18. Waves & Sound

A guitar string is supposed to vibrate at 247 Hz, but is measured to actually vibrate at 262 Hz. By what percentage should the tension in the string be changed to get the frequency to the correct value?

A particular string resonates in four loops at a frequency of 320 Hz. Name at least three other frequencies at which it will resonate. What is each called?

Two strings on a musical instrument are tuned to play at 392 Hz (G) and 494 Hz (B). What are the frequencies of the first two overtones for each string?

Two strings on a musical instrument are tuned to play at 392 Hz (G) and 494 Hz (B). If the strings, instead, have the same mass per unit length and are under the same tension, what is the ratio of their lengths (ℓ_G / ℓ_B)?

One end of a horizontal string is attached to a small-amplitude mechanical 60.0-Hz oscillator. The string’s mass per unit length is 3.9 x 10⁻ ⁴ kg/m. The string passes over a pulley, a distance ℓ = 1.50 m away, and weights are hung from this end, Fig. 15–38. What mass m must be hung from this end of the string to produce five loops of a standing wave? Assume the string at the oscillator is a node, which is nearly true.

A transverse wave pulse travels to the right along a string with a speed v = 2.4 m/s. At t = 0 the shape of the pulse is given by the function $D=\frac{4.0{\text{m}}^3}{x^2+2.0{\text{m}}^2}$, where D and x are in meters. Determine a formula for the wave pulse at any time t assuming there are no frictional losses.

The displacement of a transverse wave traveling on a string is represented by D₁ = 4.2 sin (0.84 x - 47t + 2.1), where D₁ and x are in cm and t in s. Find an equation that represents a wave which, when traveling in the opposite direction, will produce a standing wave when added to this one.

One string of a certain musical instrument is 75.0 cm long and has a mass of 8.75 g. It is being played in a room where the speed of sound is 344 m/s. (a) To what tension must you adjust the string so that, when vibrating in its second overtone, it produces sound of wavelength 0.765 m? (Assume that the break-ing stress of the wire is very large and isn't exceeded.) (b) What frequency sound does this string produce in its fundamental mode of vibration?

The lowest note on a grand piano has a frequency of 27.5 Hz. The entire string is 2.00 m long and has a mass of 400 g. The vibrating section of the string is 1.90 m long. What tension is needed to tune this string properly?

Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. When properly tuned, the note A should have a frequency of 440 Hz and the note E should be at 659 Hz. What is the frequency difference between the third harmonic of the A and the second harmonic of the E?

A 65-cm guitar string is fixed at both ends. In the frequency range between 1.0 and 2.0 kHz, the string is found to resonate only at frequencies 1.2, 1.5, and 1.8 kHz. What is the speed of traveling waves on this string?

A carbon dioxide laser is an infrared laser. A CO₂ laser with a cavity length of 53.00 cm oscillates in the m=100,000 mode. What are the wavelength and frequency of the laser beam?

In a laboratory experiment, one end of a horizontal string is tied to a support while the other end passes over a frictionless pulley and is tied to a 1.5 kg sphere. Students determine the frequencies of standing waves on the horizontal segment of the string, then they raise a beaker of water until the hanging 1.5 kg sphere is completely submerged. The frequency of the fifth harmonic with the sphere submerged exactly matches the frequency of the third harmonic before the sphere was submerged. What is the diameter of the sphere?