Standing Waves & Resonance

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Standing Waves

Standing waves are produced by two waves travelling in opposite directions along the same line with the same frequency.
This is usually achieved when a travelling wave superimposes its reflection.
The superposition produces a wave pattern where the crests and troughs only move vertically.
The waves must have the same wavelength and a similar amplitude.
Standing waves store energy, while progressive waves transfer energy.

Nodes & Antinodes

Nodes are locations of zero amplitude, separated by half a wavelength (λ/2).
Antinodes are locations of maximum amplitude.
Nodes are fixed and antinodes only oscillate in the vertical direction.
At nodes, waves are in anti-phase and destructive interference occurs.
At antinodes, waves are in phase and constructive interference occurs.
Points with an odd number of nodes between them are in anti-phase; points with an even number are in phase.
All points within a loop are in phase.

Boundary Conditions for Standing Waves

Standing waves can form on strings or in pipes when progressive waves superimpose with their reflections.
The number of nodes and antinodes depends on the frequency of the progressive waves and the boundary conditions.
For a string, the boundary condition can be fixed at both ends, free at both ends, or one end fixed and the other free.
At a free end (or open end of a pipe), the reflected wave is in phase with the incident wave, creating an antinode.
At a fixed end (or closed end of a pipe), the reflected wave is in anti-phase with the incident wave, creating a node.

Harmonics in Strings & Pipes

Harmonics are the only frequencies and wavelengths that form standing waves on strings or in pipes.
For a string fixed at both ends, the nth harmonic has (n + 1) nodes and n antinodes.
The wavelength of the nth harmonic on a string fixed at both ends is λ_n = 2L/n.
The frequency of the nth harmonic on a string is f_n = nv/(2L).
For a pipe open at both ends, the nth harmonic has (n + 1) antinodes and n nodes, with the same wavelength expression as a string: λ_n = 2L/n.
For a pipe open at one end, only odd harmonics exist, with wavelength λ_n = 4L/n where n is an odd integer.
The natural frequencies are calculated using the wave equation v = fλ.

The Nature of Resonance

Free oscillations occur when there are only internal forces and no energy input; the system oscillates at its natural frequency.
Forced oscillations are produced by a periodic external force, causing the system to oscillate at the driving frequency.
Resonance occurs when the driving frequency equals the natural frequency of the system.
At resonance, the amplitude of oscillations is at its maximum.
At resonance, energy is transferred from the driver to the oscillating system most efficiently.
If the driving frequency is slightly lower or higher than the natural frequency, the amplitude increases but to a lesser extent.

The Effect of Damping

Damping is the reduction in energy and amplitude of oscillations due to resistive forces.
Damping does not change the frequency of oscillations; the time period remains constant.
Light damping: amplitude decays exponentially with time, and oscillations continue with gradually decreasing amplitude.
Critical damping: the system returns to equilibrium in the shortest possible time without oscillating.
Heavy damping: the system returns to equilibrium slowly without oscillating.
Damping reduces the amplitude of resonance vibrations and broadens the resonance peak.
As damping increases, the resonance peak lowers, broadens, and moves slightly to the left of the natural frequency.

