Question

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?

Short answer

The points of largest sound intensity occur at distances of $2.5+n(1.14)$ m and $2.5-n(1.14)$ m from the middle point, where $n=0,1,2,3...$, with the constraint that the distances cannot exceed the length of the line connecting the two sources, i.e. 5.00 m.

Step-by-step solution

Calculate the Wavelength

Use the formula for the speed of sound ($v=f\lambda$), where $v$ is the speed of sound $=343m/s$, $f$ is frequency $=300Hz$, and $\lambda$ is the wavelength we wish to find. Solving for $\lambda$ gives $\lambda =v/f$, so the wavelength $\lambda =343/300=1.14m$.

Determine the Points of Largest Intensity

The two sources act as two coherent sources producing constructive interference where the path difference is equal to $n\lambda$, where $n$ is an integer (0, 1, 2, 3, ...). For the maximum constructive interference, the path difference should be minimum, i.e. $n=0$ leading to path difference $=0$. This will be the point exactly in the middle of the two sources. The other points are at distances of $n$ times the wavelength from this middle point. Thus the points are at $2.5+n(1.14)$ for $n=0,1,2,3...$ to right and at $2.5-n(1.14)$ for $n=0,1,2,3...$ to left of the mid-point.

Identify the Limiting Values

The points cannot exist beyond the positions of the two sources. Since the sources are 5.00 m apart, any calculated value of distance greater than 5.00 m would be impossible.

Key concepts

Sound Intensity

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.

Wavelength Calculation

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

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

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.