) + b y 2 (x, t) (where a and b are arbitrary constants) is also a solution—
this can be seen from inspection of Eq. (13.23). To be more exact, if
y 1 (x, t) = a 1 cos [k 1 (x − v t)] (13.26)represents a wave of amplitude a 1 and wavenumber k 1 which propagates in the
+x direction, and
y 2 (x, t) = a 2 cos [k 2 (x + v t)] (13.27)represents a wave of amplitude a 2 and wavenumber k 2 which propagates in the
−x direction, then
y(x, t) = y 1 (x, t) + y 2 (x, t) (13.28)is a valid solution of the wave equation, and represents the two aforementioned
waves propagating in the same region without affecting one another.
13.4 Wave-pulses
As is easily demonstrated, the most general solution of the wave equation (13.23)
is written
F(x − v t), (13.29)