Question

A transverse wave is described by the function y(x,t)=2.3cos(4.7x+12t−π/2), where distance is measured in meters and time in seconds. How long does it take for the point at x = 0 to move from a displacement of 0 m to a displacement of 1.1 m?

Answer 1

`\Delta t=4.988\ s`

Step-by-step explanation:

Given:

• equation of transverse wave,
  `\rm y(x,t)=2.3\ cos(4.7x+12t-(\pi)/(2) )`

• initial position of point x=0,
  `y_i=0\ m`

• final position of point x=0,
  `y_f=1.1\ m`

Now putting initial condition in the wave equation:

`0=2.3* cos(4.7* 0+12t_i-(\pi)/(2) )`

`cos^(-1)(0)=12t_i-(\pi)/(2)`

encountering the first occurrence:

`(\pi)/(2) =12t_i-(\pi)/(2)`

`t_i=(\pi)/(12)=0.262\ s` .........................(1)

Now putting final condition in the wave equation:

`1.1=2.3* cos(4.7* 0+12t_f-(\pi)/(2) )`

`cos^(-1)((1.1)/(2.3) )=12t_f-(\pi)/(2)`

encountering the first occurrence:

`61.43 =12t_f-(\pi)/(2)`

`t_f=5.25\ s` .........................(2)

Now time elapsed:

`\Delta t=t_f-t_i`

`\Delta t=5.25-0.262`

`\Delta t=4.988\ s`

Answer 2

0.22 second

Step-by-step explanation:

y = 2.3 Cos (4.7 x + 12 t - π/2)

at x = 0, y = 1.1 m , t = ?

Substitute the values in the given equation

1.1 = 2.3 Cos (4.7 x 0 + 12t - π/2)

Cos (12t - π/2) = 0.4783

(12t - π/2) = 1.07

12 t = 2.64

t = 0.22 second

Thus, the time taken is 0.22 second.