Two speakers vibrate in phase with one another at 523Hz. At certain points in the room, the sound waves from the two speakers interfere constructively. One such point is 2.28m from speaker #1 and is between 2m and 4m from speaker #2. How far is this point from speaker #2? Find all possible distances betweens 2m and 4m. The speed of sound in air is 343m/s. PLEASE HELP!

Question

Two speakers vibrate in phase with one another at 523Hz. At certain points in the room, the sound waves from the two speakers interfere constructively. One such point is 2.28m from speaker #1 and is between 2m and 4m from speaker #2. How far is this point from speaker #2? Find all possible distances betweens 2m and 4m. The speed of sound in air is 343m/s. PLEASE HELP!

Answer 1

To find the distance between the point and speaker #2, we can make use of the concept of phase difference and the formulas for calculating the wavelength and wave number.

Let's start by calculating the wavelength (λ) using the formula:

λ = v/f

where:
λ = wavelength
v = speed of sound in air (343 m/s)
f = frequency (523 Hz)

Substituting the given values, we find:

λ = 343 m/s / 523 Hz

Now calculate the wavelength, which gives us:

λ ≈ 0.656 m

Next, we need to determine the number of wavelengths between speaker #1 and the point of constructive interference. To do this, we divide the distance between them by the wavelength:

Number of wavelengths = Distance / λ

Since the point is 2.28 m from speaker #1, the number of wavelengths is:

Number of wavelengths = 2.28 m / 0.656 m

Now, we need to determine the number of wavelengths from speaker #2 to the point. Since the point is between 2 m and 4 m from speaker #2, we can assign variable x to represent the distance between the point and speaker #2. Therefore, the number of wavelengths is:

Number of wavelengths = x / λ

Now, since we know that the two speakers are vibrating in phase, the distance between speaker #1 and the point would be equal to the distance between speaker #2 and the point in terms of the number of wavelengths. Hence, we can equate the two expressions we derived:

2.28 m / 0.656 m = x / 0.656 m

Simplifying the equation, we find:

x ≈ 2.28 m * (0.656 m / 0.656 m)

x ≈ 2.28 m

Therefore, the distance between the point and speaker #2 is approximately 2.28 meters. This falls within the range of 2m to 4m specified in the question. Hence, the possible distances between 2m and 4m are:

2.28 m

Thus, the point is exactly 2.28m from speaker #2, and this is the only possible distance within the given range of 2m to 4m.

To find the distance between the point and speaker #2, we can make use of the concept of phase difference and the formulas for calculating the wavelength and wave number.

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