Two small stereo speakers A and B that are 1.40 m apart are sending out sound of wavelength 34 cm in all directions and all in phase. A person at point P starts out equidistant from both speakers and walks so that he is always 1.50 m from speaker B
(Fig. E35.1). For what values of x will the sound this person hears be (a) maximally reinforced. Limit your solution to the cases where x … 1.50 m

Verified step by step guidance

1. Identify the given values: distance between speakers A and B (d) is 1.40 m, wavelength (λ) is 34 cm (0.34 m), and the distance from speaker B to point P is 1.50 m.
2. Use the condition for constructive interference, which occurs when the path difference between the two waves is an integer multiple of the wavelength: ΔL = nλ, where n is an integer (0, 1, 2, ...).
3. Express the path difference ΔL in terms of the distances from point P to speakers A and B: ΔL = |PA - PB|.
4. Since PB is given as 1.50 m, express PA in terms of x, where x is the horizontal distance from the midpoint between the speakers to point P. Use the Pythagorean theorem to find PA: PA = sqrt((1.50)^2 + (x)^2).
5. Set up the equation for constructive interference: |sqrt((1.50)^2 + (x)^2) - 1.50| = n * 0.34. Solve for x for different integer values of n to find the positions where the sound is maximally reinforced.

Key Concepts

Wave Interference

Wave interference occurs when two or more waves overlap and combine to form a new wave pattern. This can result in constructive interference, where waves reinforce each other, or destructive interference, where they cancel each other out. In this scenario, the sound waves from speakers A and B will interfere based on their relative distances to point P, affecting the sound intensity heard by the listener.

Path Difference

Path difference refers to the difference in distance traveled by two waves from their sources to a common point. For constructive interference to occur, the path difference must be an integer multiple of the wavelength. In this case, the listener's position relative to the speakers determines the path difference, which is crucial for identifying points of maximum sound reinforcement.

Wavelength and Frequency

Wavelength is the distance between successive crests of a wave, while frequency is the number of wave cycles that pass a point per unit time. The relationship between wavelength and frequency is given by the equation v = fλ, where v is the wave speed. Understanding the wavelength of the sound emitted by the speakers (34 cm) is essential for calculating the conditions for constructive interference at point P.