Standing Waves & Resonance

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수업 노트

Standing Waves

Standing waves (or stationary waves) are produced when two waves travelling in opposite directions along the same line with the same frequency superpose.
This usually occurs when a travelling wave superimposes with its reflection.
For a standing wave to form, the two waves must have the same wavelength and similar amplitude.
In a standing wave, the crests and troughs only move vertically; the wave pattern does not travel along the medium.
Standing waves store energy, whereas progressive (travelling) waves transfer energy.
The principle of superposition applies to all types of waves, including transverse and longitudinal waves.

Nodes & Antinodes

Nodes are locations of zero amplitude, separated by half a wavelength (λ/2).
Antinodes are locations of maximum amplitude.
Nodes and antinodes do not move along the wave; nodes are fixed and antinodes oscillate vertically.
At nodes, the two waves are in anti-phase (crest meets trough), causing destructive interference and cancellation.
At antinodes, the two waves are in phase (crest meets crest), causing constructive interference and addition.
Two points on a standing wave are either in phase (0 phase difference) or in anti-phase (π out of phase).
Points with an odd number of nodes between them are in anti-phase; points with an even number of nodes between them are in phase.
All points within a single loop are in phase.

Boundary Conditions for Standing Waves

Standing waves form on strings or in pipes when progressive waves superimpose with their reflections.
The number of nodes and antinodes that fit depends on the frequency of the progressive waves and the boundary conditions (fixed/free ends).
For a string, boundary conditions can be: fixed at both ends, free at both ends, or one end fixed and the other free.
At a free end (or open pipe end), the reflected wave is in phase with the incident wave, producing an antinode (maximum displacement).
At a fixed end (or closed pipe end), the reflected wave is in anti-phase with the incident wave, producing a node (zero displacement).
For a pipe, possible boundary conditions are: closed at both ends, open at both ends, or closed at one end and open at the other.
When air vibrates in a pipe, longitudinal waves travel along the pipe and can form standing waves.

Harmonics in Strings & Pipes

Harmonics are the only frequencies and wavelengths that form standing waves on strings or in pipes, depending on boundary conditions.
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 λₙ = 2L/n, where L is the string length.
The frequency of the nth harmonic on a string is fₙ = nv/(2L), where v is the wave speed.
For a pipe open at both ends: the nth harmonic has n nodes and (n + 1) antinodes; the same expressions λₙ = 2L/n and fₙ = nv/(2L) apply.
For a pipe open at one end: only odd harmonics exist, with wavelength λₙ = 4L/n, where n is an odd integer (1, 3, 5...).
The first harmonic (fundamental) on a string fixed at both ends has one loop, two nodes, and one antinode, with λ₁ = 2L.
The second harmonic on a string has three nodes and two antinodes; the third harmonic has four nodes and three antinodes.

The Nature of Resonance

Free oscillations occur when there are only internal forces acting and no energy input; the system oscillates at its natural frequency.
Forced oscillations are produced by a periodic external driving force, causing the system to oscillate at the driving frequency.
The natural frequency f₀ is the frequency at which a system oscillates freely.
Resonance occurs when the driving frequency equals the natural frequency (f = f₀), resulting in maximum amplitude.
At resonance, energy is transferred most efficiently from the driver to the oscillating system.
If the driving frequency is slightly lower or higher than the natural frequency, the amplitude increases but to a lesser extent.
A child on a swing is an example of resonance: pushing at the natural frequency increases the amplitude.

The Effect of Damping

Damping is the reduction in energy and amplitude of oscillations due to resistive forces (e.g., friction, air resistance).
Damping does not change the frequency (or period) of the oscillations; the amplitude decreases over time.
Light damping: amplitude decays exponentially with time, and the system oscillates 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, taking longer than critical damping.
On a resonance curve (amplitude vs driving frequency), increased damping lowers the peak, broadens the curve, and may shift the peak slightly left.
Damping reduces the sharpness of resonance and the amplitude at the resonant frequency, but the natural frequency remains the same.

