Standing Waves and Normal Modes - UNSOLVED PRACTICE SET

Q7. Distinguish between traveling waves and standing waves with respect to:

(i) Energy transfer

(ii) Amplitude at different points

(iii) Phase relationship between points

Q8. Define node and antinode. What is the phase difference between two points on opposite sides of a node?

Q9. A string of length L is fixed at both ends. Write the expressions for the frequencies of the first three normal modes.

Q10. Explain why all harmonics are present in the vibrations of a string fixed at both ends, but only odd harmonics are present in a pipe closed at one end.

Q11. What is the difference between a harmonic and an overtone? For a string fixed at both ends, identify the first three harmonics and their corresponding overtones.

Q12. A string vibrates in its fundamental mode with frequency f. If the tension is doubled and the length is halved, what is the new fundamental frequency?

Q13. Derive the expression for the frequencies of normal modes of vibration for a string of length L fixed at both ends. Show that:

(i) The allowed wavelengths are λ_n = 2L/n

(ii) The allowed frequencies are f_n = nv/2L

(iii) All harmonics are present

Draw diagrams showing the first three normal modes and label nodes and antinodes.

Q14. Discuss standing waves in pipes:

(i) Pipe open at both ends: derive the expression for normal mode frequencies

(ii) Pipe closed at one end: derive the expression for normal mode frequencies

(iii) Compare the harmonic content in both cases

(iv) Explain why the quality (timbre) of sound differs between an open pipe and a closed pipe of the same fundamental frequency

Q15. A student investigates standing waves on a string using a vibrator and a pulley system.

(i) Describe the experimental setup.

(ii) How does she identify the fundamental mode and higher harmonics?

(iii) She observes that increasing the tension increases the frequency. Explain why.

(iv) She notices that lightly touching the string at its midpoint stops the fundamental but allows the second harmonic. Explain this observation.

Q16. A string of length 1 m and mass 5 g is stretched with a tension of 80 N.

(i) Calculate the speed of transverse waves on the string.

(ii) Calculate the fundamental frequency.

(iii) Calculate the frequencies of the first three overtones.

(iv) If the string is plucked at its midpoint, which modes are excited? Explain.

(v) The string is now stopped at its midpoint (like pressing a guitar string at the 12th fret). Calculate the new fundamental frequency.

Q17. An organ pipe is 0.5 m long.

(i) Calculate the fundamental frequency if the pipe is open at both ends. (Speed of sound = 340 m/s)

(ii) Calculate the fundamental frequency if the pipe is closed at one end.

(iii) Calculate the first three overtones for each case.

(iv) A student blows air into the open pipe with increasing intensity. She hears the fundamental and then higher notes. Explain why higher modes are excited.

(v) The temperature rises from 20°C to 30°C. How does the fundamental frequency change?

Q18. In a school music room, a tanpura has four strings of different lengths and thicknesses. The first string is 1 m long, has linear mass density 0.5 g/m, and is under 100 N tension.

(i) Calculate the fundamental frequency of this string.

(ii) The player adjusts the string to vibrate in its second harmonic. Calculate the new frequency and describe how the string vibrates.

(iii) A sympathetic string of the same length and tension but half the linear mass density is placed nearby. Calculate its fundamental frequency. Why is it called "sympathetic"?

(iv) The tanpura produces a rich sound with many overtones. Explain how the shape of the bridge (jawari) contributes to this.

(v) If the player shortens the vibrating length to 0.8 m by placing a finger on the string, what is the new fundamental frequency?