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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?

20.

A stationary wave is set up on a string that is stretched between two fixed points that are apart. At one instant, the appearance of the string is as shown.

What is the wavelength of the stationary wave?

21.

A stationary sound wave is formed in the air column inside a tube that is open at both ends. The stationary wave has three nodes. How many antinodes does it have?

22.

The sound from a loudspeaker placed above a tube causes resonance of the air in the tube. A stationary wave is formed with two nodes and two antinodes, as shown.

The tube has height . The speed of sound in the air is . What is the frequency of the sound?

23.

Two waves superpose. A resultant wave pattern is formed. Which statement about the two waves must be correct?

They have the same amplitude.
They are of the same type.
They are transverse waves.
They travel in opposite directions.

24.

Two progressive waves meet at a point. Which condition must be met for superposition of the waves to occur?

The waves must be coherent.
The waves must be of the same type.
The waves must be travelling in opposite directions.
The waves must meet in phase.

25.

In an experiment, a stationary wave is formed on a string stretched horizontally between two fixed points. Which statement about the experiment is correct?

At certain times, the string between two nodes is horizontal with all points having zero displacement.
Each point on the string between two antinodes has an oscillation of the same amplitude.
The number of nodes is equal to the number of antinodes.
Two adjacent antinodes oscillate in phase.

26.

Stationary sound waves can be formed in the air columns of pipes. One type of pipe is closed at one end and open at the other end. Another type of pipe is open at both ends. Which pipe can form a stationary sound wave with the lowest frequency?

27.

What can explain how stationary waves are formed from progressive waves?

diffraction
polarisation
superposition
the Doppler effect

28.

A pipe has a length of . It is open at one end and closed at the other end. A stationary sound wave is set up within the pipe. There are four nodes (N) and four antinodes (A) within the length of the pipe.

What is the wavelength of the sound wave?

29.

Which statement concerning a stationary wave is correct?

All the particles between two adjacent nodes oscillate in phase.
The amplitude of the stationary wave is equal to the amplitude of one of the waves creating it.
The wavelength of the stationary wave is equal to the separation of two adjacent nodes.
There is no displacement of a particle at an antinode at any time.

30.

A string is stretched between a vibration generator and a fixed point. When the vibration generator is vibrating at a frequency , a stationary wave with five nodes is created on the stretched string, as shown. There is a node at the end of the string that is attached to the vibration generator.

The frequency of vibration of the vibration generator is slowly increased. What is the next frequency that produces a stationary wave on the string?

31.

A stationary wave is formed by two progressive waves travelling in opposite directions along the same line of travel. Which statement about the two progressive waves is not correct?

They must have a constant phase difference.
They must have the same frequency.
They must have the same wavelength.
They must travel at the same speed.

32.

The diagram shows a string stretched between fixed points X and Y. There is a stationary wave on the string.

The solid curve shows the string at a position of maximum displacement at time . The dashed curve shows the other position of maximum displacement. The straight central dashed line shows the mean position of the string. Point S on the string is directly above point P. Point T on the string is directly below point Q. Which statement is correct?

A short time after , point R on the string will be displaced.
A short time after , points S and T on the string move in opposite directions.
The distance between P and Q is one wavelength.
Two moving points on the string that are equal distances from point R vibrate in phase.

33.

A pipe, closed at one end, has a loudspeaker at the open end. For some frequencies of sound from the loudspeaker, a stationary sound wave is formed in the air within the pipe with an antinode at the open end of the pipe.

The length of the pipe is . The speed of sound in air is . Which frequency of sound from the loudspeaker would not produce a stationary wave?

34.

The variation with distance of the intensity along a stationary sound wave in air is shown.

The speed of sound in air is . What is the frequency of the sound wave?

35.

In an experiment to produce a stationary sound wave in air, a fine powder is initially evenly distributed along the length of a horizontal glass tube which is closed at one end. At the open end of the tube, a loudspeaker emits a sound wave of a constant wavelength. A stationary wave is formed and the powder accumulates in regularly spaced piles, as shown.

Which statement explains the positions of the piles of powder within the tube?

The piles are where the air molecules vibrate with maximum amplitude.
The piles are where the air molecules vibrate with minimum amplitude.
The piles are where the air molecules vibrate with the highest frequency.
The piles are where the air molecules vibrate vertically.

36.

A musical instrument is made using a long tube with a mouthpiece at one end. The other end is open and flared, as shown.

A musician maintains stationary sound waves with a node at the mouthpiece and an antinode at the other end. The lowest frequency of sound that the instrument can produce is . Which different frequencies of sound can be produced by the instrument?

37.

Two waves of equal frequency and amplitude are travelling in opposite directions along a stretched string. When they meet, they form a stationary wave with three nodes and two antinodes. The frequency of both waves is doubled and a new stationary wave is formed. How many antinodes are there in the new stationary wave?

38.

Two waves of the same type overlap. When does the principle of superposition apply?

always
only when the waves have the same amplitude
only when the waves have the same frequency
only when the waves travel in opposite directions

39.

Two progressive waves overlap. What is an essential condition for the two waves to form a stationary wave?

The waves are longitudinal.
The waves are polarised.
The waves travel in opposite directions.
The waves travel in the same direction.