18. Waves & Sound

In the following figure, what is the harmonic number of the standing wave? The wavelength of the standing wave? If the frequency of the standing wave is 30 Hz, what is the speed of the waves producing the standing wave?

An unknown mass hangs on the end of a 2 m rope anchored to the ceiling when a strong wind causes the rope to vibrate and hum at its fundamental frequency of 100 Hz. If the rope has a mass of 0.15 kg, what is the unknown mass?

By whipping a string up and down, you determine the fundamental frequency to be 4 Hz. If you attached the string to a motorized oscillator and increased the frequency to 28 Hz, how many loops would this standing wave have?

One of the harmonic frequencies for a particular string under tension is 325 Hz. The next higher harmonic frequency is 390 Hz. What harmonic frequency is next higher after the harmonic frequency 195 Hz?

The figure below shows a standing wave on a 2.0-m-long string that has been fixed at both ends and tightened until the wave speed is 40 m/s. What is the frequency of this wave?

A 3m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 60 m/s. What are the wavelength and frequency of the second overtone?

(I) The fundamental frequency of a violin string is 441 Hz when unfingered. What is its fundamental frequency if it is fingered one-third of the way down from the end? (That is, only two-thirds of the string vibrates as a standing wave.)

(II) The speed of waves on a string is 96 m/s. If the frequency of standing waves is 435 Hz, how far apart are two adjacent nodes?

(II) A guitar string is 91 cm long and has a mass of 3.2 g. The vibrating portion of the string from the bridge to the support post is ℓ = 64cm and the string is under a tension of 520 N. What are the frequencies of the fundamental and first two overtones?

(II) The displacement of a standing wave on a string is given by D = 2.4 sin ( 0.60x ) cos (42t) , where x and D are in centimeters and t is in seconds. (a) What is the distance (cm) between nodes?

(II) The displacement of a standing wave on a string is given by D = 2.4 sin ( 0.60x ) cos (42t) , where x and D are in centimeters and t is in seconds. (b) Give the amplitude, frequency, and speed of each of the component waves.

(II) A particular violin string plays at a frequency of 294 Hz. If the tension is increased 22%, what will the new frequency be?

(II) When you slosh the water back and forth in a tub at just the right frequency, the water alternately rises and falls at each end, remaining relatively calm at the center. Suppose the frequency to produce such a standing wave in a 45-cm-wide tub is 0.85 Hz. What is the speed of the water wave?

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?

(I) 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).

(a) 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).

(c) 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)?

(II) 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 (c) five loops of a standing wave? Assume the string at the oscillator is a node, which is nearly true.

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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 = 4.0m³ / (x² + 2.0m²) ,

where D and x are in meters.

(b) Determine a formula for the wave pulse at any time t assuming there are no frictional losses.

II) 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. (a) Find an equation that represents a wave which, when traveling in the opposite direction, will produce a standing wave when added to this one.