Physics 1120 In-Class Problems: Standing Waves and Sound Level

Standing Waves and Resonance

1. A wire of length 4.35 m and mass 137 g is under a tension of 125 N. A standing wave has formed which has seven nodes including the endpoints. What is the frequency of this wave? Which harmonic is it? What is the fundamental frequency? The maximum amplitude at the antinodes is 0.0075 m, write an equation for this standing wave.

2. A string fixed at one end only is vibrating in its third harmonic. The wave function is y(x,t) = 0.02sin(3.13x)cos(512t), where y and x are in metres and t is in seconds. (a) What is the wavelength of the wave? (b) What is the length of the string. (c) What is the speed of the transverse wave in the string?

3. Three successive resonance frequencies for a certain string are 175, 245, and 315 Hz. (a) Find the ratio of these three modes. (b) How can you tell that this sting has an antinode at one end? (c) What is the fundamental frequency? (d) Which harmonics are these resonance frequencies? (e) If the speed of transverse waves on this string is 125 m/s, find the length of the string?

4. Having no ruler at hand, you decide to determine the length of a hollow open-ended tube by holding one end next to a sound source and varying the frequency of the sound. The lowest frequency that causes resonance in the tube is f = 420 Hz. Assuming that the speed of sound is 343 m/s, what is the length of the tube? Sketch the standing wave.
5. In an archery contest, a 1.15-m arrow sticks in the target and vibrates. If the speed of the waves in that arrow is 150 m/s, what are the three lowest resonance frequencies for the that vibration? Sketch the standing waves.

6. A narrow hollow tube of length L = 3.50 m is open at both ends and sits on two sawhorses. A wind is blowing and the tube is "moaning". What is the lowest possible frequency for the sound produced? Sketch the standing wave. Take the speed of sound to be 340 m/s.
   
   

7. While watching an archery show you taped earlier, you notice that when the 1.05 m arrows hit a target they vibrate with the following pattern. Using frame advance on your VCR you deduce that the arrow vibrates 110 times per second. What is the speed of the waves travelling through the arrow?
   

   Sound Level

8. A typical speaker diaphragm vibrates with a maximum displacement of 2.00 mm. Assuming that this is also the maximum displacement of the nearby air molecules, find the maximum pressure amplitude. Take the frequency of the speaker to be f = 3000 Hz, the density of the air to be ρ = 1.29 kg/m³, and the speed of sound to be 340 m/s.

9. When one student is doing an exam in an otherwise very quiet room, the sound level is 45 dB. What is the intensity of the noise produced by the student? If there are 30 equally noisy students in the room, and assuming that you are the same distance from all the students, what would the new sound level be?

10. You've been out very late and when you come home your parents are very angry and start shouting at you. This upsets the family dog who starts howling. The diagram below shows their positions with you in the middle. The distances are r_DAD = 1.20 m, r_MOM = 1.35 m, and r_DOG = 2.00 m. The power in their voices are respectively, P_DAD = 1.25 mW, P_MOM = 0.85 mW, and P_DOG = 1.00 mW. Find the intensity of sound from each source at your position. Treat the sources as incoherent (in the physical sense) and find the sound level. Be sure to include the effects of the normal background sound level of 40 dB. I₀ = 10^-12 W/m².

Questions?mike.coombes@kwantlen.ca