C.4 Standing waves and resonance：練習問題

A standing wave is formed on a string of length \( L \) that is fixed at both ends. Which of the following describes the distance between two adjacent nodes?

A pipe of length \( L \) is open at one end and closed at the other. If the speed of sound is \( v \), what is the frequency of the third harmonic of this pipe? (Assume no end correction.)

The diagram shows a resonance tube experiment where a tuning fork of frequency \( f \) is held over a tube. The water level is adjusted until the first resonance is heard at a distance \( L \) from the top of the tube. What is the expression for the speed of sound \( v \), ignoring end correction?

Which of the following properties is true for standing waves but not for traveling waves?

The diagram shows a standing wave on a string of length \( L \) fixed at both ends, vibrating in its second harmonic. What is the wavelength \( \lambda \) of this standing wave?

A string of length 1.2 m is fixed at both ends and held under a tension of 60 N. The linear mass density of the string is \(0.04 \text{ kg m}^{-1}\).
(a) Calculate the speed of a transverse wave on the string.
(b) Determine the frequency of the fundamental mode of vibration.
(c) State the wavelength of the second harmonic and calculate its frequency.

An organ pipe is closed at one end and open at the other. The fundamental frequency of the pipe is 220 Hz when the speed of sound is \(340 \text{ m s}^{-1}\).
(a) Calculate the length of the pipe.
(b) Sketch the displacement standing wave pattern for the next possible harmonic and calculate its frequency.
(c) The air in the pipe is replaced with helium, in which the speed of sound is \(970 \text{ m s}^{-1}\). Calculate the new fundamental frequency.