Two earthquake waves of the same frequency travel through the same portion of the Earth, but one is carrying 3.5 times the energy. What is the ratio of the amplitudes of the two waves?

Verified step by step guidance

1. Understand the relationship between the energy of a wave and its amplitude. The energy of a wave is proportional to the square of its amplitude. Mathematically, this can be expressed as: $E\; \backslash (\backslash propto\backslash )\; A^2$, where $E$ is the energy and $A$ is the amplitude.
2. Set up the ratio of the energies of the two waves. Let the energy of the first wave be $E\_1$ and the energy of the second wave be $E\_2$. According to the problem, $E\_2\; =\; 3.5\; \backslash (\backslash cdot\backslash )\; E\_1$.
3. Express the relationship between the amplitudes of the two waves using the proportionality: $\backslash (\backslash frac\{E\_2\}\{E\_1\}\backslash )\; =\; \backslash (\backslash left\backslash )(\backslash (\backslash frac\{A\_2\}\{A\_1\}\backslash )\backslash (\backslash right\backslash ))^2$. Substitute $E\_2\; =\; 3.5\; \backslash (\backslash cdot\backslash )\; E\_1$ into this equation.
4. Simplify the equation to solve for the ratio of the amplitudes: $\backslash (\backslash frac\{A\_2\}\{A\_1\}\backslash )\; =\; \backslash (\backslash sqrt\{\backslash frac\{E\_2\}\{E\_1\}\backslash )\}$. Substitute $\backslash (\backslash frac\{E\_2\}\{E\_1\}\backslash )\; =\; 3.5$ into the equation.
5. Conclude that the ratio of the amplitudes is the square root of 3.5. This gives the final expression: $\backslash (\backslash frac\{A\_2\}\{A\_1\}\backslash )\; =\; \backslash (\backslash sqrt\{3.5\}\backslash )$.

Key Concepts

Energy and Amplitude Relationship

In wave physics, the energy carried by a wave is proportional to the square of its amplitude. This means that if one wave carries more energy than another, we can determine the ratio of their amplitudes by taking the square root of the ratio of their energies.

Wave Properties

Waves are characterized by properties such as frequency, wavelength, and amplitude. Frequency refers to how many cycles occur in a given time, while amplitude measures the maximum displacement from the rest position. Understanding these properties is essential for analyzing wave behavior.

Ratio Calculation

Calculating ratios involves comparing two quantities to understand their relative sizes. In this context, we will use the relationship between energy and amplitude to find the ratio of the amplitudes of the two earthquake waves based on their energy difference.