Question

A traveling wave has displacement given by y(x, t) = (2.0 cm) × cos(2π − 4πt), where x is measured in cm and t in s.

a. Draw a snapshot graph of this wave at t = 0 s.
b. On the same set of axes, use a dotted line to show the snapshot graph of the wave at t = 1/8 s.
c. What is the speed of the wave?

Answer 1

a)
`y(x,t=0)= 2cm cos(2\pi x -4\pi *0) = 2cm cos(2\pi x)`

And we can see the plot in the first figure attached.

b)
`y(x,t=1/8)= 2cm cos(2\pi x -4\pi *(1)/(8)) = 2cm cos(2\pi x-(\pi)/(2))`

And we can see the result on the second figure attached.

As we can see we have a translation on the x axis for this new function.

c)
`(dx)/(dt)=-8\pi cm sin(2\pi x -4\pi t)`

Step-by-step explanation:

For this case we have the following function given (assumed):

`y(x,t) = 2 cm cos (2\pi x -4\pi t)`

Where x is in cm and t in seconds

Part a

For this case we need to replace the value of t =0 and we got:

`y(x,t=0)= 2cm cos(2\pi x -4\pi *0) = 2cm cos(2\pi x)`

And we can see the plot in the first figure attached.

Part b

For this case we just need to replace the value of t =1/8 s and we have the following function:

`y(x,t=1/8)= 2cm cos(2\pi x -4\pi *(1)/(8)) = 2cm cos(2\pi x-(\pi)/(2))`

And we can see the result on the second figure attached.

As we can see we have a translation on the x axis for this new function.

Part c

The velocity on this case is given by the first derivate of the position respect to the x axis and we got:

`(dx)/(dt)= -2cm * (4\pi) sin (2\pi x -4\pi t)`

`(dx)/(dt)=-8\pi cm sin(2\pi x -4\pi t)`