A loudspeaker at the origin emits a 120 Hz tone on a day when the speed of sound is 340 m/s. The phase difference between two points on the x-axis is 5.5 rad. What is the distance between these two points?

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Determine the wavelength of the sound wave using the formula:

λ = \frac{v}{f}

, where

v

is the speed of sound (340 m/s) and

f

is the frequency (120 Hz).

Relate the phase difference to the distance between the two points using the formula:

Δϕ = \frac{2 π d}{λ}

, where

Δϕ

is the phase difference (5.5 rad),

d

is the distance between the points, and

λ

is the wavelength.

Rearrange the formula to solve for

d

d = \frac{Δϕ λ}{2 π}

Substitute the known values for

Δϕ

(5.5 rad) and

λ

(calculated in step 1) into the equation.

Simplify the expression to find the distance

d

between the two points.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Wave Frequency

Frequency is the number of cycles of a wave that occur in a unit of time, typically measured in Hertz (Hz). In this context, the loudspeaker emits a tone at 120 Hz, meaning it produces 120 sound wave cycles per second. This frequency is crucial for determining the wavelength of the sound wave, which is essential for calculating the distance between points based on phase difference.

Speed of Sound

The speed of sound is the rate at which sound waves propagate through a medium, which in this case is air. Given as 340 m/s, this value is used to relate frequency and wavelength through the equation v = fλ, where v is the speed of sound, f is the frequency, and λ is the wavelength. Understanding this relationship is key to solving for the distance between points based on their phase difference.

Phase Difference

Phase difference refers to the difference in the phase of two waves at a given point in time, measured in radians. A phase difference of 5.5 radians indicates how far one wave is ahead or behind another. This concept is critical for determining the distance between two points on the x-axis, as the distance can be calculated using the formula Δx = (φ/2π)λ, where φ is the phase difference and λ is the wavelength.