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There are two different point sources which produce spherical waves with the same power, amplitude, ω, wavenumber and phase.

I can calculate the intensity of each wave in a point: $$ I_1 = P / (4 \pi r_1^2) $$ $$ I_2 = P / (4 \pi r_2^2) $$ But how can I calculate the intensity of the resulting wave in that point?

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Are the sources at the same point or seperated? Where are you trying to find the intesity? –  Michael Brown Feb 20 '13 at 21:04
    
@MichaelBrown The sources are separated. And I want to find the intensity in a point where there is constructive interference (but I'm curious about finding the intensity in a random point too) –  Oriol Feb 20 '13 at 21:19
1  
Well then you just need to add the two amplitudes together and square that. It's important that you add the amplitudes, not the intensities. Formula for spherical waves can be found here en.wikipedia.org/wiki/Wave_equation#Spherical_waves –  Michael Brown Feb 20 '13 at 21:34
    
@MichaelBrown Why? I don't understand wikipedia article –  Oriol Feb 20 '13 at 22:10
2  
Waves add by their amplitudes, not their intensities. Otherwise there would be no interference. The amplitude for a general spherical wave is given in the article. For a pure (single frequency) wave it can be written $ A \cos(k r - \omega t)/r $ where $k$ is the wavenumber, $r$ is the distance from the source, $\omega$ is the frequency, $t$ is the time and $A$ is the overall scale of the amplitude (loud or quiet etc.). You have one of these expressions for each source and you must add them together. This gives the amplitude at any point in space. You square that to get the intensity. –  Michael Brown Feb 20 '13 at 23:00

1 Answer 1

I found the solution, but forgot to add it here. But since a considerable number of people has seen the question, the answer could be useful.

Notation

  • $r_1$: distance between the given point and the focus 1.
  • $r_2$: distance between the given point and the focus 2.
  • $A_1$: amplitude in the given point of the wave generated by focus 1.
  • $A_2$: amplitude in the given point of the wave generated by focus 2.
  • $A_r$: amplitude in the given point of the resulting wave.
  • $I_1$: intensity in the given point of the wave generated by focus 1.
  • $I_2$: intensity in the given point of the wave generated by focus 2.
  • $I_r$: intensity in the given point of the resulting wave.
  • $P$: power of the initial waves.
  • $\varphi$: phase difference between the initial waves in the given point.
  • $k$: wavenumber of the initial and resulting waves.
  • $\omega$: angular frequency of the initial and resulting waves.
  • $v$ : velocity of the initial and resulting waves.

Preface

I will use the formula

$$ I = \frac{1}{2}\rho v \omega^2 A^2 $$

The resulting wave will be a wave of amplitude $A_r$ with the same $k$ and $\omega$ than the initial waves. Moreover, $\rho$ and $v$ will also be the same because they only depend on the environment.

Then,

$$ \begin{cases} I_r = \frac{1}{2}\rho v \omega^2 A_r^2 \\ I_2 = \frac{1}{2}\rho v \omega^2 A_2^2 \\ \end{cases} \implies I_r = I_2 \left(\frac{A_r}{A_2}\right)^2 = \frac{P}{4\pi r_2^2} \left(\frac{A_r}{A_2}\right)^2 $$

In order to express $I_r$ in terms of $r_1$ and $r_2$ instead of $A_1$ and $A_2$, I will use that the amplitude of an spherical wave is inversely proportional to the distance to the focus. That is:

$$ \frac{A_1}{A_2} = \frac{r_2}{r_1} $$

Answer

In a point with destructive interference ($\varphi = \pi$)

The resulting amplitude will be the difference of amplitudes:

$$ A_r = |A_1 - A_2| $$

Then, the resulting intensity is

$$ I_r = \frac{P}{4\pi r_2^2} \left(\frac{A_r}{A_2}\right)^2 = \frac{P}{4\pi r_2^2} \left(\frac{A_1 - A_2}{A_2}\right)^2 = \frac{P}{4\pi r_2^2} \left(\frac{A_1}{A_2}-1\right)^2 = \frac{P}{4\pi r_2^2} \left(\frac{r_2}{r_1}-1\right)^2 = \frac{P}{4\pi} \left(\frac{r_2-r_1}{r_1 r_2}\right)^2 $$

In a point with constructive interference ($\varphi = 0$)

The resulting amplitude will be the sum of amplitudes:

$$ A_r = A_1 + A_2 $$

Then, the resulting intensity is

$$ I_r = \frac{P}{4\pi r_2^2} \left(\frac{A_r}{A_2}\right)^2 = \frac{P}{4\pi r_2^2} \left(\frac{A_1 + A_2}{A_2}\right)^2 = \frac{P}{4\pi r_2^2} \left(\frac{A_1}{A_2}+1\right)^2 = \frac{P}{4\pi r_2^2} \left(\frac{r_2}{r_1}+1\right)^2 = \frac{P}{4\pi} \left(\frac{r_1+r_2}{r_1 r_2}\right)^2 $$

In general

The resulting amplitude will be ($\varphi$ is the phase difference):

$$ A_r = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos{\varphi}} $$

Then, the resulting intensity is

$$ I_r = \frac{P}{4\pi r_2^2} \left(\frac{A_r}{A_2}\right)^2 = \frac{P}{4\pi r_2^2} \frac{A_1^2 + A_2^2 + 2 A_1 A_2 \cos{\varphi}}{A_2^2} = \frac{P}{4\pi r_2^2} \left(\left(\frac{A_1}{A_2}\right)^2 + 1 + 2 \frac{A_1}{A_2} \cos{\varphi}\right) = \frac{P}{4\pi r_2^2} \left(\left(\frac{r_2}{r_1}\right)^2 + 1 + 2 \frac{r_2}{r_1} \cos{\varphi}\right) = \frac{P}{4\pi} \frac{r_1^2 + r_2^2 + 2 r_1 r_2 \cos{\varphi}}{(r_1 r_2)^2} $$

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