Resonance and Wave Phenomena

Push a child on a swing at just the right moment — matching the natural rhythm of the swing — and the amplitude builds with each push. Push at the wrong rhythm, and the swing barely moves no matter how hard you push.

That dramatic difference is resonance: the amplification of oscillations when an external driving force matches the system’s natural frequency. Resonance is everywhere — guitar bodies, MRI machines, the song that always seems to rattle one specific window in your room, the famous Tacoma Narrows bridge collapse. This section also wraps up the chapter with the three other classic wave behaviors at boundaries: reflection, refraction, and diffraction.

Resonance

Every object that can vibrate has a natural frequency (or set of natural frequencies) determined by its physical properties — mass, stiffness, shape, size. When an external periodic force drives the system at that natural frequency, energy transfers very efficiently into the oscillation, and amplitude grows dramatically.

Examples of Resonance

• Musical instruments. A guitar body resonates at certain frequencies, amplifying the sound from the strings. Different instruments have different resonant frequency profiles → different timbres.
• Tacoma Narrows Bridge (1940). Wind-driven oscillations matched the bridge’s natural torsional frequency, causing total collapse. (Iconic black-and-white footage worth a quick search.)
• MRI. Radio-frequency pulses tuned to the Larmor frequency of hydrogen nuclei cause resonance, producing the signal that builds the image.
• Microwave oven. Microwaves at ~2.45 GHz resonate with rotational modes of water molecules — heating water-containing food much more efficiently than dry food.

Wave Reflection

When a wave hits a boundary between two media, part of it bounces back (reflection) and part continues into the new medium (transmission). Behavior at the boundary depends on whether the end is fixed or free.

• Fixed end (hard boundary): the reflected wave is inverted (flipped upside down). A crest comes back as a trough. Think of a rope tied to a wall — a pulse sent toward the wall returns upside down.
• Free end (soft boundary): the reflected wave is upright (same orientation). A crest comes back as a crest. Think of a rope tied to a ring that slides freely on a pole.

Wave Refraction

Refraction is the bending of a wave as it crosses from one medium to another where it has a different speed. When a wave enters a slower medium, it bends toward the normal (the imaginary line perpendicular to the boundary). When it enters a faster medium, it bends away from the normal.

Why? Different parts of the wavefront cross the boundary at different times. The part that hits the new medium first slows down (or speeds up), while the rest of the wavefront is still in the original medium — the result is that the whole wavefront pivots and changes direction. (Same reason a row of marchers pivots when one end hits a muddy patch.)

Wave Diffraction

Diffraction is the spreading of waves as they pass through an opening or around an obstacle. When a wave hits a gap comparable in size to its wavelength, it spreads out a lot. When the gap is much larger than the wavelength, the wave passes through with minimal spreading.

This is why you can hear someone talking around a corner (sound wavelengths are comparable to doorway sizes — they diffract well) but you can’t see them (visible light wavelengths are vastly smaller than doorway sizes, so light barely diffracts and travels in nearly straight lines).

Summary of Wave Behaviors at Boundaries