To find the frequency of the wave in the second rope, we can use the wave equation:
\[ v = f \cdot \lambda \]
where:
- \( v \) is the wave speed,
- \( f \) is the frequency,
- \( \lambda \) is the wavelength.
First, we need to calculate the wave speed in the first rope:
For the first rope:
- Frequency (\( f_1 \)) = 1.5 hertz
- Wavelength (\( \lambda_1 \)) = 6 meters
Calculating the wave speed (\( v_1 \)):
\[ v_1 = f_1 \cdot \lambda_1 = 1.5 , \text{Hz} \cdot 6 , \text{m} = 9 , \text{m/s} \]
Since the two ropes are identical, the speed of the wave in the second rope (\( v_2 \)) will also be 9 m/s.
Now we can apply this to the second rope:
For the second rope:
- Wavelength (\( \lambda_2 \)) = 2 meters
Using the wave speed we've calculated:
\[ v_2 = f_2 \cdot \lambda_2 \]
Setting \( v_2 = 9 , \text{m/s} \) and substituting for \( \lambda_2 \):
\[ 9 = f_2 \cdot 2 \]
Now solving for \( f_2 \):
\[ f_2 = \frac{9}{2} = 4.5 , \text{Hz} \]
So the frequency of the wave in the second rope is \( \boxed{4.5 , \text{hertz}} \).