Two sources of sound are separated by a distance of \(5.00 \mathrm{~m}\). They both emit sound at the same amplitude and frequency,\(300 \mathrm{~Hz}\), but they are \(180^{\circ}\) out of phase. At what points along the line connecting them will the sound intensity be the largest?

Sound intensity is a measure of the power per unit area carried by a sound wave. It's the energy that the sound wave transports per second over an area of one square meter. It's important to note that sound intensity is directly related to the amplitude of the sound wave, where a higher amplitude results in greater intensity.

When two sound waves from coherent sources interfere, the resulting sound intensity at a point can increase (constructive interference) or decrease (destructive interference). Constructive interference occurs when the waves are in phase or have a phase difference of multiples of 360 degrees, leading to a larger amplitude and thus greater sound intensity. In the context of the exercise, constructive interference is sought to locate where the sound intensity would be maximized along the line connecting two sound sources.

The wavelength of a sound wave is the distance over which the wave's shape repeats. It is determined by the speed of sound in the medium and the frequency of the sound wave. You can calculate the wavelength (\( \text{λ} \) ) with the simple formula: \( \text{λ} = \frac{v}{f} \) where \(v\) represents the speed of sound in the medium and \(f\) is the frequency of the sound.

In this exercise, the wavelength was calculated using the speed of sound in the air, which is typically around 343 meters per second, and the given frequency of 300 Hz. Understanding how to calculate the wavelength is critical because it helps determine where constructive interference will occur between two sound waves.

Coherent sources are those sound or light sources that have a constant phase difference and the same frequency. These sources are essential in creating consistent interference patterns. For interference to occur, the waves emitted by the sources must maintain a fixed phase relationship.

In the context of our problem, the two sound sources emit waves that are 180 degrees out of phase but have the same frequency, which makes them coherent, albeit they are in anti-phase. Because of their coherency, they can produce a predictable interference pattern of sound intensity along the line that connects them, enabling us to calculate where the points of maximum intensity (constructive interference) are located.

Path difference refers to the difference in distance traveled by two waves from their respective sources to a common point. It plays a crucial role in determining the type of interference that occurs at that point. For constructive interference, the path difference must be an integer multiple of the wavelength (\( n\text{λ} \) ), where \(n\) is a whole number (0, 1, 2, 3, ...).

During the exercise, a path difference of zero or multiples of the wavelength corresponds to points along the line connecting the two sources where the sound intensity is largest. These points of constructive interference are where the sound waves reinforce each other, resulting in a louder combined sound. The calculation of path difference is fundamental in predicting the interference pattern produced by coherent sources.