Recognizing Standing Waves: Simple Equation Analysis

[nil1996]

Homework Statement

Hello everyone,
I am stuck with a basic concept.How do we can recognize if a given equation is wave or not?For example we know that the standing waves equation is y=2Acos(kx)sin(ωt). In my textbook there is a problem which says that two waves having only different amplitudes are traveling on opposite directions on x-axis. They have asked if the waves form standing waves.

Homework Equations

The Attempt at a Solution

First of all i wrote the following equations.
y₁=A₁sin(kx-ωt) ; y₁=A₂sin(kx+ωt)
i added them and got
y₁+y₂= (A₂+A₁)sin(kx)cos(ωt)+(A₂-A₁)coskxsinωt

how do i recognize if this equation represents a standing wave or not??

 

[NihalSh]

The standing waves have pretty neat definition.

Standing wave is formed when two waves of the same amplitude and frequency, traveling with the same speed in opposite directions interfere (add up).

I guess, this gives you the answer!

For a given function $h$ to be a traveling wave, it must be of the form $h(x-vt)$, where v is velocity of wave.

 

[nil1996]

The standing waves have pretty neat definition.



I guess, this gives you the answer!

For a given function $h$ to be a traveling wave, it must be of the form $h(x-vt)$, where v is velocity of wave.

i was thinking the same but my book says that it forms standing waves and also transports energy

 

[NihalSh]

i was thinking the same but my book says that it forms standing waves and also transports energy

standing waves in general form pattern in which point of minimum and maximum displacement don't change with time, moreover in general there is no transport of energy over large distance.

Maybe author of your book wants to make a point!

You can check this out at: http://www.physics.buffalo.edu/claw/Page15/ProjectCLAW-P15.html

Edit: The equation you found can be considered to be superposition of two standing waves. But it won't reproduce pattern characteristic of standing waves!

 

[nil1996]

Edit: The equation you found can be considered to be superposition of two standing waves. But it won't reproduce pattern characteristic of standing waves!

I got it from a simulation at http://www.uAlberta.ca/~pogosyan/teaching/PHYS_130/FALL_2010/lectures/lect12/lecture12.html

if the amplitudes are not equal then the wave formed is a wave both traveling and standing.This wave transmites energy as it is partially traveling.

 

[NihalSh]

I got it from a simulation at http://www.uAlberta.ca/~pogosyan/teaching/PHYS_130/FALL_2010/lectures/lect12/lecture12.html

if the amplitudes are not equal then the wave formed is a wave both traveling and standing.This wave transmites energy as it is partially traveling.

That is an ambiguous phrase!...glad you figured it out.

Edit: a wave cannot be both traveling as well as standing because both are complementary to each other!

 

[nil1996]

a wave cannot be both traveling as well as standing because both are complementary to each other!

I meant to say partially traveling.
thanks for guidance

 

[NihalSh]

I meant to say partially traveling.
thanks for guidance

no problem!