# Travelling Pulses (Wave Packets)

Travelling Pulses (Wave Packets)

Various travelling pulses (or wave packets) are displayed; they have similar central shapes but different decay rates. They are given different velocities, so they separate when the time, , is not zero. The Gaussian curve (with velocity zero) is always shown; the amplitudes of the others can be 0, .5, or 1. The rising arm is labelled if the amplitude is 1.

t

The "rational pulse" has a square-law decay with an effectively infinite range. The "power sinusoid" (πx/(2n)) approaches the Gaussian curve as ; (x/3.56) (using 3.56 in place of ) can be seen to provide a close approximation in the central region (though it is a train of pulses separated by "almost zero" regions—try , ). The and pulses are "single soliton" solutions to the MKdV and KdV equations; they are the limiting cases of the and pulse trains (the parameter has been adjusted to give suitable periods). Their "skirts" are wider than the rapidly decaying (short range) Gaussian pulse, but they have a limited range as the decay is exponential.

2

cos

n→∞

12

cos

12/π~3.82

xf=12

t=0

sech

2

sech

JacobiCN(x-vt,K)

2

JacobiCN(x-vt,K)

K