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Solved problems for the chapter:

Stationary Waves

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Stationary transverse wave

1. Transverse wave reflection

2. Stationary (Standing) waves

2.1. Fundamental oscillation frequency and harmonic components

2.1.1. Rope clamped on both sides

2.1.2. Elastic plate clamped on one side

2.1.3. Free elastic tile on both sides

3. Forum

↓ Show theory [1] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:stationary_waves/4067/

Example 1.

Two strings of equal length and material differ only in diameter. The ratio of their diameters is 1 : 3. What is the ratio of their fundamental frequencies with which they oscillate?

Example 2.

Two identical strings are fixed at both ends. What is the fraction of the tensions on both strings in order for the fundamental frequency of the first string to be equal to the second harmonic frequency of the second string?

The string of a guitar sounds with a fundamental frequency of 392 Hz. The length of the string is 90 cm and the mass of the string is 0.2 g. What is the tension of the string?

A guitar has six strings of equal length and tension. The strings oscillate with fundamental frequencies 164.81 Hz, 220.00 Hz, 293.66 Hz, 392.00 Hz, 493.88 Hz, and 659.26 Hz respectively. What is the ratio of the mass of each string to the mass of the first string with the lowest fundamental frequency?

A 6 m long rope is clamped at one end. At the other end, it is plucked by hand with a small amplitude. The wave speed on the rope is . At what frequency do we have to swing our hand to form a wave node in the middle of the rope?

A company of soldiers marches rhythmically along a 25 m long bridge. At what marching speed can the soldiers bring the bridge into resonance if the speed of the transverse wave on the bridge is and the stride length of a soldier is 0.8 m?

Fixed at both ends, a 0.5 m long steel wire oscillates at a frequency of 400 Hz.

With what tensile stress is the wire tensioned?
If the wire is then cooled by 15 K, by how much does the natural oscillation frequency change?

The density of steel is , the young modulus of steel is , the linear expansivity of steel is .

A string with a mass of 0.25 g and a length of 60 cm is attached at one end to a tuning fork oscillating at a frequency of 440 Hz. If a block is hung on it at the other end via a pulley as shown in the figure below, what is the mass of this block if there are three antinodes on the string?

Standing transverse wave

A half-open pipe has a length of .

What is its fundamental frequency?
If the pipe is then fully opened, how does the fundamental frequency change?

The speed of sound in air is .

10.

An ultrasonic wave propagates from air to water. What is the maximum angle of incidence at which the wave can still penetrate the water? The wave velocity in water is and in air .

11.

We listen to music from a speaker in an empty room measuring and without wall coverings that would dampen the sound. At what fundamental frequencies can the speaker emit the sound so that a stationary wave is generated in the room? How do we avoid it? The speed of sound propagation is .

12.

An ultrasonic signal passes from water to air at an angle of 45 degrees. In which direction does it spread into the air? The wave velocity in water is and in air .

13.

An organ consists of a semi-open pipe. The fundamental frequency of the organ is . How long must the pipe be to transmit this frequency? The speed of sound is .

14.

The air in a pipe oscillates at a fundamental frequency of . The temperature of the air in the pipe is . At what frequency does it oscillate if the temperature of the air is increased by ?

15.

The auditory canal of the human ear is about long. What is the fundamental frequency of air oscillations in the canal? Consider the auditory canal as a half-open pipe. The speed of sound in air is .

16.

What is the ratio of the fundamental oscillation frequencies of two pipes of equal length if the first pipe is open at both ends and the second is opened only at one end?

17.

Approximately what must be the length of a pipe open at both ends so that the third harmonic frequency and the fundamental frequency differ by . The speed of longitudinal wave in the air is .

18.

There is a concert in the town square. In front of the stage is a grandstand for the audience, and behind the grandstand is the palace that reflects the sound back towards the audience. A listener at some distance from the stage hears the music in such a way that each tone emitted by the orchestra is repeated after a time of 0.8 s. How far is the palace from the listener? The speed of sound is 340 m/s.

19.

The lowest tone emitted by an organ is called C and has a frequency of . Which pipe would you choose for this tone and what would be its length?