To estimate the length of the vibrating portion of the string and calculate the wave speed, we'll break the problem down into manageable steps.

Step 1: Estimate the length of the vibrating portion of the string

For a violin, the vibrating portion of the string refers to the length of the string between the bridge and the nut. The typical length of this vibrating portion on a violin is about 32 cm (0.32 m), though it can vary slightly depending on the violin. We’ll use this as a reasonable estimate for the length of the vibrating portion of the string.

Step 2: Relate frequency to wave speed

The fundamental frequency of the vibrating string is related to the wave speed $v$, the string's length $L$, and the frequency $f$ by the following relationship for the fundamental mode of vibration:

$f = \frac{v}{2L}$

where:

• $f$ is the frequency of the wave (659 Hz for the open E5 string),
• $L$ is the length of the vibrating portion of the string (0.32 m),
• $v$ is the wave speed on the string, which we need to calculate.

Rearranging the formula to solve for $v$:

$v = 2L \times f$

Substitute the known values:

$v = 2 \times 0.32 \, \text{m} \times 659 \, \text{Hz}$

$v = 2 \times 0.32 \times 659 = 421.12 \, \text{m/s}$

Step 3: Conclusion

The estimated wave speed on the string for the E5 note vibrating in its fundamental mode is approximately 421 m/s.

Would you like further details on this calculation, or do you have any questions? Here are some related topics you might be interested in:

1. How does the tension in the string affect the wave speed?
2. How does changing the frequency of the string affect its vibration and wave speed?
3. How can the length of a string on a violin be adjusted to tune it to a specific note?
4. What other factors influence the wave speed on a violin string besides tension and length?
5. How does the wave speed relate to the harmonics and overtones of the vibrating string?

Tip: The wave speed $v$ is directly proportional to the square root of the tension $T$ and inversely proportional to the square root of the mass per unit length $μ$ of the string: $v = \sqrt{T/\mu}$.