Explanation of Stationary Waves

When a wave travelling in one direction along a string meets a rigid boundary, it reflects back into the original medium and interferes there with the wave still arriving from behind it. Superposing two harmonic progressive waves of equal amplitude $A$ and equal angular frequency $\omega$, moving in opposite directions -- $y_1=A\sin(kx-\omega t)$ travelling rightward (the incident wave) and $y_2=A\sin(kx+\omega t)$ travelling leftward (the reflected wave) -- and adding them using trigonometric identities gives the combined result $y(x,t)=y_1+y_2=2A\cos(\omega t)\sin(kx)$. This can be rewritten in the compact form $y(x,t)=A'\cos(\omega t)$, with position-dependent amplitude $A'=2A\sin(kx)$, showing that every particle at a given fixed position $x$ still executes simple harmonic motion in time, but with an amplitude $A'$ that itself varies from position to position along the string rather than being the same everywhere. Unlike a genuine travelling wave, this combined pattern does NOT itself move forward or backward through the medium at all -- it remains steady in place, oscillating up and down without ever advancing -- which is exactly why it is called a stationary wave or standing wave. The amplitude $A'$ reaches its maximum possible value, $2A$, at points called antinodes, located where $\sin(kx)=1$, i.e. at positions $x=(2m+1)\lambda/4$ for $m=0,1,2,\ldots$; and it falls to exactly zero at points called nodes -- which therefore never vibrate at all -- located where $\sin(kx)=0$, i.e. at position …