Calculating resultant amplitude, given 2 ampli's and phase diff

In summary, the OP's equation is a good approximation for small phase angles and similar amplitudes, but is not accurate for larger phase angles and different amplitudes.
  • #1
RKOwens4
33
0

Homework Statement



Two waves of the same frequency have amplitudes 1.02 and 2.27. They interfere at a point where their phase difference is 59.5°. What is the resultant amplitude?


Homework Equations



Resultant Amp = (Amp1 + Amp2)cos(theta/2)

The Attempt at a Solution



Okay this seemed like a simple plug and chug problem. I ended up getting an answer of 2.86, but this is marked wrong. I then did the "practice another version" thing on webassign and worked another version of the same exact problem (just with different values of the two amps and a different theta), but used the exact same formula and guess what... it was correct! I've spent the last 15 minutes trying to figure out why the formula works for one version of the problem but not the one that matters. Can anyone help?
 
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  • #2
RKOwens4 said:
Resultant Amp = (Amp1 + Amp2)cos(theta/2)

Are you sure that this formula is true?

ehild
 
  • #3
ehild said:
Are you sure that this formula is true?

ehild

It looks like it holds true when the frequencies of the two signals are identical.
 
  • #4
Can someone help, please? I don't expect anyone to just solve the problem and give me the answer like they do on cramster, but statements like the two above (with all due respect) don't really help me one single iota. No, I'm not sure if that equation works 100% of the time (obviously, it doesn't). But, what formula should I try instead?
 
  • #5
RKOwens4 said:
Can someone help, please? I don't expect anyone to just solve the problem and give me the answer like they do on cramster, but statements like the two above (with all due respect) don't really help me one single iota. No, I'm not sure if that equation works 100% of the time (obviously, it doesn't). But, what formula should I try instead?

The formula is correct. If your result, 2.856, or rounded to two sig figs, 2.86 is marked incorrect, then either:

1. There are some units that should have been specified
2. There is a typo in the question (no way to fix that!)
3. The marking program is incorrectly programmed for this question (call your instructor)
 
  • #6
Try the following formula for the resultant amplitude :

A=sqrt[A11+A22-2A1A2cos(θ)].

ehild
 
  • #7
attachment.php?attachmentid=37444&stc=1&d=1311367039.gif
 

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  • #8
The interfering waves are A1sin(wt-kx) and A2sin(wt-kx+θ). Their resultant is a single wave with amplitude A. Denoting wt-kx by α, we have the sum

A1sin(α)+A2sin(α+θ)=Asin(α+φ)*.

sin(α+θ)=sin(α)cos(θ)+cos(α)sin(θ), and sin(α+φ)=sin(α)cos(φ)+cos(α)sin(φ).

Replacing into eq. * :

sin(α)[A1+A2cos(θ)]+cos(α)A2sin(θ)=A[sin(α)cos(φ)+cos(α)sin(φ)]

The equation is valid for any angles, that is for any values of sin(α) and cos(α) between -1 and 1. Therefore

A1+A2cos(θ)=Acos(φ) and A2sin(θ)=Asin(φ)

Square both equation and add up:

A2=A12+A22+2A1A2cos(θ).

ehild
 
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  • #9
gneill said:
It looks like it holds true when the frequencies of the two signals are identical.

Let be the phase difference θ=pi. Then the resultant amplitude is always 0, according to the formula in the original post which is certainly not true when the amplitudes are different.

ehild
 
  • #10
gneill said:
attachment.php?attachmentid=37444&stc=1&d=1311367039.gif
Is the resultant amplitude 2.86, according to the plot? It is close but different.

ehild
 
  • #11
I've been looking more closely at the OP's formula. It would appear to be a good approximation for small phase angles, say less than 50 degrees or so. I'm currently investigating a derivation that will provide an error term.
 
  • #12
ehild said:
Is the resultant amplitude 2.86, according to the plot? It is close but different.

ehild

The plot maxima are at amplitude 2.92 ! So yes, close to 2.86, but not spot on. Your formula delivers the goods (as it should, based upon the derivation!).
 
  • #13
I wonder why not the exact formula is used? It is derived in every books on General Physics.

ehild
 
  • #14
ehild said:
I wonder why not the exact formula is used? It is derived in every books on General Physics.

ehild

I suppose it's one of those shortcuts that engineers use :smile:
Makes the math simpler.

I've derived an expression for the signal sum that looks as follows:

[tex] (A + B) cos\left(\frac{\phi}{2}\right) cos(\omega t) + (A - B)sin\left(\frac{\phi}{2}\right) sin(\omega t) [/tex]

The amplitude of the left term is clearly the OP's formula. The right term represents a deviation from that formula. Apparently the OP's formula works best for similar amplitudes and small angles.
 
  • #15
gneill said:
The amplitude of the left term is clearly the OP's formula. The right term represents a deviation from that formula. Apparently the OP's formula works best for similar amplitudes and small angles.

I see now where the OP's formula came from. Thanks gneil! But it is really a rough approximation.

ehild
 
  • #16
ehild said:
But it is really a rough approximation.

I can't argue with you there! :smile:
 
  • #17
What if students try to use the resultant for A1=3 A2=1 and phase difference of 180°? They see only a formula to plug in data. Applying the correct formula is not much more work than using the wrong one.Why do they teach the wrong one then?

ehild
 
  • #18
ehild said:
What if students try to use the resultant for A1=3 A2=1 and phase difference of 180°? They see only a formula to plug in data. Applying the correct formula is not much more work than using the wrong one.Why do they teach the wrong one then?

ehild

To be fair, we don't know the background of where that formula comes from. For all we know it was a perfectly valid formula in a particular example (equal amplitude waves, perhaps), that was lifted out of context. We would have to as the OP where he found it.
 

1. What is the formula for calculating resultant amplitude?

The formula for calculating resultant amplitude is: R = √(A1^2 + A2^2 + 2A1A2cosθ), where A1 and A2 are the two given amplitudes and θ is the phase difference between them.

2. What does the phase difference between two amplitudes represent?

The phase difference represents the difference in the starting point of the two waves. It can be measured in degrees or radians and determines the interference pattern between the two waves.

3. How do I determine the direction of the resultant amplitude?

The direction of the resultant amplitude can be determined by using the law of cosines. If the phase difference is 0 or a multiple of 2π, the waves are in phase and the resultant amplitude will be in the same direction as the larger amplitude. If the phase difference is π or an odd multiple of π, the waves are in antiphase and the resultant amplitude will be in the opposite direction of the larger amplitude.

4. Can the resultant amplitude be negative?

Yes, the resultant amplitude can be negative if the two waves are in antiphase and the smaller amplitude is subtracted from the larger amplitude. This means that the two waves are canceling each other out and the resulting amplitude will be in the opposite direction of the larger amplitude.

5. How does the resultant amplitude change when the phase difference between the two waves is varied?

The resultant amplitude will vary in a sinusoidal manner as the phase difference between the two waves is changed. It will reach its maximum value when the two waves are in phase (phase difference of 0 or a multiple of 2π) and its minimum value when the two waves are in antiphase (phase difference of π or an odd multiple of π).

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