Problems about interference of two wave

1. Oct 7, 2013

samtim

1. The problem statement, all variables and given/known data
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.)

2. Relevant equations

y=Asin(kx-wt)

3. 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.)