To determine the distance from the wall where a plane sound wave has an antinode, we need to understand the behavior of sound waves when they reflect off surfaces. Let's break this down step-by-step. ### Step 1: Understanding Sound Waves and Antinodes 1. **Sound Waves**: Sound waves are longitudinal waves that consist of compressions and rarefactions. When a sound wave travels and hits a wall, it reflects back. 2. **Nodes and Antinodes**: In a standing wave pattern, nodes are points where there is no movement (minimum amplitude), and antinodes are points where the amplitude is maximum. For a sound wave reflecting off a wall, the wall acts as a node because the air particles at the wall cannot move (the wall is a fixed boundary). ### Step 2: Wave Properties Given: - Frequency (f) = 85.5 Hz - Velocity (v) = 342 m/s We can calculate the wavelength (λ) of the sound wave using the formula: \[ v = f \cdot \lambda \] Rearranging gives: \[ \lambda = \frac{v}{f} \] Substituting the values: \[ \lambda = \frac{342 \, \text{m/s}}{85.5 \, \text{Hz}} \approx 4 \, \text{m} \] ### Step 3: Distance to Antinode In a standing wave formed by the reflection of a sound wave, the distance from the wall (node) to the first antinode is a quarter of the wavelength (λ/4). This is because the first antinode occurs at a distance of λ/4 from the node (the wall in this case). Calculating the distance to the first antinode: \[ \text{Distance to first antinode} = \frac{\lambda}{4} = \frac{4 \, \text{m}}{4} = 1 \, \text{m} \] ### Conclusion Thus, the distance from the wall where the wave has an antinode is **1 meter**. ### Explanation of Options - **Option A (2m)**: This is incorrect because it does not correspond to the position of the first antinode, which is at λ/4. - **Option B (4m)**: This is incorrect as it represents a full wavelength, which would correspond to a node, not an antinode. - **Option C (1m)**: This is the correct answer, as we calculated that the first antinode occurs at a distance of 1 meter from the wall. - **Option D (3m)**: This is incorrect because it does not correspond to any specific node or antinode position in the standing wave pattern. ### Summary - The wavelength of the sound wave is calculated to be 4 meters. - The first antinode occurs at a distance of λ/4 from the wall, which is 1 meter. - The wall acts as a node, and the first antinode is located 1 meter away from it. - The correct answer is **C (1m)**. This understanding of wave behavior is crucial for solving similar problems in wave mechanics.