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

In summary, the equation ψ(y,t)= -2A sinky sin wt represents a standing wave, also known as a stationary wave, which does not propagate through space but rather oscillates in time. This phenomenon occurs when two identical waves traveling in opposite directions interfere with each other, creating a disturbance that appears to stand still. This is why it is called a standing wave.
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noobmaster69
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Homework Statement
Optics help
Relevant Equations
.
ψ(y,t)= -2A sinky sin wt
Why is this called a standing wave?
 
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"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
 
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noobmaster69 said:
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$$\psi(y_0,t)=A \sin(ky_0-\omega t)+A\sin(ky_0-\omega t-\pi).$$
 
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FAQ: Optics homework: Why is this equation called a standing wave?

1. Why is this equation called a standing wave?

The equation for a standing wave is called so because it describes a wave pattern that appears to be standing still, rather than moving forward or backward. This is because the wave is created by the interference of two waves with the same frequency and amplitude, traveling in opposite directions.

2. What is the significance of a standing wave in optics?

Standing waves are important in optics as they can be used to describe the behavior of light, which is an electromagnetic wave. They also have practical applications, such as in the creation of laser beams and in understanding the properties of different materials.

3. How is a standing wave different from a traveling wave?

A standing wave differs from a traveling wave in that it does not propagate or move through a medium. Instead, it appears to be stationary and is created by the superposition of two waves traveling in opposite directions with the same frequency and amplitude.

4. What are the characteristics of a standing wave?

A standing wave has several distinct characteristics, including nodes and antinodes, which are points of minimum and maximum displacement respectively. It also has a fixed wavelength and amplitude, and the nodes and antinodes are located at specific intervals along the wave.

5. How is the equation for a standing wave derived?

The equation for a standing wave is derived from the principles of wave interference and superposition. By combining the equations for two waves traveling in opposite directions with the same frequency and amplitude, we can arrive at the equation for a standing wave.

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