# 17.9 RESONANCE Resonance is a phenomenon that occurs when a driving force is applied to an oscillator at a frequency equal to its natural frequency. This results in a dramatic increase in the amplitude of the oscillator's vibrations due to the efficient transfer of energy. ### Defining Resonance * **Driving Force:** An external force required to initiate and sustain the oscillations of a system. The driving force has its own frequency. * **Natural Frequency:** The frequency at which a system oscillates when not subjected to a continuous or repeated external force. * **Resonance:** The phenomenon where the amplitude of an oscillator's vibration reaches its maximum value. This occurs when the frequency of the driving force becomes equal to the natural frequency of the oscillator. * **Resonance Frequency:** The natural frequency of the oscillator, at which resonance occurs. At this frequency, the transfer of energy from the driving force to the oscillator is most efficient. In the context of RLC series circuits, resonance occurs when the inductive reactance ($X_L$) equals the capacitive reactance ($X_C$). **Demonstration with Pendulums:** As shown in Figure 17.10, several pendulums are hung from a common rod. If one pendulum (E) is set to oscillate, only those pendulums with the same length (and therefore the same natural frequency) as E will begin to oscillate with a large amplitude. This demonstrates that energy is efficiently transferred only when the driving frequency (from pendulum E) matches the natural frequency of the other pendulums. ### Effect of Damping on Resonance Damping (frictional forces) reduces the amplitude of oscillations at the resonance frequency. The sharpness of the resonance peak is affected by damping: * **Small Damping:** Results in a sharp, high-amplitude resonance peak. * **Heavy Damping:** Results in a broad, low-amplitude resonance peak. This is illustrated in the graph below, where amplitude is plotted against the driving frequency for different levels of damping. In electrical circuits, resistance $R$ acts as the damping factor. ### Applications of Resonance Resonance can be both useful and destructive. **Useful Applications:** | Application | Description | | :--- | :--- | | **Tuning a Radio** | The radio's circuit is tuned to change its natural frequency. When it matches the frequency of a specific radio station, resonance occurs, and the signal from that station is received strongly. | | **Microwave Ovens** | Microwaves with a frequency matching the natural frequency of water molecules are used. This causes the water in food to vibrate vigorously, generating heat for cooking. | | **Magnetic Resonance Imaging (MRI)** | Atomic nuclei (like hydrogen) are made to resonate by strong radio waves. The pattern of energy absorption is used to create detailed images of internal body structures. | | **Musical Instruments** | In a guitar, the vibrating strings cause the hollow wooden body to resonate, amplifying the sound. | **Destructive Applications (to be avoided):** | Situation | Description | | :--- | :--- | | **Bridges** | Rhythmic marching by soldiers can match the bridge's natural frequency, causing large oscillations that could lead to collapse. The Tacoma Narrows Bridge collapse is a famous example caused by wind-induced resonance. | | **Airplane Wings** | If the frequency of wind gusts matches the natural frequency of an airplane's wing, the vibrations can become large enough to cause structural failure. | ### Resonance and Standing Waves **Rubens Tube:** A tube with holes that leaks flammable gas. When sound waves at a resonant frequency create a standing wave inside, the gas pressure varies. Flames are higher at pressure antinodes and lower at pressure nodes, visually demonstrating the standing wave. **Acoustic Levitation:** High-intensity sound waves create a standing wave. The pressure nodes are stable points where small objects can be suspended in the air against gravity. **Chladni Plates:** A metal plate is vibrated at a resonant frequency. Sand sprinkled on the plate moves away from the antinodes (maximum vibration) and collects at the nodes (zero vibration), forming intricate patterns that visualize the vibrational modes.