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연습 문제

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1.What is the phase relationship between the two waves at a node of a standing wave?
Easy
- AThey are in anti-phase, so destructive interference occurs
- BThey are in phase, so constructive interference occurs
- CThey are in quadrature (90° out of phase)
- DThey have a phase difference that varies with time
2.A standing wave is set up on a string fixed at both ends. The distance between two consecutive nodes is 0.25 m. What is the wavelength of the standing wave?
Medium
- A0.125 m
- B0.25 m
- C0.50 m
- D1.00 m
3.A pipe is open at one end and closed at the other. Which of the following frequencies could NOT form a standing wave in this pipe?
Medium
- A150 Hz
- B300 Hz
- C450 Hz
- D750 Hz
4.A standing wave is formed in a pipe of length L that is open at both ends. Which expression gives the wavelength of the second harmonic?
Medium
- Aλ = 2L
- Bλ = L
- Cλ = L/2
- Dλ = 4L
5.A standing wave is formed on a stretched string. Which of the following correctly describes the energy of the wave?
Medium
- AIt transfers energy along the string
- BIt stores energy and does not transfer it along the string
- CIt transfers energy but at half the speed of a travelling wave
- DIt loses all its energy at the nodes
6.The frequency of a damped oscillation changes as the amplitude decreases.
Easy
True or false?
7.In a pipe that is open at one end, only odd harmonics can be formed.
Easy
True or false?
8.Match each term with its correct definition.
Medium
- Node
- Antinode
- Resonance
- Damping
- A point of zero amplitude on a standing wave
- A point of maximum amplitude on a standing wave
- When the driving frequency equals the natural frequency, giving maximum amplitude
- The reduction in energy and amplitude of oscillations due to resistive forces

