DP IB Physics: SL
C. Wave Behaviour
C.4 Standing waves and resonance
DP IB Physics: SLC. Wave BehaviourC.4 Standing waves and resonanceUnderstandings | |
|---|---|
| a) | The nature and formation of standing waves in terms of superposition of two identical waves travelling in opposite directions |
| b) | Nodes and antinodes, relative amplitude and phase difference of points along a standing wave |
| c) | Standing waves patterns in strings and pipes |
| d) | The nature of resonance including natural frequency and amplitude of oscillation based on driving frequency |
| e) | The effect of damping on the maximum amplitude and resonant frequency of oscillation |
| f) | The effects of light, critical and heavy damping on the system. |
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a) The Nature and Formation of Standing Waves
- A standing wave is a wave pattern that does not appear to move along the medium. Instead, it oscillates in place with fixed regions of no motion (nodes) and maximum motion (antinodes).
- They are formed by the superposition (overlap) of two identical waves traveling in opposite directions in the same medium.
- These waves must have:
- – The same frequency
- – The same amplitude
- – The same wavelength
- When these two waves meet, they interfere at every point in space.
- Where they always cancel out → nodes
- Where they always reinforce → antinodes
- The result is a stationary pattern with no net energy transfer along the medium.
- ⇒ Examples of Standing Waves:
- A guitar string plucked at the middle
- Air column in a flute or pipe
- Vibrating membrane in a drum
- Microwaves inside an oven (standing EM waves)
- Figure 1 Standing waves
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b) Nodes and Antinodes, Amplitude, and Phase
- ⇒ Nodes
- Points of zero displacement (no motion).
- Caused by destructive interference.
- Located where the two waves always cancel out.
- Nodes appear stationary — the medium doesn’t move there at all.
- ⇒ Antinodes
- Points of maximum displacement (maximum motion).
- Caused by constructive interference.
- Located where the two waves reinforce each other.
- At antinodes, the oscillation is largest (maximum amplitude).
- Figure 2 Nodes and Anti-node
- Relative Amplitude Along a Standing Wave
- Amplitude varies along the wave:
- – Zero at nodes
- – Maximum at antinodes
- The amplitude at each point stays constant in time, though the wave oscillates.
- ⇒ Phase Difference Along a Standing Wave
- All points between two nodes (in the same segment) move in phase:
- – They reach their peaks and troughs together.
- Adjacent segments (separated by a node) move completely out of phase:
- – If one section goes up, the next goes down.
- So:
- Same side of a node → in phase
- – Opposite sides of a node → 180° out of phase
- Figure 3 Phase difference (In-phase and out of phase
- ⇒ Standing Waves on a String (Fixed Ends)
- Must have nodes at both ends.
- Fundamental (1st harmonic): 1 loop (½λ)
- 2nd harmonic: 2 loops (λ)
- 3rd harmonic: 3 loops (3/2 λ), and so on…
- ⇒ Standing Waves in Air Columns:
- Closed-end must be a node
- Open-end must be an antinode
- So, the standing wave pattern depends on whether the pipe is open or closed at each end.
- Figure 4 Standing waves in Air Columns
- ⇒ Applications of Standing Waves:
- Musical instruments (strings, pipes)
- Resonance in buildings and bridges
- Tuning microwave ovens and antennas
- Testing materials (non-destructive testing using ultrasonic standing waves)
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(c) Standing Wave Patterns in Strings and Pipes
- Standing Waves in Strings
- When a string (like a guitar or violin string) is fixed at both ends and vibrates, it forms standing waves.
- ⇒ Conditions:
- Nodes at both ends (because the string can’t move at the endpoints)
- The wave reflects back and forth along the string
- ⇒ Harmonics (Modes of Vibration):
- Each harmonic (also called a mode) shows a different number of loops:
- Figure 5 Standing wave Patterns
- ⇒ Harmonics in a String:
| Harmonic | Pattern | Wavelength | Frequency |
|---|---|---|---|
| 1st (fundamental) | One loop (½λ) | λ=2L | [math]f_1 = \frac{v}{2L}[/math] |
| 2nd harmonic | Two loops (1λ) | λ=L | [math]f_2 = \frac{v}{L} = 2f_1[/math] |
| 3rd harmonic | Three loops (3/2λ) | [math]λ = \frac{2L}{3}[/math] | [math]f_3 = \frac{3v}{2L} = 3f_1[/math] |
- Where:
- – L = length of the string
- – v = speed of wave on string
- ⇒ Standing Waves in Pipes (Air Columns)
- Air columns in tubes (like flutes, organ pipes, or horns) also form standing waves, but:
- – Open ends = Antinodes (air vibrates freely)
- – Closed ends = Nodes (air is trapped)
- ⇒ Two Types of Pipes:
- (i) Open at Both Ends
| Harmonic | Pattern | Wavelength | Frequency |
|---|---|---|---|
| 1st (fundamental) | ½λ | λ=2L | [math]f_1 = \frac{v}{2L}[/math] |
| 2nd harmonic | 1λ | λ=L | [math]f_2 = \frac{v}{L} = 2f_1[/math] |
| 3rd harmonic | 3/2λ | [math]λ = \frac{2L}{3}[/math] | [math]f_3 = \frac{3v}{2L} = 3f_1[/math] |
- Figure 6 First Harmonic
- (ii) Closed at One End, Open at the Other
- Only odd harmonics are present.
