Preview
IB DP Physics — Unit C: Wave Behaviour | SL & HL | C.4 Standing Waves and Resonance
📋 What You Will Learn
- Observe and describe how oscillating systems respond to periodic driving forces, using real-world examples such as bridges, swings, and musical instruments
- Define natural frequency, driving frequency, resonance, and damping, and explain the physical meaning of each
- Explain qualitatively and quantitatively how damping affects the amplitude and energy of an oscillating system over time
- Analyse resonance curves (amplitude vs. driving frequency graphs) and interpret the effect of damping on their shape and peak position
- Derive and apply the exponential decay of amplitude for a damped oscillator
- Evaluate real-world applications and consequences of resonance and damping in engineering, medicine, and everyday life
🌍 Real-World Context On 7 November 1940, the Tacoma Narrows Bridge in Washington State, USA, began to oscillate violently in a steady wind of only 67 km/h. Within hours, the bridge twisted itself apart and collapsed into the gorge below. Engineers had not anticipated that the wind could push the bridge at a frequency matching the bridge's own natural frequency of vibration — a phenomenon known as resonance. This catastrophic event is one of the most dramatic demonstrations in engineering history of what happens when a system is driven at just the right frequency, and it permanently changed how bridges are designed. Understanding resonance — and how to control it through damping — is one of the most practically important topics in all of physics.
🔍 Guiding Question How does the frequency of a periodic driving force determine the amplitude of oscillation of a system, and how does damping modify this relationship? Why is resonance both useful and potentially destructive?
Physical Setup: Free, Forced, and Damped Oscillations
Imagine pushing a child on a swing. If you push at random intervals, the swing barely moves. But if you time each push to match the natural back-and-forth rhythm of the swing, the amplitude grows dramatically with each push. You are doing very little work, yet the swing reaches impressive heights. This is the essence of resonance — a phenomenon that emerges from the interplay between a system's natural tendency to oscillate and an external periodic force.
Before exploring resonance, we need to distinguish between three types of oscillation: free oscillations, forced oscillations, and damped oscillations. Each describes a different physical situation, and understanding the differences is essential for the IB exam.
In a free oscillation, the system oscillates at its own natural frequency \( f_0 \) with no energy input or loss — a theoretical ideal. In a forced oscillation, an external periodic force drives the system at a frequency \( f_d \) that may or may not match \( f_0 \). In a damped oscillation, energy is continuously removed from the system (by friction, air resistance, or other dissipative forces), causing the amplitude to decrease over time. Real oscillations are always damped to some degree.
Core Concepts and Definitions
When a system is left to oscillate freely — without any driving force and without energy loss — it oscillates at a specific frequency determined entirely by its physical properties. For a pendulum, this depends on its length and the gravitational field strength. For a mass-spring system, it depends on the mass and the spring constant. This intrinsic frequency is called the natural frequency of the system.
📖 Key Definition — Natural Frequency The natural frequency \( f_0 \) of a system is the frequency at which it oscillates when displaced from equilibrium and released, in the absence of any driving force or damping. It is determined solely by the physical properties of the system (e.g. mass, spring constant, pendulum length).
Now consider what happens when an external agent applies a periodic force