A sound source is located somewhere along the x-axis. Experiments show that the same wave front simultaneously reaches listeners at x = -7.0 m and x = ＋3.0 m. What is the x-coordinate of the source?

Verified step by step guidance

1. The problem involves determining the x-coordinate of the sound source based on the symmetry of the wave front. A wave front propagates outward in all directions from the source, so the source must be equidistant from the two given points.
2. Let the x-coordinate of the source be denoted as \( x_s \). The two listeners are located at \( x_1 = -7.0 \ \text{m} \) and \( x_2 = +3.0 \ \text{m} \). The source must lie at the midpoint between these two points.
3. The formula for the midpoint between two points \( x_1 \) and \( x_2 \) is given by: \( x_s = \frac{x_1 + x_2}{2} \).
4. Substitute the values \( x_1 = -7.0 \ \text{m} \) and \( x_2 = +3.0 \ \text{m} \) into the formula: \( x_s = \frac{-7.0 + 3.0}{2} \).
5. Simplify the expression to find the x-coordinate of the source. This will give the location of the sound source along the x-axis.

Key Concepts

Wave Fronts

A wave front is an imaginary surface representing points of a wave that oscillate in phase. In the context of sound waves, it indicates the position of the wave at a given moment in time. When a wave front reaches multiple observers simultaneously, it suggests that they are equidistant from the source, allowing for the determination of the source's location.

Sound Propagation

Sound propagates as a longitudinal wave through a medium, such as air, where particles oscillate parallel to the direction of wave travel. The speed of sound in air is approximately 343 m/s at room temperature. Understanding how sound travels helps in analyzing the distances from the source to the listeners based on the time it takes for the sound to reach them.

Equidistant Points

Equidistant points are locations that are the same distance from a reference point, in this case, the sound source. If two listeners receive the sound wave simultaneously, the source must lie on the perpendicular bisector of the line segment connecting the two listeners. This geometric principle is crucial for calculating the position of the sound source along the x-axis.