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Chapter 3: Problem 66

Show that an earthquake with Richter magnitude \(R\) has seismic waves of size \(S_{0} 10^{R},\) where \(S_{0}\) is the size of the seismic waves of an earthquake with Richter magnitude \(0 .\)

Short Answer

Expert verified
The size of seismic waves for an earthquake with Richter magnitude 0 is equal to S鈧�. Using the given formula, we can calculate the size of seismic waves for an earthquake with any Richter magnitude R as \(S = S_0 \cdot 10^R\), which proves the given statement.

Step by step solution

01

Understand the Richter magnitude scale

The Richter magnitude scale is a logarithmic scale that measures the energy released by an earthquake. It was developed by Charles F. Richter in 1935 and is now widely used across the world. The formula for the Richter magnitude scale is given by: \[R = \log_{10}(\frac{A}{A_0})\] Where R is the Richter magnitude, A is the amplitude of the seismic waves, and A鈧� is the amplitude of seismic waves at a reference level.
02

Calculate the size of seismic waves for earthquake with Richter magnitude 0

Using the formula given in the exercise, let's calculate the size of seismic waves for an earthquake with Richter magnitude 0. \[S = S_0 \cdot 10^R\] Since the Richter magnitude (R) for the earthquake is 0, the formula becomes: \[S = S_0 \cdot 10^0\] And as \(10^0 = 1\), we have: \[S = S_0 \cdot 1\] Thus, the size of seismic waves for an earthquake with Richter magnitude 0 is equal to S鈧�.
03

Calculate the size of seismic waves for earthquake with Richter magnitude R

Now, let's apply the given formula to find the size of seismic waves for an earthquake with any Richter magnitude R. Using the formula: \[S = S_0 \cdot 10^R\] To find the size of seismic waves, we can plug in the values of S鈧� and R into the formula. Since we are not given any specific value for S鈧�, the formula remains: \[S = S_0 \cdot 10^R\] This is the given formula in the exercise, and we have shown that earthquakes with any Richter magnitude R have seismic waves of the size S鈧� \(\cdot\) 10^R, where S鈧� is the size of the seismic waves of an earthquake with Richter magnitude 0.

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