Derive the following displacement relation for a plane progressive wave.y(x,t)=asin(kx-wt+theta)
Answers
y(x, t) = a sin (kx + ωt + φ )
Explanation:
A wave which travels continuously in a medium in the same direction without the change in its amplitude is called a traveling wave or a progressive wave.
Let y(x,t) be the displacement of an element at a position x and time t about the y-axis. Considering the wave as periodic and sinusoidal, the displacement of the element at a position x and time t, from the y-axis will be:
y (x, t ) = a sin (kx – ωt + φ ) -------------(1)
Writing the equation using sine and cosine functions, we get:
y (x, t) =A sin (kx – ωt ) + B cos (kx – ωt ) --------------(2)
Taking a = √(A² + B²) and ∅ = tan⁻¹ B/A
The equations (1) and (2) represent the transverse wave moving along the X-axis.
The shape of the wave can be determined at any given time.
y(x, t) = a sin (kx + ωt + φ )
The above equation represents a transverse wave moving along the negative direction of the X-axis.
Please refer the attached picture for the diagram.
