Question

The displacement of a plane progressive wave in a medium, traveling towards the positive x-axis with velocity 4m/s at t=0 is given by $y=3sin2\pi(-\frac{x}{3})$. Then the expression for the displacement at a later time t=4 sec will be

• A. $y=3\,sin2\pi(-\frac{x-16}{3})$
• B. $y=3\,sin2\pi(\frac{-x-16}{3})$
• C. $y=3\,sin2\pi(\frac{-x+1}{3})$
• D. $y=3\,sin2\pi(\frac{-x-1}{3})$

Answer 1

Answer: (A)

$y=3\,sin2\pi(-\frac{x-16}{3})$

Answer 2

The given wave equation is $y=3sin(2π(−3x​))$, representing a plane progressive wave traveling towards the positive x-axis with a velocity of 4 m/s at t=0.
To justify the answer $y=3sin(2π(−3x−16))$, we need to account for the wave's velocity. The general form of a plane wave equation is $y=Asin(kx−ωt+ϕ)$, where k is the wave number, ω is the angular frequency, and ϕ is the phase angle.
Given that the wave's velocity is 4 m/s, we know that $v=kω$​, and in this case, v=4. By equating v with the phase velocity kω ​, we can find the relationship between ω and k. Then, we substitute these values into the general equation and adjust the phase angle ϕ to match the initial condition at t=0.
This process leads to the answer $y=3sin(2π(−3x−16))$, which correctly represents the displacement of the wave as it travels towards the positive x-axis with the given velocity and initial condition.
The correct option is(A): $y=3\,sin2\pi(-\frac{x-16}{3})$