Problems about interference of two wave

[samtim]

Homework Statement

Consider two point sources S1 and S2 which emit waves of the same frequency f and amplitude A. The waves start in the same phase, and this phase relation at the sources is maintained throughout time. Consider point P at which r1 is nearly equal to r2. a) Show that the superposition of these two waves gives a wave whose amplitude ym varies with the position P approximately according to
$y_m=\frac{2A}{r}cos(\frac{k}{2}(r_1-r_2))$ in which $r=\frac{r_1+r_2}{2}$. b) Then show that total cancellation occurs when $r_1-r_2=(n+0.5)\lambda$, n being any integer, and that total reinforcement occurs when $r_1-r_2=n\lambda$. The locus of points whose difference in distance from two fixed points is a constant is a hyperbola, the fixed points being the foci. Hence each value of n gives a hyperbolic line of distructive interference. At points at which r1 and r2 are not approximately equal (as near the sources), the amplitudes of the waves from S1 and S2 differ and the cancellations are only partial. (This is the basis of the OMEGA navigation system.)

Homework Equations

y=Asin(kx-wt)

The Attempt at a Solution

Part a)
I let
$y_1=Asin(kx-\omega t)$
$y_2=Asin(kx-\omega t)$
At P, $y_(resultant)=y_1+y_2$
and this gives $y_(reusltant)=2Acos(\frac{k}{2}(r_1-r_2))sin(kx-\omega t)$
which is wrong($y_m=2Acos(\frac{k}{2}(r_1-r_2))$, which step I did is wrong and how to achieve the answer?

Part b)
I directly substitute $r_1-r_2=(n+0.5)\lambda$ into the proved ans in a) and successfully get 0.
but for the reinforcement part I substitute $r_1-r_2=n\lambda$, I get $y_m=2A/r$ , and did i do wrong and does it directly show total reinforcement occur?

Finally, I want to ask the question is ended already or not because I don't understand what is the meaning of the question since "The locus of points whose difference in distance from two fixed points......OMEGA navigation system.)

 

[mark726]

"

it is important to fully understand the question before attempting to solve it. In this case, the question is asking you to mathematically show the phenomenon of superposition and interference in a system of two point sources emitting waves. It also asks you to explain the concept of hyperbolic lines of destructive interference and how it relates to the OMEGA navigation system.

In your attempt at a solution, you correctly used the equation for a wave (y=Asin(kx-wt)) and substituted it into the superposition equation (y_(resultant)=y_1+y_2). However, your mistake was in assuming that the two waves have the same phase at all points. In reality, the phase difference between the two waves will change as the waves travel to point P. This is why the correct equation for the resultant amplitude is y_m=\frac{2A}{r}cos(\frac{k}{2}(r_1-r_2)).

For part b, you correctly substituted the values for total cancellation and reinforcement into the equation. However, your interpretation of the results is incorrect. When r_1-r_2=(n+0.5)\lambda, the waves are completely out of phase and cancel each other out, resulting in a zero amplitude. When r_1-r_2=n\lambda, the waves are in phase and reinforce each other, resulting in a maximum amplitude.

To fully answer the question, you should also explain the concept of hyperbolic lines of destructive interference. These lines represent the points where the difference in distance from the two fixed points (the two point sources) is a constant. This constant difference in distance results in a constant phase difference between the two waves, leading to destructive interference. This concept is used in the OMEGA navigation system, which uses hyperbolic lines of destructive interference to determine the position of a receiver.

In conclusion, the question is not yet fully answered. You should revise your solution and include an explanation of hyperbolic lines of destructive interference and their role in the OMEGA navigation system.