NEED HELP WITH ONLY PART C
Two 135.1.0-W speakers, A and B, are separated by a distance D = 3.6 m. The speakers emit in-phase sound waves at a frequency f = 11800.0 Hz. Point P1 is located at x1= 4.50 m and y1 = 0 m; point P2 is located at x2 = 4.50 m and y2 = –y.
a) Neglecting speaker B, what is the intensity, IA1 (in W/m2), of the sound at point P1 due to speaker A? Assume that the sound from the speaker is emitted uniformly in all directions.
b) What is this intensity in terms of decibels (sound level, ßA1)?
c) When both speakers are turned on, there is a maximum in their combined intensities at P1. As one moves toward P2, this intensity reaches a single minimum and then becomes maximized again at P2. How far is P2 from P1, that is, what is ?y? You may assume that L >> y and that D >> y, which will allow you to simplify the algebra by using (a±b)1/2 ˜ a1/2 ± 0.5b/a1/2 when a >> b.
I don't see any indicaton of the x,y coordinates of the speakers.
To answer part c of your question, we need to find the distance between point P1 and point P2 (represented by the value of y).
To do this, we can consider the interference pattern that occurs when sound waves from two speakers combine.
At point P1, the two speakers are emitting sound waves that are in-phase, which means the waves combine constructively and the intensity is maximized.
As we move towards point P2, the path length difference between the waves from the two speakers changes. This path length difference determines whether the waves will combine constructively or destructively.
At a certain distance, known as the minimum point, the path length difference results in destructive interference, causing a minimum in intensity.
After the minimum point, the path length difference starts increasing again, resulting in constructive interference and the intensity increases.
To find the distance between P1 and P2 (y), we can use the concept of the minimum and maximum points in the interference pattern.
The minimum point occurs when the path length difference is equal to an odd multiple of half the wavelength (λ/2). This can be represented as:
D = (n + 0.5)λ/2
where D is the distance between the speakers (3.6 m) and n is an integer.
The maximum point occurs when the path length difference is equal to an even multiple of the wavelength (λ). This can be represented as:
D = nλ
where n is an integer.
Knowing the frequency (f = 11800.0 Hz), we can find the wavelength (λ) using the equation:
λ = c / f
where c is the speed of sound in air, which is approximately 343 m/s.
Substituting the values, we can find the wavelength:
λ = 343 m/s / 11800.0 Hz
Now, we can solve for y using the equations for the minimum and maximum points.
Starting from the maximum point at P1, we want to find the distance to the first minimum point. This corresponds to n = 1 in the equation for the minimum point.
So, for n = 1:
D = (1 + 0.5)λ/2
Substituting the values, we can solve for y:
3.6 m = (1.5)(λ/2)
Simplifying the equation:
y = (3.6 m)(2) / (1.5)(λ)
Now, substitute the value of λ:
y = (3.6 m)(2) / (1.5)(343 m/s / 11800.0 Hz)
Simplifying the equation:
y = (3.6 m)(2)(11800.0 Hz) / (1.5)(343 m/s)
Finally, calculate the value of y to find the distance between P1 and P2.