Travelling Pulses (Wave Packets)

time

-4

0

4

8

12

initial x

-8

-6

-4

final x

2

4

8

12

rational

0

0.5

1

Jacobi

0

0.5

1

sech

0

0.5

1

exp

0

0.5

1

2

Jacobi

0

0.5

1

2

sech

0

0.5

1

12

cos

0

0.5

1

Gauss

0.5

1

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,

t

, 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.

The "rational pulse" has a square-law decay with an effectively infinite range. The "power sinusoid"

2

cos

(πx/(2n))

approaches the Gaussian curve as

n→∞

;

12

cos

(x/3.56)

(using 3.56 in place of

12/π~3.82

) 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

xf=12

,

t=0

). The

sech

and

2

sech

pulses are "single soliton" solutions to the MKdV and KdV equations; they are the limiting cases of the

JacobiCN(x-vt,K)

and

2

JacobiCN(x-vt,K)

pulse trains (the parameter

K

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.