Two small stereo speakers are driven in step by the same variable-frequency oscillator. Their sound is picked up by a microphone arranged as shown in Fig. E16.39. For what frequencies does their sound at the speakers produce constructive interference?

Verified step by step guidance

1. Identify the path difference between the sound waves reaching the microphone from the two speakers. This is crucial for determining the conditions for constructive interference.
2. Use the Pythagorean theorem to calculate the distance from each speaker to the microphone. For the left speaker, the distance is \( \sqrt{(4.50)^2 + (2.00)^2} \) meters. For the right speaker, the distance is 4.50 meters.
3. Calculate the path difference \( \Delta d \) between the two sound waves. This is the absolute difference between the two distances calculated in the previous step.
4. For constructive interference, the path difference \( \Delta d \) must be an integer multiple of the wavelength \( \lambda \), i.e., \( \Delta d = m \lambda \), where \( m \) is an integer (0, 1, 2, ...).
5. Relate the wavelength \( \lambda \) to the frequency \( f \) using the speed of sound \( v \) in air: \( \lambda = \frac{v}{f} \). Substitute this into the constructive interference condition to find the frequencies that satisfy \( \Delta d = m \frac{v}{f} \).

Key Concepts

Constructive Interference

Constructive interference occurs when two or more waves meet in such a way that their crests and troughs align, resulting in a wave of greater amplitude. This phenomenon is crucial in sound waves, as it enhances the sound intensity at certain points, such as where the microphone is placed in this scenario. The condition for constructive interference is that the path difference between the waves from the two sources must be an integer multiple of the wavelength.

Path Difference

Path difference refers to the difference in distance traveled by two waves from their sources to a common point, such as the microphone in this case. For constructive interference to occur, this path difference must equal nλ, where n is an integer (0, 1, 2, ...) and λ is the wavelength of the sound. Understanding how to calculate the path difference based on the geometry of the setup is essential for determining the frequencies that produce constructive interference.

Wavelength and Frequency Relationship

The relationship between wavelength (λ) and frequency (f) of a wave is given by the equation v = fλ, where v is the speed of sound in air. This relationship is fundamental in solving the problem, as knowing the speed of sound allows us to calculate the wavelength for specific frequencies. By determining the wavelengths that correspond to the path differences for constructive interference, we can find the frequencies at which the speakers produce enhanced sound at the microphone.