더 많은 문제

9.Place the following steps in the correct order to describe how a standing wave is formed on a string fixed at both ends.
Medium
- A wave travels along the string and reflects at the fixed end
- The incident and reflected waves travel in opposite directions
- The waves superpose, producing points of zero and maximum amplitude
- A standing wave pattern with nodes and antinodes is established
10.Which of the following are conditions required for the formation of a standing wave? (select all that apply)
Medium
- AThe two waves travel in opposite directions
- BThe two waves have the same frequency
- CThe two waves have the same amplitude
- DThe two waves are polarised in the same plane
- EThe two waves travel in the same direction
11.Which of the following statements about nodes and antinodes on a standing wave are correct? (select all that apply)
Medium
- AAt a node, the two waves are in anti-phase and destructive interference occurs
- BAt an antinode, the two waves are in phase and constructive interference occurs
- CNodes are separated by half a wavelength
- DAntinodes are points of zero amplitude
- EThe distance between a node and the next antinode is one wavelength
12.A pipe is open at one end and closed at the other. Which of the following statements are true? (select all that apply)
Medium
- AOnly odd harmonics can be formed
- BThere is a node at the closed end
- CThere is an antinode at the open end
- DThe wavelength of the first harmonic is 2L
- EThe wavelength of the third harmonic is 4L/3
13.Which of the following are key features of a displacement–time graph for a lightly damped oscillating system? (select all that apply)
Medium
- AThe amplitude decreases exponentially with time
- BThe frequency of the oscillations remains constant
- CThe time period of the oscillations stays the same
- DThe displacement returns to zero without oscillating
- EThe amplitude decreases linearly with time
14.Which of the following statements about resonance are correct? (select all that apply)
Medium
- AResonance occurs when the driving frequency equals the natural frequency
- BAt resonance, the amplitude of oscillations is at a maximum
- CAt resonance, energy is transferred most efficiently from the driver to the system
- DResonance occurs when the driving frequency is much greater than the natural frequency
- EDamping increases the sharpness of the resonance peak
15.As the degree of damping on an oscillating system is increased, which of the following changes occur to its resonance curve? (select all that apply)
Hard
- AThe peak of the curve lowers
- BThe resonance peak broadens
- CThe peak moves slightly to the left of the natural frequency when heavily damped
- DThe natural frequency of the oscillator changes
- EThe peak becomes sharper
16.A standing wave is formed on a stretched string fixed at both ends. Which of the following statements about its harmonics are correct? (select all that apply)
Medium
- AThe nth harmonic has (n + 1) nodes and n antinodes
- BThe wavelength of the nth harmonic is given by λ_n = 2L/n
- CThe second harmonic has three nodes and two antinodes
- DThe first harmonic has one node and one antinode
- EThe frequency of the nth harmonic is f_n = nv/(2L)
17.A free oscillation occurs when there are only internal forces acting and no energy input.
Easy
True or false?
18.The frequency of a damped oscillation decreases as the amplitude decreases.
Easy
True or false?
19.In a pipe open at both ends, the nth harmonic has (n + 1) antinodes and n nodes.
Easy
True or false?
20.In a pipe open at one end, only odd harmonics can be formed.
Medium
True or false?
21.Damping reduces the amplitude of resonance vibrations but does not affect the natural frequency of the oscillator.
Medium
True or false?
22.Match each boundary condition with the correct wavelength expression for the nth harmonic.
Medium
- String fixed at both ends
- Pipe open at both ends
- Pipe open at one end
- λ_n = 2L/n
- λ_n = 4L/n (n odd)
- λ_n = 2L/n (n integer)
23.Match each type of damping with its correct description.
Medium
- Light damping
- Critical damping
- Heavy damping
- Amplitude decays exponentially and the system oscillates with gradually decreasing amplitude
- Returns to equilibrium in the shortest possible time without oscillating
- Takes a long time to return to equilibrium without oscillating
24.Place the following steps in the correct order to calculate the frequency of the nth harmonic on a string fixed at both ends.
Medium
- Identify the length L of the string and the wave speed v
- Use λ_n = 2L/n to find the wavelength of the nth harmonic
- Use the wave equation v = fλ to find the frequency
- Rearrange to f_n = nv/(2L)
25.Place the following steps in the correct order to describe the effect of increasing damping on a resonance curve.
Medium
- The amplitude of resonance vibrations decreases
- The peak of the curve lowers
- The resonance peak broadens
- The peak moves slightly to the left of the natural frequency when heavily damped
26.Which expression gives the wavelength of the third harmonic?
Medium
- A4L/3
- B2L/3
- C4L
- D2L
27.A standing wave is formed in a pipe open at both ends. The pipe has a length of 0.45 m and the speed of sound in the gas inside is 381 m/s. What is the wavelength of the fourth harmonic?
Medium
- A0.225 m
- B0.45 m
- C0.90 m
- D1.80 m
28.A loudspeaker is positioned above a vertical pipe that is gradually filled with water. The loudness reaches a maximum when the distance from the loudspeaker to the water surface is 55 cm, and the next maximum occurs at 33 cm. The speed of sound in air is 330 m/s. What is the frequency of the sound emitted by the loudspeaker?
Medium
- A750 Hz
- B1500 Hz
- C375 Hz
- D300 Hz
29.A standing wave is formed on a violin string of length 32 cm fixed at both ends. The string vibrates in its first harmonic at a frequency of 440 Hz. What is the speed of the wave on the string?
Medium
