In a standing wave, the amplitude of a point does not vary with time because it is determined by the spatial distribution of the wave, rather than the time-dependent component of the wave.
A standing wave is formed by the superposition of two waves of the same frequency and amplitude traveling in opposite directions. The resulting pattern appears stationary or "standing" because the constructive and destructive interference between the waves creates regions of maximum and minimum amplitudes at specific locations.
The mathematical expression for a standing wave can be written as:
y(x, t) = A sin(kx) cos(ωt)
In this equation, A represents the amplitude of the wave, k is the wave number (related to the wavelength), x is the spatial coordinate, ω is the angular frequency, and t is the time. The term cos(ωt) represents the time-dependent component, while sin(kx) represents the spatial distribution.
If we consider a fixed point along the wave, the amplitude at that point is given by A sin(kx). This part of the equation depends only on the spatial coordinate x and not on time. The time-dependent component cos(ωt) affects how the wave varies with time at any given point, but it does not influence the amplitude at a fixed point along the wave.
In summary, the amplitude of a point in a standing wave is determined solely by the spatial distribution of the wave and remains constant over time, regardless of the presence of the cos(ωt) term in the equation.