Two speakers are separated by a distance of 3.4 m. A point P is placed at 5.7 m from one of the speakers so that they form a right triangle. If the speed of sound in this situation is 340 m/s and the speakers are in phase, what is the lowest frequency for which the intensity at P is:

maximum:
minimum:

To find the lowest frequency for which the intensity at point P is maximum or minimum, we need to consider the interference of the sound waves from the two speakers.

When two sound waves interfere, they can either constructively interfere (maximum intensity) or destructively interfere (minimum intensity) depending on the phase relationship between them.

In this case, the two speakers are in phase, which means their sound waves have the same frequency and are synchronized. This implies that the two sound waves arrive at point P in phase, resulting in constructive interference.

To find the lowest frequency for maximum intensity at point P, we can use the concept of the path difference between the two sound waves.

Let's assume that the wavelength of the sound wave is λ and the distance between the speakers is d = 3.4 m. To achieve constructive interference, we need the path difference between the two sound waves to be an integer multiple of the wavelength.

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Applying this to the given situation:

d^2 = (5.7)^2 + x^2

Simplifying the equation:

(x^2) = (d^2) - (5.7)^2

Substituting the values:

(x^2) = (3.4)^2 - (5.7)^2

Now, we can use the formula for the path difference:

Path Difference = x = m * λ

where m is an integer.

Substituting the values of x and λ:

(3.4)^2 - (5.7)^2 = m * λ

Simplifying this equation gives us:

m = [(3.4)^2 - (5.7)^2] / λ

Since we are looking for the lowest frequency for maximum intensity at point P, we need the minimum value of m. The minimum value for m would occur when the denominator λ (wavelength) is maximum.

The wavelength (λ) can be calculated using the formula:

λ = v / f

where v is the speed of sound (340 m/s) and f is the frequency.

Since we are looking for the lowest frequency, the wavelength will be maximum when f is the lowest. Therefore, for the minimum value of m, we need the lowest frequency for which the intensity at P is maximum.

To find this frequency, we set m equal to 1 and solve for f:

1 = [(3.4)^2 - (5.7)^2] / (v / f)

Simplifying this equation for f, we get:

f = (v * [(3.4)^2 - (5.7)^2]) / [(2 * (5.7)^2)]

Plugging in the values:

f = (340 * [(3.4)^2 - (5.7)^2]) / [(2 * (5.7)^2)]