Question
Question: A progressive wave whose displacement equation is given by: \({\text{y = A sin(2}}\omega {\text{t - ...
A progressive wave whose displacement equation is given by: y = A sin(2ωt - kx/2) falls on a wall normal to its surface and gets reflected. At what minimum distance in front of the wall the particles of air will be vibrating with maximum amplitude?
A. kπ
B. k2π
C. 2kπ
D. 4kπ
Solution
The wave which travels continuously in a medium in the same direction without any change in its amplitude is known as a travelling wave or a progressive wave. Amplitude can be defined as the magnitude of maximum displacement of a particle in a wave from the equilibrium position.
Complete step by step answer:
A progressive wave can be transverse or longitudinal. During the propagation of a wave through a medium, if the particles of the medium vibrate simply harmonically about their mean positions, then the wave is called a plane progressive wave.
Generally, the displacement of a sinusoidal wave propagating in the positive direction of x-axis is given by: y(x,t)=asin(kx−ωt+ϕ)
Where a=amplitude of the wave, k=angular wave number and ω=angular frequency.
(kx−ωt+ϕ)=phase
ϕ=phase angle
The equation given in the question is y = A sin(2ωt - kx/2).
At time t=0:
±A = A sin( - kx/2)
⇒sin( - kx/2) = ±1
⇒2kx=2π
∴x=kπ
Therefore, option A is the correct answer.
Note: The phase describes the state of motion of the wave. The points on a wave that travel in the same direction and rise and fall together are said to be in phase with each other. The points on a wave which travel in opposite directions to each other such that one is falling and another one is rising, are said to be in anti-phase with each other.