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!

how did you get this value for part a? I tried your numbers and it gave me 0.554

To find the distance from the speaker to the first relative minimum, you need to use the formula for the distance between consecutive minima or maxima in interference patterns.

The formula for the distance between consecutive minima or maxima in an interference pattern is given by:

d_minmax = λ/(2*sin(θ))

where λ is the wavelength of the sound wave and θ is the angle between the line joining the speakers and the line connecting the observer to the speakers.

In this case, the observer is walking along a line perpendicular to the line joining the speakers. At the position of the first relative minimum, the angle θ is 90 degrees.

So, the equation becomes:

d_minmax = λ/(2*sin(90))

As sin(90) = 1, the equation simplifies to:

d_minmax = λ/2

Substituting the value of λ = 0.5119 m into the equation, we get:

d_minmax = 0.5119/2 = 0.25595 m

Therefore, the observer will be approximately 0.25595 m away from the speaker when the first relative minimum is detected in the intensity.

In order to find the correct answer for part (b), we need to consider the concept of constructive and destructive interference of sound waves.

(a) To determine the distance the observer must walk before reaching a relative maximum in intensity, you correctly calculated the wavelength of the sound wave using the formula: wavelength = speed of sound / frequency.

wavelength = 343 m/s / 670 Hz = 0.5119 m

Next, we need to find the path length difference between the two speakers. Since the observer originally starts at one of the speakers and begins to walk perpendicular to the line joining the speakers, the path length difference can be calculated using the following formula:

path length difference = (distance between speakers)^2 - (wavelength / 2)^2
= (0.9 m)^2 - (0.5119 m / 2)^2
= 0.81 m - 0.1307 m
= 0.6793 m

Therefore, the observer must walk a distance of 0.6793 meters before reaching a relative maximum in intensity.

(b) Now, to find the distance from the speaker to the observer when the first relative minimum is detected in the intensity, we need to consider that for a relative minimum of intensity to occur, there must be destructive interference between the sound waves from the two speakers.

To calculate this distance, we can use the following formula:

distance = (path length difference + wavelength / 2) / 2

Let's substitute the values into the formula:

distance = (0.6793 m + 0.5119 m / 2) / 2
= (0.6793 m + 0.25595 m) / 2
= 0.93525 m / 2
= 0.467625 m

Therefore, the observer will be approximately 0.468 meters away from the speaker when the first relative minimum is detected in the intensity.

On B, it is a right triangle

you want the distances to be half wavelength apart, or .512/2 meters different.

d= distance walked
x= distance from far speaker
x-d=1/2 wavelength. or x=.512/2 + d
Now the right traiangle:
.9^2+d^2=x^2 or
.9^2+d^2=(.256+d)^2
.9^2+d^2=.256^2+d^2+512d
(.81-.256^2 )/.512 = d

which agrees with your answer.