Two loudspeakers, A and B (Fig. E16.35), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 m to the right of speaker A. Consider point Q along the extension of the line connecting the speakers, 1.00 m to the right of speaker B. Both speakers emit sound waves that travel directly from the speaker to point Q. What is the lowest frequency for which constructive interference occurs at point Q?

Verified step by step guidance

1. Identify the path difference between the sound waves from speakers A and B to point Q. The distance from A to Q is 3.00 m (2.00 m + 1.00 m), and the distance from B to Q is 1.00 m.
2. Calculate the path difference: ΔL = L_AQ - L_BQ = 3.00 m - 1.00 m = 2.00 m.
3. For constructive interference, the path difference must be an integer multiple of the wavelength: ΔL = mλ, where m is an integer (m = 0, 1, 2, ...).
4. To find the lowest frequency, use the speed of sound in air, v = 343 m/s, and the relationship between speed, frequency, and wavelength: v = fλ. Rearrange to find the wavelength: λ = v/f.
5. Substitute the expression for wavelength into the constructive interference condition: 2.00 m = m(343 m/s)/f. Solve for the frequency f for the smallest integer m that satisfies the equation.

Key Concepts

Constructive Interference

Constructive interference occurs when two or more waves meet in phase, resulting in a wave of greater amplitude. For sound waves, this happens when the path difference between the waves arriving at a point is an integer multiple of the wavelength. In this scenario, the condition for constructive interference at point Q must be determined based on the distances from speakers A and B.

Path Difference

The path difference is the difference in distance traveled by two waves from their sources to a common point. In this case, the path difference between the sound waves from speakers A and B to point Q is crucial for determining whether constructive or destructive interference occurs. For constructive interference, this path difference must equal nλ, where n is an integer and λ is the wavelength of the sound.

Frequency and Wavelength Relationship

The frequency of a wave is inversely related to its wavelength, described by the equation v = fλ, where v is the speed of sound, f is the frequency, and λ is the wavelength. To find the lowest frequency for constructive interference at point Q, one must first calculate the wavelength corresponding to the path difference that satisfies the constructive interference condition, and then use this relationship to determine the frequency.