Two speakers that are 15.0 m apart produce in-phase sound waves of frequency 250.0 Hz in a room where the speed of sound is 340.0 m>s. A woman starts out at the midpoint between the two speakers. The room's walls and ceiling are covered with absorbers to eliminate reflections, and she listens with only one ear for best precision. (c) How far from the center must she walk before she first hears the sound maximally enhanced?

Verified step by step guidance

1. Calculate the wavelength (\(\lambda\)) of the sound waves using the formula \(\lambda = \frac{v}{f}\), where \(v\) is the speed of sound and \(f\) is the frequency of the sound waves.
2. Understand that the woman will first hear the sound maximally enhanced when she is at a position where the path difference between the sound waves from the two speakers is equal to one wavelength (\(\lambda\)).
3. Set up the equation for the path difference between the two speakers to the woman's position. Let \(d\) be the distance she walks from the center. The path difference can be expressed as \(\Delta L = \sqrt{(7.5 + d)^2 + y^2} - \sqrt{(7.5 - d)^2 + y^2}\), where \(y\) is the perpendicular distance from the line joining the speakers to the woman.
4. Since the woman starts at the midpoint and moves along the line joining the speakers, set \(y = 0\) in the path difference equation. Simplify the equation to find \(d\) when \(\Delta L = \lambda\).
5. Solve the simplified equation for \(d\) to find the distance she must walk from the center to first hear the sound maximally enhanced.

Key Concepts

Interference of Sound Waves

Interference occurs when two or more sound waves overlap, resulting in a new wave pattern. In this scenario, the speakers produce in-phase waves, meaning their peaks and troughs align. This leads to constructive interference, where the sound intensity is maximized at certain points, known as nodes and antinodes, depending on the distance from the sources.

Path Difference

Path difference refers to the difference in distance traveled by sound waves from two sources to a given point. For constructive interference to occur, the path difference must be an integer multiple of the wavelength. In this case, the woman must walk a distance that creates a path difference that results in the sound waves reinforcing each other, leading to a maximally enhanced sound.

Wavelength and Frequency Relationship

The wavelength of a sound wave is inversely related to its frequency, described by the equation λ = v/f, where λ is the wavelength, v is the speed of sound, and f is the frequency. For the given frequency of 250 Hz and speed of sound at 340 m/s, the wavelength can be calculated, which is essential for determining the points of constructive interference as the woman moves away from the midpoint.