What Determines the Position of Antinodes in Standing Waves?

[Nubs]

The eq. of standing waves is y(x,t) = 2Asin(kx)cos(wt). Derive the eq. for postion of antinodes.

I know I can figure it out by assuming sin(kx) = 1, but I can't figure out how to get from there to final eq.

Any help? Thanks.

 

[Redbelly98]

Welcome to PF

Can you solve this related equation:

sin θ = 1​

That will help with finding the solution to sin(kx)=1.

 

[nlightner]

To derive the equation for the position of antinodes, we first need to understand what antinodes are in the context of standing waves. Antinodes are points along a standing wave where the amplitude of the wave is at its maximum. In other words, they are points of constructive interference where the crests and troughs of the wave align.

In the standing wave equation provided, y(x,t) = 2Asin(kx)cos(wt), the term sin(kx) represents the vertical displacement of the wave at a given point x and time t. This displacement is directly related to the amplitude of the wave, represented by A.

To find the position of antinodes, we need to find the values of x where sin(kx) = 1. This can be accomplished by solving for x in the equation sin(kx) = 1. We can rewrite this equation as kx = π/2, since sin(π/2) = 1. Solving for x, we get x = π/(2k).

Therefore, the position of antinodes along the standing wave is given by x = π/(2k), where k is the wave number. Substituting this into the original equation, we get y(π/(2k),t) = 2Acos(wt).

In summary, the equation for the position of antinodes is x = π/(2k) and the corresponding vertical displacement is given by y(x,t) = 2Acos(wt). This equation tells us that the position of antinodes is directly proportional to the wave number, and the vertical displacement at these points is equal to twice the amplitude of the wave.