For the standing wave, the parts of the body oscillate in harmonic motion at the same angular frequency ω and with the same phase constant ϕ. Thus y (x, t) should have the same time dependence, sin (ωt + ϕ), for all particles i.e., for all x. In this case, the amplitude of vibration at x can be written as a continuous function of x denoted by A (x). We can write the general expression for standing wave as
y(x, t) = A (x) sin (ωt + ϕ).
Substituting this into the wave equation, we get
Thus, the general solution for the displacement in the standing wave is
y(x, t)=(A sin kx + B cos kx) sin (ωt + ϕ).
For the standing waves the space and time variables are no longer associated together as (x ± vt).