| Harmonic | Pattern | Wavelength | Frequency |
|---|---|---|---|
| 1st (fundamental) | ¼λ | λ=4L | [math]f_1 = \frac{v}{4L}[/math] |
| 3rd harmonic | ¾λ | [math]λ = \frac{4L}{3}[/math] | [math]f_3 = \frac{3v}{4L} = 3f_1[/math] |
| 5th harmonic | 5/4λ | [math]λ = \frac{4L}{5}[/math] | [math]f_5 = \frac{5v}{4L} = 5f_1[/math] |
- Figure 7 Closed at one end and open at the other
- ⇒ Summary of Standing Wave Patterns:
| System | Boundary Conditions | Harmonics Present | Node/Antinode Positions |
|---|---|---|---|
| String (both ends fixed) | Node–Node | All (1st, 2nd, 3rd…) | Nodes at ends |
| Open pipe (both ends open) | Antinode–Antinode | All | Antinodes at ends |
| Closed pipe (one end closed) | Node–Antinode | Odd only (1st, 3rd, 5th…) | Node at closed, antinode at open |
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(d) The Nature of Resonance
- Resonance occurs when a system is driven at a frequency that matches its natural frequency, resulting in a huge increase in amplitude.
- – Natural frequency: The frequency at which a system oscillates freely (without external force)
- – Driving frequency: Frequency of an external periodic force applied to the system.
- – When driving frequency matches natural frequency → resonance happens.
- ⇒ Amplitude vs. Frequency Graph:
- Figure 8 Amplitude and frequency graph
- ⇒ Real-World Examples of Resonance:
| Example | Description |
|---|---|
| Guitar string | Pluck it → vibrates at natural frequency |
| Tacoma Narrows Bridge | Collapsed due to wind-induced resonance |
| MRI machines | Use magnetic resonance to image tissues |
| Tuning forks | Ring loudly when another fork of same pitch vibrates nearby |
| Glass shattering | A singer can break glass by matching its resonant frequency |
- ⇒ Important Effects of Resonance:
- Large amplitude oscillations (can be dangerous or useful)
- Occurs only at specific frequencies
- Energy input is most efficiently transferred at resonance
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(e) The Effect of Damping on Maximum Amplitude and Resonant Frequency
- Damping is the process by which energy is gradually lost from an oscillating system, usually due to friction, air resistance, or other resistive forces.
- – It causes the amplitude of oscillations to decrease over time.
- ⇒ Damping and the Resonance Curve
- In a damped system, the resonance curve (amplitude vs. driving frequency) is affected in two main ways:
- 1. Effect on Maximum Amplitude
- – More damping → lower maximum amplitude
- – At resonance, the system can’t build up as much energy due to losses.
- 2. Effect on Resonant Frequency
- The resonant frequency shifts to a slightly lower value as damping increases.
- In lightly damped systems, this shift is small.
- In heavily damped systems, it becomes significant.
- ⇒ Resonance Curve Comparison:
- Figure 9 Resonance curve
| Damping | Maximum Amplitude | Resonant Frequency |
|---|---|---|
| None | Highest | Natural frequency |
| Light | Slightly reduced | Slightly lower |
| Heavy | Greatly reduced | Significantly lower |
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(f) The Effects of Light, Critical, and Heavy Damping
- ⇒ Types of Damping Explained:
- Let’s compare all three cases using an example of a mass on a spring.
- 1. Light Damping (Underdamped)
- Oscillations continue but amplitude gradually decreases over time.
- System still oscillates around the equilibrium.
- Energy is slowly lost.
- Applications: Car suspensions, musical instruments (strings), microphones.
- Graph (displacement vs. time):
- Figure 10 Graph between different parameters
- 2. Critical Damping
- The system returns to equilibrium in the shortest time possible without oscillating.
- Just enough damping to prevent overshooting.
- Applications: Car shock absorbers, door dampers, analog meters.
- 3. Heavy Damping (Overdamped)
- No oscillations
- The system returns to equilibrium very slowly.
- Damping is more than necessary.
- Applications: Slow-return systems, like certain gas springs, mechanical meters.
- Graph:
- Figure 11 Critical and heavy damping
| Type of Damping | Behavior | Time to Return | Oscillations | Applications |
|---|---|---|---|---|
| Light (underdamped) | Slow decay in amplitude | Moderate | Yes | Musical instruments, suspension systems |
| Critical damping | Fastest return to rest | Fast | No | Shock absorbers, doors |
| Heavy (overdamped) | Very slow return | Slow | No | Sensitive instruments, slow-closing devices |
- ⇒ Bonus: Damping in Real Life
- – Violin: light damping = beautiful sustained note
- – Car suspension: critical damping = no bounce after bump
- – Galvanometer: overdamped = needle returns slowly without overshooting
| Concept | Effect |
|---|---|
| Damping | Reduces amplitude of oscillation |
| More damping | Lower amplitude, lower resonant frequency |
| Light damping | Oscillations slowly fade |
| Critical damping | Quickest return, no oscillation |
| Heavy damping | Very slow return, no oscillation |