Question

An acoustic waveguide consists of a long cylindrical tube with radius r designed to channel sound waves, A tone with frequency f is emitted from a small source at the center of one end of this tube. Depending on the radius of the tube and the frequency of the tone, pressure nodes can develop along the tube axis where rays reflected from the periphery constructively interfere with direct rays1) The tube has radius 25.0 cm and the temperature is 20∘C. If the tone has frequency 2.30 kHz, how many nodes exist?2) At what distance d are these nodes located?3) If the tube were filled with helium rather than air, how many nodes would exist?4) At what value of d are these nodes located?

Answer 1

The number of nodes and their distances in an acoustic waveguide can be calculated using the principles of standing waves, with adjustments for the medium's properties such as the speed of sound.

Step-by-step explanation:

To determine the number of nodes and their location in an acoustic waveguide (in this case, a cylindrical tube), we use the principle of a standing wave in a tube that is closed at one end and open at the other. The standing wave pattern created in such a tube consists of a node at the closed end and an antinode at the open end. Since the length of the tube is equal to ¼ of the wavelength (λ) of the standing wave, knowing the speed of sound in air at 20℃, we can calculate the wavelength using the formula λ = v/f, where v is the speed of sound and f is the frequency. For air at 20°C, the speed of sound is approximately 343 m/s. We can then determine the number of nodes from the total length available in the tube for the standing waves to form.

For helium, the process is similar but the speed of sound is different because the speed of sound in helium at the same temperature is higher, approximately 927 m/s. This affects the wavelength and consequently the number of nodes and their distance apart, as they will change with the speed of sound in the medium.