Equation of a Wave Homework: Intensity Variation w/ Phase Difference

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In summary, the equation of a wave is a = a°sin(ωt - kx + φ), where φ is the phase of the wave. When two waves with phases φ1 and φ2 interfere, the intensity I = a² varies as a function of the phase difference φ1 - φ2. Using the double angle formula cos(2x) = cos²(x) - sin²(x), we can obtain the result that the amplitude of the interference is given by 2a°cos(0.5(φ1-φ2))sin(0.5(2ωt - 2kx + φ1 + φ2)). When φ1 = φ2, the
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Homework Statement



Equation of a wave is a = a°sin(ωt - kx + φ)

where φ is the phase of a wave. if 2 waves with phases φ1 and φ2 interfere, show how the intensity I =a² varies as a function of the phase difference φ1 - φ2. Use one of the trigonometric double angle forumula or otherwise to obtain your result.



Homework Equations


The double angle formulas



The Attempt at a Solution



Well am I supposed to map I =a² onto the equation?

If so then the only double angle formula is cos(2x) = cos²(x) - sin²(x)

But I get a really stupid answer when I square the wave equation..

What do I do?
 
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  • #2
show your work

start by writing the sum of the amplitude of the 2 waves with same fequency but different phase & work from there...

also intuitively, what do you expect will happen?
 
  • #3
I have 2 waves with a phase difference:

a1 = a°sin(ωt - kx + φ1)
a2 = a°sin(ωt - kx + φ2)

If the waves combine, then an interference occurrs...

a1 + a2 = a°sin(ωt - kx + φ1) + a°sin(ωt - kx + φ2)

K apparently, sin a + sin b = 2cos 0.5(a - b) sin 0.5(a + b)

So in relation:

a1 + a2 = 2a°cos 0.5(φ1-φ2) sin 0.5(2ωt - 2kx + φ1 + φ2)

a3 = 2a°cos 0.5(φ1-φ2) sin 0.5(2ωt - 2kx + φ1 + φ2)

Now I =a²

But I'm unsure of how to square this expression I have, assuming it is even right..
 
  • #4
Unto said:
Use one of the trigonometric double angle forumula or otherwise to obtain your result.

If so then the only double angle formula is cos(2x) = cos²(x) - sin²(x)

Hi Unto! :smile:

I think by a "double angle formula" they mean like sin(A + B) or (sinA + sinB) etc …

these are trigonometric identities which you must learn. :wink:
 
  • #5
using a few diffenrt trigonamteric identities you can show the identity you used
[tex] sin(a) + sin(b) = 2cos(\frac{a-b}{2})sin(\frac{a+b}{2}) [/tex]
Unto said:
a3 = 2a°cos 0.5(φ1-φ2) sin 0.5(2ωt - 2kx + φ1 + φ2)
QUOTE]
so you get the amplitude as
[tex] 2cos(\frac{\phi_1-\phi_}{2})sin(\omega t - k x \frac{\phi_1+\phi_}{2}) [/tex]
which looks reasonable to me

before you even consider intensity, have a look at what you've got with the amplitude
- the sin term looks similar to the input waves just with a different phase factor
- now the cos term, what happens when phi_1 = phi_2? and for what z, does cos(z) = 0?
- then think about the periodicity of the functions
 
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FAQ: Equation of a Wave Homework: Intensity Variation w/ Phase Difference

What is the equation of a wave?

The equation of a wave is given by: y = A*sin(kx - wt), where A is the amplitude of the wave, k is the wave number, x is the position of the wave, w is the angular frequency, and t is the time.

How is intensity variation related to phase difference in a wave?

The intensity of a wave is directly proportional to the square of the amplitude, which is affected by the phase difference between two waves. When two waves have a phase difference of 0 or an integer multiple of 2π, they are in phase and the intensity is at its maximum. On the other hand, when the phase difference is a half-integer multiple of 2π, the waves are in antiphase and the intensity is at its minimum.

Can the intensity of a wave vary with time?

Yes, the intensity of a wave can vary with time. This is because the amplitude and phase of a wave can change with time, which affects the intensity. For example, in a standing wave, the intensity varies periodically between maximum and minimum values as the waves interfere with each other.

How does the phase difference affect the interference of waves?

The phase difference between two waves determines whether they will interfere constructively or destructively. If the phase difference is 0 or an integer multiple of 2π, the waves will interfere constructively, resulting in a larger amplitude and higher intensity. On the other hand, if the phase difference is a half-integer multiple of 2π, the waves will interfere destructively, resulting in a smaller amplitude and lower intensity.

How is the intensity of a wave measured?

The intensity of a wave can be measured using a variety of instruments such as a photometer, spectrophotometer, or lux meter. These instruments measure the power of the wave passing through a specific area, and the intensity is then calculated by dividing the power by the area. The unit of intensity is watts per square meter (W/m2).

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