Optics homework: Why is this equation called a standing wave?

[noobmaster69]

Homework Statement

Optics help

Relevant Equations

.

ψ(y,t)= -2A sinky sin wt



Why is this called a standing wave?

 

[docnet]

"In physics, a standing wave, also known as a stationary wave, is a wave which oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with time, and the oscillations at different points throughout the wave are in phase."

source: https://en.wikipedia.org/wiki/Standing_wave

 

[kuruman]

Homework Statement:: Optics help
Relevant Equations:: .
ψ(y,t)= -2A sinky sin wt
Why is this called a standing wave?

The equation that you wrote is mathematically equivalent to the sum of two waves of identical wavenumber
$k$ and angular frequency $\omega$ traveling in opposite directions, $$ \psi(y,t)=A \sin(ky-\omega t)+A\sin(-ky-\omega t).$$Many people erroneously believe that, when two identical waves traveling in opposite directions are added, they "cancel each other out". This is not the case. Think of waves as disturbances of the medium they travel in. In contradistinction to producing no disturbance at all, two identical disturbances traveling in opposite directions produce a disturbance that goes nowhere, i.e. a standing wave.

To "cancel each other out" at a given point in space $y_0$, the added identical disturbances must be traveling in the same direction and have a phase difference of π at all times. The name for this is "destructive interference" and is mathematically described as the sum of two waves thusly$$\psi(y_0,t)=A \sin(ky_0-\omega t)+A\sin(ky_0-\omega t-\pi).$$