# Why standing waves are called standing?

## Question:

Why standing waves are called standing?

## Waves

A wave is a disturbance of a physical quantity traveling in space. The simplest form of a wave is a sine wave propagating along one dimension {eq}x {/eq} in a time {eq}t {/eq}, characterized by {eq}A(x,t) = A_0 \, \sin{(k \, x - \, \omega \, t)} {/eq}, where A can stand for displacement (wave on a string or on water), pressure (sound wave) etc. The characteristic parameters for the propagation are the wave number {eq}k = 2 \pi / \lambda {/eq} and the angular frequency {eq}\omega = 2 \pi \, f {/eq}. From this, we get the practically relevant quantities frequency ({eq}f {/eq}) and wavelength ({eq}\lambda {/eq}). The wave propagates at a speed {eq}v = \lambda \cdot f {/eq}.

## Answer and Explanation:

A forward-moving wave (positive x-direction) and a backward-moving wave (negative x-direction) of equal amplitude, frequency, and wavelength interfere such that a wave pattern results, which does not move into x-direction at all. Not moving equals standing, therefore the name. Mathematically, this reads: $$A \, \sin{(k \, x - \, \omega \, t)} + A \, \sin{(- k \, x - \, \omega \, t)} = A \, \sin{(\omega \, t)} \, \sin{(k \, x)}$$ We see that time and location are separated now, so that each point at a fixed {eq}x {/eq} undergoes an oscillation that is not traveling anymore.