Stationary Waves: Equations for Standing Waves

[Abhishekdas]

Stationary waves...

Homework Statement

Which of the following equations can form stationary waves...
1. y=Asin(wt-kx)
2. y=Acos(wt-kx)
3. y=Asin(wt+kx)
4. y=Acos(wt+kx)

Homework Equations

The Attempt at a Solution

Answer is 1,3 and 2,4 which is obviously correct...But why can't other combinations be possible as long as they are traveling in opposite directions (like 2 and 3)?
And you can standing waves be formed by waves of different amplitudes?

 

[zorro]

2 & 3 are traveling in the same direction :)

 

[Abhishekdas]

No... they are opposite... 2 is in the +ve x direction and 3 is in the -ve x direction...

 

[zorro]

Why do you think so? (your explanation)
Hint: The functions are not same...one is sine and the other is cosine

 

[Abhishekdas]

The direction of wave is determined by the sign of the quantity (coefficient of w/coefficient of x) it its is positive then the wave is traveling in -ve x direction and vice versa...That is my explanation...

 

[zorro]

the wave is traveling in -ve x direction and vice versa...

As long as the function defining wave remains the same!
You can write 2. y=cos(wt-kx) as y=sin(π/2-wt+kx)=sin(w't+kx)

 

[Abhishekdas]

As long as the function defining wave remains the same!
You can write 2. y=cos(wt-kx) as y=sin(π/2-wt+kx)=sin(w't+kx)

But how can one write it as sin(w't+kx)...w has to remain same and has to be positive...
And ya...how does it explain why 2 and 3 can not form a standing wave?

 

[zorro]

But how can one write it as sin(w't+kx)...w has to remain same and has to be positive...

My mistake.
Its y=sin(π/2-wt+kx)=sin(Φ-wt+kx). Compare this with y=sin(kx+wt).

And ya...how does it explain why 2 and 3 can not form a standing wave?

2. and 3. are traveling in the same direction. They can't produce a standing wave.

 

[Abhishekdas]

I am still not clear..how are they traveling in the same direction? if the velocities have opposite sign...

 

[zorro]

Sorry for the late reply.
Yes you were correct earlier, 2 and 3 form standing waves. I was confused in the direction too. You can check the sign of kx if the function remains same, like y=Asin(wt-kx) and y=Asin(wt+kx). Note that if you write y=Asin(wt-kx) as y=-Asin(kx-wt), it does not change the direction, but reflects the wave w.r.t x-axis.

So to avoid the confusion, there is a fundamental method. If f(x,t) is the function representing a wave, then df/dt=0 (if wave shape remains constant, which is usually the case). Find out the sign of dx/dt from each equation and compare