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1.Which line correctly describes the conditions for superposition to produce a standing wave?
Easy
- AOne wave travelling in water with a constant speed
- BTwo waves travelling in opposite directions with the same frequency
- CTwo waves travelling near each other with the same frequency
- DThree or more waves travelling in the same direction with different frequency and speed
2.Which pair of statements both correctly describe properties of standing waves?
Easy
- AAll points have the same amplitude in turn; Points which are one wavelength apart are in phase
- BEach point has a different amplitude; Points between nodes are in phase
- CAll points have the same amplitude in turn; Points between nodes are in phase
- DEnergy is moved from one point to another; The wave has nodes and antinodes
3.The diagram shows a stationary wave. Which line correctly identifies the labelled sections?
Easy
- AP: antinode, Q: node, R: length of string
- BP: node, Q: antinode, R: wavelength
- CP: node, Q: antinode, R: frequency
- DP: wavelength, Q: wavelength, R: length of string
4.A stationary wave is formed in a pipe which is open at both ends. Which statement must be correct?
Easy
- ANodes form at the ends of the wave.
- BAntinodes form at both ends of the wave.
- CDisplacement is a maximum at the nodes.
- DDisplacement is at a minimum at the antinodes.
5.A stationary wave forms on a string of length L. Which harmonic has a wavelength where λ = 2L/3?
Medium
- AFirst harmonic
- BSecond harmonic
- CThird harmonic
- DFourth harmonic
6.Which is the correct general equation for the wavelength of the nth harmonic in a pipe of length L which is open at one end?
Medium
- Af_n = nv/(2L)
- Bλ_n = 2L/n
- Cv = fλ
- Dλ_n = 4L/n
7.Which one of the following options is not possible for longitudinal waves?
Medium
- APolarisation
- BInterference
- CSuperposition
- DDiffraction
8.Which one of the following options is always true regarding the energy transferred along a standing wave and its amplitude?
Medium
- AEnergy transferred: None; Amplitude: Varies
- BEnergy transferred: None; Amplitude: Is constant
- CEnergy transferred: Some energy transferred; Amplitude: Varies
- DEnergy transferred: Some energy transferred; Amplitude: Is constant

