Question

Let ?(?) be a unit step function defined as:

?(?)=1, −1≤?≤1

0,   ??ℎ??????

a) Sketch ?(?−??), where ?=0.5 for ?=0 ,   ?=1,   ?=2 on the same z-axis versus ?(?−??) graph. What do you observe as t increases? How this can be related to the wave motion?
b) Sketch ?(?+??), where ?=0.5 for ?=0 ,   ?=1,   ?=2 on the same z-axis versus ?(?+??) graph. Comment on the differences with part (a).
c) Sketch ?(?−??) where ?=1 for ?=0 ,   ?=1,   ?=2 on the same z-axis versus ?(?−??) graph. What do you observe for different v values? How this can be related to the speed of the propagation?

Answer 1

solution:

Solution of the problem is:

a)

To sketch u(z-Vt)

where V=0.5

at=>t=0

u(z-0.5*0)=u(z)

*u(z) will be the same as we plotted above

at t=1

u(z-Vt) becomes

u(z-0.5*1)=u(z-0.5)

at t=2

*u(z-o.5*2)=u(z-1)

at t=0

at t=1

The graph shifted 0.5 unit towards right on z-axis

at t=2

*The graph shifted towards right by 1 unit on z-axis.

*As we increase t, the graph shifts towards right more and more

*Thus we can say that it is a wave travelling in +z direction

#Conclusion from the above observation is that u(z-Vt) is a wave travelling in +z direction

b)

u(z+Vt)

at t=0

v(z+0.5*0)=u(z)

at t=1

u(z+0.5*1)=u(z+0.5)

at t=2

u(z+0.5*2)=u(z+1)

*Now plotting for all

t=0,t=1 and t=2.

at t=0,1 and 2

at t=1

*Thus we can observe that the graph is travelling in -ve z direction like a wave with speed 0.5

#Conclusion:

It (u(z+Vt)) represents a wave travelling in -z direction

*The wave u(z-Vt) was travelling in +z direction and the wave in part(b) u(z+Vt) is travelling in -z direction

#(c) To sketch

u(z-Vt)

where V=1 at t=0,1,2

at t=0

u(z-Vt)=u(z-1*0)

=u(z)

at t=1

u(z-Vt)=u(z-1*1)

=u(z-1)

at t=2

u(z-Vt)=u(z-1*2)

=u(z-2)

Now plotting for each value of t

*As we increase the value of V the speed of shifting of graph increases

*at V=1 the graph shifts or travels with 1 unit per second

*Thus 'V' behaves like the speed of propagation of wave.