Problem 1 - University Physics with Modern Physics

Question

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, (b) cancelled? Limit your solution to the cases where x \(\leq\) 1.50 m.

Step-by-Step Solution

Answer

Maximally reinforced: x = 0 m. Cancelled: x \approx 1.11 m.

Step 1: Determine the Path Difference

To find when sound is maximally reinforced or cancelled, we start by determining the path difference from the point \( P \) to speakers \( A \) and \( B \). Since the person at point \( P \) is always 1.50 m from speaker \( B \), the distance from \( P \) to \( A \) is \( \sqrt{(x + 0.70)^2 + 0.98^2} \), where \( 0.70 = 1.40 / 2 \) and \( 0.98 = \sqrt{1.50^2 - 0.70^2} \).

Step 2: Maximally Reinforced Sound (Constructive Interference)

For constructive interference, the path difference must be an integer multiple of the wavelength: \( \Delta d = n\lambda \). This gives:\[ \sqrt{(x + 0.70)^2 + 0.98^2} - 1.50 = n \times 0.34 \]Solve for \( x \) using the possible values \( n=0,1,2,... \).

Step 3: Cancelled Sound (Destructive Interference)

For destructive interference, the path difference must be an odd half multiple of the wavelength: \( \Delta d = (n+0.5)\lambda \). This leads to:\[ \sqrt{(x + 0.70)^2 + 0.98^2} - 1.50 = (n+0.5) \times 0.34 \]Solve for \( x \) using possible values for \( n \).

Step 4: Solve the Equations within Limits

Solve the equations from Step 2 and Step 3 while limiting solutions to \( x \leq 1.50 \) m:For constructive (maximally reinforced):1. \( n = 0 \); solve the equation.2. \( n = 1 \); solve the equation, and so forth.For destructive (cancelled):1. \( n = 0 \); solve the equation.2. \( n = 1 \); solve the equation, and so forth.

Key Concepts

Constructive Interference

In physics, constructive interference occurs when two or more waves overlap and combine to form a wave with a higher amplitude. This phenomenon happens when the path difference between the waves is an integer multiple of their wavelength.
For instance, if you have two sound waves that are in phase, meaning their peaks and troughs align perfectly, they amplify each other. The formula to find this is \[ \Delta d = n\lambda \]where

• \( \Delta d \) is the path difference,
• \( n \) is an integer (0, 1, 2, ...), and
• \( \lambda \) is the wavelength of the sound wave.

When you stand in certain positions where this condition is met, you hear the sound much louder because the sound waves add up to reinforce each other. Think of it as spots where the sound feels just right and becomes richer and stronger.

Destructive Interference

Destructive interference is the opposite of constructive interference, where two waves interfere in such a way that they cancel each other out, leading to a lower amplitude.
This happens when the path difference is an odd multiple of half the wavelength, which can be expressed with the formula \[ \Delta d = (n+0.5)\lambda \] Here,

• \( \Delta d \) is the path difference,
• \( n \) is an integer (0, 1, 2, ...), and
• \( \lambda \) is the wavelength.

When waves meet that are out of phase, meaning one wave crests where the other troughs, they effectively reduce the sound, sometimes to the point of silence.
In practical terms, this explains why you might hear little to no sound at certain points in a room with multiple sound sources.

Path Difference

Path difference is a key concept when studying wave interference. It refers to the difference in distance that two waves travel to reach a common point.
In the context of sound waves coming from two speakers, the path difference can be the difference in distance from each speaker to the listener's position. This distance affects how the waves combine.The formulae for determining if interference is constructive or destructive depend on this path difference:

• For constructive interference: \( \Delta d = n\lambda \)
• For destructive interference: \( \Delta d = (n+0.5)\lambda \)

By understanding and calculating the path difference, one can predict spots where sound will be loudest or quietest, a principle that's useful in designing auditoriums or speaker systems.

Sound Waves

Sound waves are a type of mechanical wave that travels through a medium, such as air, liquid, or solid. These waves are produced by vibrating objects and travel via compressions and rarefactions.Sound waves have characteristics such as

• Wavelength (\( \lambda \)): the distance between consecutive points of phase (e.g., crest to crest).
• Frequency: how many wavelengths pass a point in one second.
• Amplitude: the height of the wave, which determines loudness.

Sound waves interact with each other and the environment, leading to interference patterns like constructive and destructive interference. These interactions are critical when positioning speakers for optimal sound quality or in understanding sound acoustics in a given space.