Plus de questions

9.An open pipe has sound waves sent through it causing it to have a standing wave of the third harmonic occurring within it. What is the nature of oscillations at the centre of the pipe (point X) and at its edge (point Y)?
Medium
- AX: A node, Y: A node
- BX: A node, Y: An anti-node
- CX: An anti-node, Y: A node
- DX: An anti-node, Y: An anti-node
10.The fundamental frequency for a pipe which is closed at one end, but open at the other, is 380 Hz. What is the frequency of its third harmonic?
Medium
- A127 Hz
- B380 Hz
- C760 Hz
- D1140 Hz
11.An open pipe has both ends clear and has a standing wave formed within it. Which of the following statements is always true for this standing wave?
Medium
- AThe number of nodes added to the number of antinodes is an odd number
- BThere is always a central anti-node in the wave
- CThe number of nodes added to the number of antinodes is an even number
- DThere is always a central node in the wave
12.A wave of a certain frequency is set up on a string with two fixed ends which act as nodes. If the frequency is doubled, but nothing else is changed, how many nodes will be on the string?
Medium
- ATwo
- BThree
- CFive
- DSeven
13.Organs such as those found in churches often have many different kinds of pipes with both open and closed ends. For a single pipe that has a specific length, what is the ratio of the fundamental frequency with both ends open to the fundamental frequency with one end closed?
Medium
- A1:2
- B1:4
- C2:1
- D4:1
14.A pipe with a single closed end has a standing wave set up within it. If the length of the pipe is 2 m, which value below is closest to the wavelength of the seventh harmonic?
Medium
- A0.345 m
- B0.782 m
- C0.983 m
- D1.14 m
15.A standing sound wave is set up in an open pipe (both ends open) and the frequency of the first harmonic is 400 Hz. What is the approximate length of this pipe? (Assume the speed of sound is 343 m/s).
Medium
- A0.14 m
- B0.43 m
- C1.27 m
- D4.81 m
16.A pipe of length L is open at one end and closed at the other end of the pipe. What is the frequency of the fifth harmonic? (Assume the speed of sound in air is 343 m/s).
Medium
- A150 ÷ L
- B281 ÷ L
- C429 ÷ L
- D611 ÷ L
17.A standing wave is set up on a string with both ends fixed. The frequency of the first harmonic is 150 Hz. Determine the approximate length of the string, L, and for the second harmonic, the approximate distances between two successive nodes, N, and two successive antinodes, A. The speed of sound on the string is 250 m/s.
Hard
- AL = 1.10 m, N = 0.55 m, A = 0.55 m
- BL = 1.10 m, N = 0.24 m, A = 0.55 m
- CL = 0.83 m, N = 0.42 m, A = 0.42 m
- DL = 0.83 m, N = 0.42 m, A = 0.56 m
18.A guitar string is stretched and fixed at each end so that the length is 75 cm. It is found to vibrate with resonant frequencies at 240 Hz and 360 Hz. There are no other resonant points between these two. What is the lowest resonant frequency which can oscillate on the guitar string?
Hard
- A60 Hz
- B120 Hz
- C180 Hz
- D240 Hz
19.A copper pipe of diameter 15 mm and length 45 cm is sealed closed at one end. A string is attached such that the copper pipe can be quickly spun around, causing the effect of a wind blowing across the open end at different frequencies. What is the wavelength of the third harmonic?
Hard
- A30 cm
- B34 cm
- C45 cm
- D60 cm
20.A standing wave on a string between two fixed points sounds its second harmonic at frequency f. Which expression, where n is an integer, gives the frequencies of harmonics that have an antinode at the centre of the string?
Hard
- A1/2 nf
- Bnf
- C2nf
- D(2n + 1)f
21.A pipe of length L₁ has one open end and one closed end. A second pipe, of length L₂, is open at both ends. The frequency of the first harmonic of both pipes is the same. What is the ratio L₁/L₂?
Hard
- A1/2
- B1/4
- C3/2
22.A flute is a musical instrument with a typical length, L, of 67 cm. It has a hole in the lip plate at one end which the musician blows across, and is open at the other end. The length of the air column is varied by opening and closing a series of keys along the side. The velocity of the air entering the lip plate is controlled by the musician. To create a frequency f the musician creates a wave with velocity v, holding down the keys such that the column of air is half the available length. If she wants to increase the frequency to three times the original pitch, which changes to v and L would achieve this?
Hard
- Av × 9, L × 2
- Bv × 3, L × 1/2
- Cv stays the same, L × 3
- Dv × 1/2, L × 1/6
23.A thin wire is held at tension with length L so that the speed of the waves on the string is equal to the speed of sound in air. At this tension the fundamental frequency of oscillation produces resonance in a pipe which is closed at one end. Which of the pipes shown resonates?
Hard
- AA pipe of length L/4
- BA pipe of length L/2
- CA pipe of length L
- DA pipe of length 2L
24.Which of the following statements about standing waves are correct? (Select all that apply)
Medium
- ANodes are points of zero amplitude.
- BAntinodes are points of maximum amplitude.
- CThe distance between adjacent nodes is one wavelength.
- DEnergy is transferred along a standing wave.
- EAll points between two nodes oscillate in phase.
25.In a standing wave, the distance between two successive nodes is half a wavelength.
Easy
