DP IB Physics: SL
C. Wave behavior
C.4 Standing waves and resonance
DP IB Physics: SLC. Wave behaviorC.4 Standing waves and resonance
Guiding questions: | |
|---|---|
| a) | What distinguishes standing waves from travelling waves? |
| b) | How does the form of standing waves depend on the boundary conditions? |
| c) | How can the application of force result in resonance within a system? |
a) What distinguishes standing waves from travelling waves?
- Solution:
- The main difference between travelling and standing waves is how energy is transmitted. Unlike stationary waves, travelling waves move through a medium and transfer energy from one place to another. Instead, energy is trapped between nodes and antinodes as standing waves fluctuate in place.
- Figure 1 Travelling waves Vs Standing waves
- ⇒ Traveling waves:
- Energy Transfer:
- As they move across a medium, they bring energy with them.
- Wave Propagation:
- Through the medium, the wave pattern moves or travels.
- – Examples of travelling waves include light waves, sound waves, and water waves.
- ⇒ Standing waves:
- No Energy Transfer:
- Energy oscillates inside a small area rather of being transferred from one place to another.
- Fixed Pattern:
- With nodes (points of zero displacement) and antodes (points of maximum displacement), the wave pattern seems to be motionless.
- Formation:
- Two travelling waves of the same amplitude and frequency moving in opposing directions frequently interact to produce them.
- – Examples of standing waves include resonance in pipes and vibrations in musical instruments such as guitar strings.
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b) How does the form of standing waves depend on the boundary conditions?
- Solution:
- The system’s boundary conditions have a direct impact on a standing wave’s shape. In particular, the wavelengths and frequency at which standing waves can occur are determined by the boundary conditions.
- A string that is fixed at both ends, for instance, may only sustain standing waves with nodes (points of zero displacement) at the endpoints, which results in certain frequencies and wavelengths.
- Figure 2 Standing waves on an infinite length string
- Fixed Boundaries:
- A wave inverts and forms a node at a fixed end when it reflects off of it. Nodes must be present at both ends of a string in order for standing waves to form on it.
- This limits the range of wavelengths to those that fall inside the string’s length by an integer number of half-wavelengths.
- Free Boundaries:
- An antinode, or point of greatest displacement, arises when a wave reflects off a free end without inverting.
- In systems with free ends, standing waves will have antinodes at those points, which again results in certain permitted frequencies and wavelengths.
- Mixed Boundaries:
- To further define the potential standing wave patterns, systems with one fixed end and one free end will have an antinode at the free end and a node at the fixed end.
- Wavelength and Frequency:
- The permitted standing wave wavelengths (λ) are determined by the boundary conditions.
- The formula v = fλ where v is the wave speed, provides the connection between wavelength and frequency (f). Consequently, the permitted frequencies for standing waves in a particular system are similarly determined by the boundary conditions.
- Resonance:
- Resonant frequencies are the particular frequency at which standing waves take place. These frequencies match the wavelengths that satisfy the system’s boundary criteria.
c) How can the application of force result in resonance within a system?
- Solution:
- When an external force is applied at a system’s inherent frequency, resonance takes place, leading to a notable increase in oscillation amplitude.
- This phenomenon occurs as a result of the external force constantly adding energy to the system at a pace that corresponds with its inherent oscillation propensity. This causes vibrational energy to accumulate and the amplitude response to increase.
- Figure 3 Analysis and stability assessment of the vibratory motion
- Natural Frequency:
- All mechanical, electrical, and acoustic systems have a natural frequency at which they usually vibrate or oscillate when anything disturbs them. The physical characteristics of the system—such as mass, stiffness, and damping in mechanical systems, or inductance and capacitance in electrical circuits—determine this frequency.
- Forcible Oscillations:
- A system may oscillate when an outside force is applied to it.
- The oscillations will be relatively modest and the energy transfer will be inefficient if the driving frequency (the frequency of the external force) differs from the natural frequency of the system.
- Resonance:
- This phenomenon happens when the driving frequency and the natural frequency coincide. In this condition, energy is efficiently transferred to the system by the external force pushing it continually at the desired pace.
- The oscillations’ amplitude dramatically increases as a result, frequently surpassing the generating force’s magnitude.
- ⇒ Examples:
- Musical Instruments:
- By singing a note at its inherent frequency, a vocalist may break a glass by making it vibrate more intensely until it breaks.
- Swinging:
- When a youngster is pushed on a swing, the child swings higher and higher due to the rhythmic pushes at the swing’s natural frequency.
- Bridges:
- The collapse of the Tacoma Narrows Bridge is a well-known instance of resonance, in which wind vibrations matched the bridge’s inherent frequency.
- Radio Receivers:
- To choose certain radio frequencies, radio receivers employ resonance. The receiver can enhance that signal since its circuit is set to resonate with the desired frequency.