The correct option is B 54+9(x−4t+2)2
Given, at t=0
y(x,0)=54+(3x+6)2
Re-writing the above equation, we get
y(x,0)=54+9(x+2)2 .......(1)
If the speed of the pulse is v, then the pulse will travel to the right by a distance vt in time t.
Assuming the shape of the pulse does not change with time, the transverse position of the particles for all positions and times, measured in a stationary frame with the origin at O can be written as
y(x,t)=y(x−vt,0)
Therefore, replacing x with x−vt in the equation (1), we get
y(x,t)=54+9(x−vt+2)2
Given that , speed of the wave v=4 m/s
∴y(x,t)=54+9(x−4t+2)2
Thus, option (b) is the correct answer.