Plotting Transverse Wave Pulse: f(x,t) at t=0 & t=2s

[Squires]

A transvers wave pulse travels to the right (+x direction) along a string with a speed v=3m/s. At t=0, the shape of the pulse has the form:

f(x)=Ae^-(x^2/b^2), where A=3m and b=2m.

a) Plot the variation of f(x) with x at t=0.
b) What will the mathematical description of the pulse as a function of time, f(x,t), assuming that the pulse moves without changing its shape.
c)Sketch the profile of the wave pulse at t=2s.


I'm guessing the position with time is dependant on the velocity, but I'm totally confused at what the variation will even look like, as I've never come across an exponential to x^2 function before.

Thanks for looking.

 

[NateD]

a) The variation of f(x) with x at t=0 can be plotted by substituting t=0 into the equation. This gives:f(x)=Ae^-(x^2/b^2), where A=3m and b=2m.This gives us a graph of f(x) = 3e^-(x^2/4).b) The mathematical description of the pulse as a function of time, f(x,t), assuming that the pulse moves without changing its shape is given by:f(x,t) = Ae^-( (x-vt)^2/b^2 )Where v is the speed of the wave pulse.c) The sketch of the wave pulse at t=2s can be obtained by substituting t=2s into the equation from part (b). This gives us:f(x,t) = Ae^-( (x-2v)^2/b^2 )The profile of the wave pulse at t=2s will be a bell-shaped curve with its peak located at x=6m (2v).