A pair of speakers separated by distance d = 0.900 m are driven by the same oscillator at a frequency of 670 Hz. An observer originally positioned at one of the speakers begins to walk along a line perpendicular to the line joining the speakers as in the figure below.

(a) How far must the observer walk before reaching a relative maximum in intensity?

Here I calculated the wavelength 343/670= 0.511940

Then I did (0.9)^2 - (0.5119)^2/(2*0.5119) and got 0.535m as my answer, which is correct.

(b) How far will the observer be from the speaker when the first relative minimum is detected in the intensity?

I can't seem to find the right answer for part b.

I did [(0.9)^2-((0.5119)^2/4)]/0.5119
to get 1.45

but that is not the correct answer. Can someone tell me what I did wrong? thank you!

To find the distance at which the first relative minimum in intensity is detected, you need to consider the phase difference between the waves arriving at the observer from the two speakers.

The general formula for calculating the phase difference at a point due to wave interference is:
Δφ = 2π(Δx / λ)

Where:
Δφ is the phase difference,
Δx is the path difference between the two waves, and
λ is the wavelength of the wave.

In this case, the observer is moving along a line perpendicular to the line joining the speakers, so the path difference between the two waves is simply the distance the observer has traveled (let's call it x).

For the first relative minimum in intensity, the phase difference should be 180° or π radians.

Therefore, we can set up the equation:
2π(x / λ) = π

Simplifying the equation, we have:
2(x / λ) = 1

Rearranging this equation to solve for x:
x = 0.5λ

Now, substituting the value of λ that you calculated in the previous part (λ = 0.5119 m), we can find the distance the observer will be from the speaker at the first relative minimum in intensity:
x = 0.5 * 0.5119 = 0.25595 m

So, the correct distance at which the observer will be from the speaker when the first relative minimum is detected is approximately 0.256 m.