Question

An observer measures an intensity of \(1.13 \mathrm{~W} / \mathrm{m}^{2}\) at an unknown distance from a source of spherical waves whose power output is also unknown. The observer walks \(5.30 \mathrm{~m}\) closer to the source and measures an intensity of \(2.41 \mathrm{~W} / \mathrm{m}^{2}\) at this new location. Calculate the power output of the source.

Short answer

The power output of the source is approximately \(792 W\).

Step-by-step solution

Express the intensities in terms of distances and power output

The intensity \(I\) at a distance \(r\) from a source of power \(P\) is given by: \(I = \frac{P}{4πr^{2}}\). Thus:Intensities at the two locations can be represented as follows:\(I_{1} = \frac{P}{4πr_{1}^{2}}\)\(I_{2} = \frac{P}{4πr_{2}^{2}}\)

Solve the equation for the power output

Divide the first equation by the second equation. The power output \(P\) cancels out, leaving:\(\frac{I_{1}}{I_{2}} = \frac{r_{2}^{2}}{r_{1}^{2}}\)We can rearrange this equation to get:\(\frac{r_{2}^{2}}{r_{1}^{2}} = \frac{I_{1}}{I_{2}}\)Remembering that \(r_{2} = r_{1} - 5.30 m\), we get:\(\frac{(r_{1} - 5.30)^{2}}{r_{1}^{2}} = \frac{1.13}{2.41}\)After cross-multiplying and simplifying, we obtain a quadratic equation for \(r_{1}\). We only consider the positive root, since distance cannot be negative:\(r_{1} ≈ 13.6 m\)

Calculate the power output

Substitute the obtained value for \(r_{1}\) into the first equation:\(P = 4πr_{1}^{2}I_{1} ≈ 792 W\), which is the power output of the source.

Key concepts

Wave Intensity

Wave intensity is a measure of the power of a wave as it travels through a specific area. In the context of spherical waves, such as sound or light emanating from a point source, intensity describes how much energy is passing through a surface perpendicular to the direction of wave propagation. The formula for intensity \( I \) is given by:\[I = \frac{P}{4\pi r^2}\]where:

• \( I \) is the wave intensity (in watts per square meter \( \mathrm{W/m^2} \)).
• \( P \) is the power output of the wave source (in watts \( \mathrm{W} \)).
• \( r \) is the distance from the wave source (in meters \( \mathrm{m} \)).

Wave intensity diminishes with increasing distance from the source. This is due to the spreading out of the wavefronts as they cover a larger surface area. Understanding this concept is vital, as it helps to explain how energy transferred by waves decreases over distance.

Power Output

The power output of a wave source refers to the total energy emitted per unit time. In exercises like the one discussed, it is key to connect the power output with intensity to measure how energetic the wave is over a given area. Power output \( P \) in the formula can be rearranged to solve for it when other variables are known:\[P = 4\pi r^2 I\]Here, \( P \) represents the power being dispersed through a sphere at a distance \( r \), spreading the energy evenly. This concept is crucial for identifying energy sources' efficiency and behavior, which can be applied in real-world situations such as sound systems, light sources, and even environmental acoustics.

Quadratic Equation

Quadratic equations are fundamental tools in various fields of science and mathematics. In this exercise, the quadratic equation arises from the need to find distances between the observer and the source. When rearranging relationships of intensity and distance, a quadratic form can emerge:\[(r_1 - 5.30)^2 = \frac{1.13}{2.41} r_1^2\]Rewriting it results in a typical quadratic equation that can be solved using various methods such as factoring, completing the square, or using the quadratic formula. Quadratic equations often have two solutions; however, in physical scenarios like this, only one, typically non-negative, solution is meaningful. The quadratic solution then informs us of the physical measurements, such as distance, required for further calculations.

Distance Measurement

Accurate distance measurement is crucial in the study of wave phenomena because it allows us to deduce how intensity values change with position. In spherical waves, distance \( r \) plays a significant role in the formulas that define intensity, influencing how we calculate and interpret these values. In this exercise, two distances are key: one unknown \( r_1 \), and the other \( r_2 = r_1 - 5.30 \, \mathrm{m} \), creating a link between observations at different positions. The careful measurement of distance ensures that calculations remain accurate and dependable. Understanding this relationship helps explain how phenomena such as attenuation and dispersion operate, affecting everything from mobile signal strength to how we hear sound in open spaces.