Solved Examples: Standing Waves on a String with Fixed Ends

📚 Quick Study Guide

• 📏 Definition: Standing waves are formed when two waves of the same frequency interfere with each other while traveling in opposite directions.
• 🧱 Fixed Ends: For a string with fixed ends, the ends of the string must be nodes (points of zero displacement).
• 🔢 Wavelengths: The possible wavelengths for standing waves on a string of length $L$ are given by $\lambda_n = \frac{2L}{n}$, where $n = 1, 2, 3, ...$ is the mode number.
• 📈 Frequencies: The corresponding frequencies are given by $f_n = \frac{nv}{2L}$, where $v$ is the speed of the wave on the string.
• 🎻 Wave Speed: The speed of a wave on a string is determined by $v = \sqrt{\frac{T}{\mu}}$, where $T$ is the tension in the string and $\mu$ is the linear mass density (mass per unit length).
• 🎼 Harmonics: The fundamental frequency (n=1) is the first harmonic, n=2 is the second harmonic (first overtone), n=3 is the third harmonic (second overtone), and so on.

Practice Quiz

1. Question 1: A string of length 2m is fixed at both ends. What is the wavelength of the third harmonic?
   1. A. 1.33 m
   2. B. 4 m
   3. C. 0.67 m
   4. D. 2 m
2. Question 2: A string fixed at both ends has a length of 3m and a wave speed of 150 m/s. What is the frequency of the second harmonic?
   1. A. 25 Hz
   2. B. 50 Hz
   3. C. 75 Hz
   4. D. 100 Hz
3. Question 3: If the tension in a string is quadrupled, how does the speed of the wave on the string change?
   1. A. It doubles
   2. B. It quadruples
   3. C. It halves
   4. D. It remains the same
4. Question 4: A string of length $L$ is vibrating at its fundamental frequency. If the length of the string is doubled, what happens to the fundamental frequency (assuming tension remains constant)?
   1. A. It doubles
   2. B. It quadruples
   3. C. It halves
   4. D. It remains the same
5. Question 5: What is the distance between two consecutive nodes in a standing wave pattern?
   1. A. $\lambda$
   2. B. $\lambda/2$
   3. C. $\lambda/4$
   4. D. $2\lambda$
6. Question 6: A string has a linear mass density of 0.01 kg/m and is under a tension of 100 N. What is the speed of the wave on the string?
   1. A. 10 m/s
   2. B. 100 m/s
   3. C. 1000 m/s
   4. D. 0.1 m/s
7. Question 7: For a string fixed at both ends, which of the following statements is true at a node?
   1. A. Displacement is maximum
   2. B. Displacement is minimum
   3. C. Velocity is maximum
   4. D. Acceleration is maximum