True or false?
26.Resonance occurs when the driving frequency is equal to the natural frequency of the oscillator.
Easy
True or false?
27.Damping increases the amplitude of resonance vibrations.
Easy
True or false?
28.Match the term to its correct description.
Medium
- Node
- Antinode
- Resonance
- Damping
- Point of zero amplitude
- Point of maximum amplitude
- Maximum amplitude when driving frequency equals natural frequency
- Reduction in amplitude due to resistive forces
29.Arrange the following steps to describe how a standing wave forms on a string fixed at both ends.
Medium
- A progressive wave travels along the string.
- The wave reflects at the fixed end.
- The incident and reflected waves superpose.
- A standing wave pattern with nodes and antinodes is produced.
30.Which of the following are necessary for a standing wave to form when two waves superpose? (Select all that apply)
Medium
- AThey travel in opposite directions
- BThey have the same frequency
- CThey have the same wavelength
- DThey have the same amplitude
- EThey have a similar amplitude
31.Which of the following statements about standing waves on a string fixed at both ends are correct? (Select all that apply)
Medium
- AThe distance between adjacent nodes is λ/2
- BThe distance between adjacent antinodes is λ
- CAll points between two adjacent nodes are in phase
- DPoints separated by one node are in antiphase
- EThe number of nodes is always one more than the number of antinodes
32.Which of the following are true about resonance? (Select all that apply)
Medium
- AIt occurs when the driving frequency equals the natural frequency
- BAt resonance, the amplitude of oscillation is maximum
- CDamping increases the amplitude of resonance vibrations
- DThe natural frequency of an oscillator changes with damping
- EAt resonance, energy is transferred most efficiently from the driver to the oscillator
33.Match each term with its correct description.
Medium
- Node
- Antinode
- Resonance
- Damping
- Location of zero amplitude
- Location of maximum amplitude
- Maximum amplitude when driving frequency equals natural frequency
- Reduction in amplitude due to resistive forces
34.Match each harmonic to its wavelength for a pipe of length L open at both ends.
Medium
- First harmonic
- Second harmonic
- Third harmonic
- Fourth harmonic
- λ = 2L
- λ = L
- λ = 2L/3
- λ = L/2
35.Arrange the following harmonics in order of increasing frequency for a pipe open at one end.
Medium
- First harmonic
- Third harmonic
- Fifth harmonic
- Seventh harmonic
36.For a pipe open at one end, only odd harmonics can form.
Medium
True or false?
37.A wave profile shows a leading edge that is straight and a trailing edge that is sinusoidal. The wave reflects off a barrier and the incident and reflected waves superimpose. Which statement correctly describes the result?
Medium
- AA standing wave is formed because the half-cycles are symmetrical
- BA standing wave is not formed because the half-cycles are not symmetrical
- CA standing wave is formed because the waves have the same frequency
- DA standing wave is not formed because the wave is travelling in one direction only
38.Transverse waves travel along a stretched wire 100 cm long. The speed of the waves is 250 m/s. A person can hear frequencies up to 15 kHz. What is the maximum harmonic detectable by this person?
Medium
- A60
- B120
- C240
- D600
39.A mechanical weighing scale has a needle that sometimes oscillates before settling, making the reading difficult. Which type of damping should be applied so the needle settles as quickly as possible without oscillating?
Medium
- ALight damping
- BCritical damping
- CHeavy damping
- DNo damping
40.Which of the following scenarios are examples of forced oscillations? (Select all that apply)
Medium
- AStriking a tuning fork
- BBreaking a glass with a high-pitched sound
- CThe interior of a car vibrating at a particular speed
- DPlaying a clarinet
- EA pendulum swinging freely after being displaced
41.Complete the sentence about standing waves.
Easy
The distance between two successive nodes in a standing wave is equal to ____ of the wavelength.
42.A pipe open at both ends has a length of 1.2 m. Calculate the wavelength of the second harmonic in metres.
Medium
43.What is the name for a point in a standing wave where the amplitude is always zero?
Medium
44.Complete the sentence about resonance.
Medium
Resonance occurs when the driving frequency is equal to the ____ frequency of the oscillator, resulting in ____ amplitude of oscillation.
45.A pipe open at both ends has a length of 1.5 m. Calculate the wavelength of the second harmonic in metres.
Medium
46.A pipe closed at one end has a length of 0.8 m. Calculate the wavelength of the first harmonic in metres.
Hard
47.Explain why a standing wave does not transfer energy along its length.
Hard
48.A pipe open at both ends has a length of 1.2 m. Calculate the wavelength of the second harmonic in metres.
Medium
- A2.2 m
- B3.2 m
- C0.2 m
- D1.2 m
49.A pipe open at both ends has a length of 1.5 m. Calculate the wavelength of the second harmonic in metres.
Medium
- A2.5 m
- B3.5 m
- C1.5 m
- D0.5 m
50.A pipe closed at one end has a length of 0.8 m. Calculate the wavelength of the first harmonic in metres.
Hard
- A4.2 m
- B2.2 m
- C3.2 m
- D5.2 m

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