- A141 m/s
- B282 m/s
- C440 m/s
- D880 m/s
30.A stretched string of length 100 cm carries transverse waves at a speed of 250 m/s. What is the highest harmonic that a person who can hear frequencies up to 15 kHz could detect?
Medium
- A60th harmonic
- B120th harmonic
- C240th harmonic
- D150th harmonic
31.A pipe of length 45 cm is open at both ends and is filled with an unknown gas. The frequency of the first harmonic is 381 Hz. The pipe is then submerged and completely filled with water, where the speed of sound is 1500 m/s. What is the period of the first harmonic in the water-filled pipe?
Medium
- A6.0 × 10⁻⁴ s
- B1.2 × 10⁻³ s
- C2.4 × 10⁻³ s
- D3.0 × 10⁻⁴ s
32.A standing wave is formed in a column of air in a pipe of length L that is open at one end. The standing wave has 2.5 full wavelengths within the length L. What is the wavelength of this standing wave in terms of L?
Medium
- Aλ = L / 2.5
- Bλ = 2.5L
- Cλ = 2L / 5
- Dλ = 5L / 2
33.A tube of length L is open at both ends. What is the wavelength of the first harmonic standing wave formed in the tube?
Medium
- Aλ = L/2
- Bλ = L
- Cλ = 2L
- Dλ = 4L
34.A standing wave is formed in a pipe of length L that is open at one end and closed at the other. What is the wavelength of the third harmonic?
Medium
- Aλ = 4L/3
- Bλ = 2L/3
- Cλ = 3L/4
- Dλ = L/3
35.A standing wave is formed on a string fixed at both ends. The string has a length of 0.80 m and the wave speed is 120 m/s. What is the frequency of the third harmonic?
Medium
- A75 Hz
- B150 Hz
- C225 Hz
- D300 Hz
36.A pipe of length 0.25 m is open at one end and closed at the other. The speed of sound in air is 340 m/s. What is the frequency of the third harmonic?
Medium
- A340 Hz
- B680 Hz
- C1020 Hz
- D1360 Hz
37.A standing wave is formed in a pipe open at both ends. The pipe has a length of 1.2 m and the speed of sound is 343 m/s. What is the frequency of the first harmonic?
Medium
- A143 Hz
- B286 Hz
- C572 Hz
- D71.5 Hz
38.A pipe is open at one end and closed at the other. Which of the following statements are correct? (select all that apply)
Medium
- AOnly odd harmonics can be formed.
- BThe closed end is a node and the open end is an antinode.
- CThe wavelength of the third harmonic is 4L/3.
- DThe wavelength of the second harmonic is L.
- EThe frequency of the first harmonic is v/(4L).
39.A standing wave is formed on a stretched string fixed at both ends. Which of the following statements are correct? (select all that apply)
Hard
- AThe nth harmonic has (n + 1) nodes and n antinodes.
- BThe wavelength of the nth harmonic is 2L/n.
- CThe frequency of the nth harmonic is nv/(2L).
- DThe distance between adjacent nodes is λ/4.
- EThe wave speed on the string depends on the tension and mass per unit length.
40.In a pipe open at one end, the frequency of the third harmonic is three times the frequency of the first harmonic.
Easy
True or false?
41.Complete the sentence about standing waves.
Easy
In a standing wave, points that are ____ apart have a phase difference of π.
42.What is the name for the maximum displacement point on a standing wave?
Medium
43.A pipe is open at both ends and has a length of 0.45 m. The speed of sound in the pipe is 343 m/s. Calculate the frequency of the first harmonic in Hz.
Medium
44.A standing wave is formed on a stretched string fixed at both ends. The string has a length of 0.80 m and the wave speed is 120 m/s. Calculate the frequency of the second harmonic in Hz.
Medium
45.A pipe is open at one end and closed at the other. The pipe has a length of 0.25 m and the speed of sound in air is 340 m/s. Calculate the frequency of the third harmonic in Hz.
Hard
46.A standing wave is formed in a pipe open at both ends. The pipe has a length of 1.2 m and the speed of sound is 343 m/s. Calculate the frequency of the first harmonic in Hz.
Medium
47.Complete the sentence about resonance.
Medium
Resonance occurs when the driving frequency equals the ____ frequency, and the amplitude of oscillations is at its ____.
48.Name the three degrees of damping.
Medium
The three degrees of damping are ____, ____, and ____.
49.Complete the sentence about harmonics in a pipe open at one end.
Medium
In a pipe open at one end, only ____ harmonics can be formed, and the wavelength of the nth harmonic is given by λ____n = ____.
50.What is the name for the point on a standing wave where the amplitude is zero?
Medium
51.What is the name for the point on a standing wave where the amplitude is maximum?
Medium
52.What is the name of the phenomenon that occurs when the driving frequency of an oscillating system equals its natural frequency?
Medium
53.State three conditions required for the formation of a standing wave.
Hard
54.State what is meant by damping.
Medium
55.State three conditions which are required for the formation of a standing wave.
Hard
56.Outline one difference between standing waves and travelling waves.
Hard
57.Explain how a standing wave is formed on a stretched string fixed at both ends.
Hard
58.A pipe is open at both ends and has a length of 0.45 m. The speed of sound in the pipe is 343 m/s. Calculate the frequency of the first harmonic in Hz.
Medium
- A343 Hz
- B419 Hz
- C381 Hz
- D457 Hz
59.A standing wave is formed on a stretched string fixed at both ends. The string has a length of 0.80 m and the wave speed is 120 m/s. Calculate the frequency of the second harmonic in Hz.
Medium
- A165 Hz
- B135 Hz
- C180 Hz
- D150 Hz
60.A pipe is open at one end and closed at the other. The pipe has a length of 0.25 m and the speed of sound in air is 340 m/s. Calculate the frequency of the third harmonic in Hz.
Hard
- A1020 Hz
- B1224 Hz
- C1122 Hz
- D918 Hz
61.A standing wave is formed in a pipe open at both ends. The pipe has a length of 1.2 m and the speed of sound is 343 m/s. Calculate the frequency of the first harmonic in Hz.
Medium
- A143 Hz
- B157 Hz
- C129 Hz
- D171 Hz

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