Problem 3 - Physics

Question

Two in-phase sources of waves are separated by a distance of \(4.00 \mathrm{~m}\). These sources produce identical waves that have a wave length of \(5.00 \mathrm{~m}\). On the line between them, there are two places at which the same type of interference occurs. (a) Is it constructive or destructive interference, and (b) where are the places located?

Step-by-Step Solution

Answer

(a) Constructive. (b) At the midpoint and two equal parts at half of 4.0 m.

Step 1: Understand Interference Types

In-phase sources can produce either constructive or destructive interference based on their path difference. Constructive interference occurs when the path difference is a multiple of the wavelength. Destructive interference happens when the path difference is an odd multiple of half the wavelength.

Step 2: Determine the Type of Interference

Given the sources are in-phase and the problem states two places of the same type of interference occur, it's more likely to be constructive interference as this occurs at both integer multiples and half-integer multiples. Destructive interference would require the sources to be half a wavelength apart, which doesn't align well with the scenario provided repeatedly.

Step 3: Calculate Potential Positions for Constructive Interference

The general condition for constructive interference is that the path difference is a multiple of the wavelength \( n\lambda \). Given that \( \lambda = 5.00 \text{ m} \), we are looking for positions \( x \) such that the path difference \( |r_1 - r_2| = n \times 5.00 \text{ m} \).

Step 4: Use Geometric Symmetry for Source Positions

We know the two sources are \(4.00 \text{ m}\) apart. Let's place one source at \(0\) and the other at \(4.00\). This means a point \( x \) experiences a path difference of \( |x| + |4.00 - x| = 4.00 \). For constructive interference, the path difference should be \(0, 5.00, 10.00, ...\).

Step 5: Conclude Location of Constructive Interference

For \( |x| + |4.00 - x| = 4.00\), possible values of \(x\) which result in multiples of \(5.00\) are symmetry points: the middle point and the ends of the spread where the simplest constructive interference case \(0\) difference happens due to the setup. There will be constructive interference at these points.

Key Concepts

Constructive Interference

Constructive interference occurs when waves from two sources align in such a way that their amplitudes add together. This happens when the path difference between the waves is a multiple of the wavelength.
This means their peaks (crests) and troughs (valleys) sync up perfectly, creating a larger wave.

• Path difference: multiple of the wavelength \,\( n\lambda \).
• Results in a wave with increased amplitude.

Understanding constructive interference helps explain why certain areas along the path of waves are prone to experiencing amplified sounds or light. In the scenario of two in-phase sources, constructive interference is highlighted when the waves travel paths that result in integer multiples of the wavelength, leading to places where the waves reinforce each other.

Destructive Interference

Destructive interference occurs when two waves meet and effectively cancel each other out. This happens when the path difference between them is an odd multiple of half the wavelength.
In this case, the crest of one wave meets the trough of another, leading to reduced or zero amplitude.

• Path difference: odd multiple of half the wavelength \,\( (n+1/2)\lambda \).
• Results in a wave with decreased or nullified amplitude.

In practical terms, destructive interference can cause areas where sound becomes markedly softer or even silent, and similarly, where light dims. However, in the problem above, consistent constructive interference was more likely because it is more straightforward to achieve the needed exact multiples with given parameters.

Path Difference

Path difference is a crucial concept when discussing wave interference. It refers to the difference in the distance traveled by two waves from their sources to a particular point.
The path difference determines the type of interference - whether constructive or destructive.

• If the path difference is a whole number multiple of the wavelength \,\( n\lambda \), constructive interference happens.
• If it's an odd multiple of half the wavelength \,\( (n+1/2)\lambda \), destructive interference occurs.

In real-world applications, engineers and scientists manipulate path differences to achieve desired interference patterns. In the exercise mentioned, recognizing the path difference helps identify points where specific interference occurs based on geometric positions of the wave sources.

Wave Sources

Wave sources are locations or points from which waves originate. The behavior of these waves as they travel depends significantly on the attributes of the sources, including phase and distance between them.
Let's break down some key aspects:

• Phase: Sources in phase produce waves that are synchronized. These waves have crests and troughs that line up, making interference analysis straightforward, particularly when identifying patterns such as constructive interference.
• Separation Distance: The distance affects potential path differences, determining what kind of interference occurs—constructive or destructive.

In our problem, the sources are in phase and separated by 4 meters. This setup allows for the evaluation of path differences and thus predicts zones of constructive interference between the sources.

Wavelength

Wavelength is the distance between successive crests, troughs, or identical points of a wave. It is a fundamental property that influences wave behavior and interference.
Measured in meters (or other units of length), wavelength links directly to path difference and thus dictating the interference type.

• A larger wavelength means waves will overlap over greater distances before repeating, affecting the location of interference patterns.
• Given \,\( \lambda = 5.00 \, \text{m} \) in this exercise, this determines the points at which waves constructively interfere based on integer multiples.

By understanding wavelength, you get a powerful insight into the regularity and locales of interference patterns, crucial for interpreting the behavior of waves between the sources in the provided exercise.