Oscillations & Waves · WaveLab

Spring Oscillator

A weight hangs from the lower end of a vertical spring. Pull the weight down a little and release it: it moves up and down about its equilibrium position. The restoring force is proportional to the displacement and opposite in direction:

F = −kx

Such motion is called simple harmonic motion. A pen on the weight traces an x-t graph on a paper tape moving at constant speed, and the trace is a sine curve.

T = 2π√(m/k)

The period does not depend on the amplitude; it is determined only by the oscillator's mass m and the spring constant k. Vary m and k, record the period, and plot a T-√m graph.

Simple Pendulum

A small ball is tied to the lower end of a thin string. When the amplitude angle is small, the ball moves in approximately simple harmonic motion. A photogate records each moment the ball passes the lowest point, which gives the period:

T = 2π√(L/g)

The period of a simple pendulum does not depend on the bob's mass or (for small angles) the amplitude; it depends only on the pendulum length L and the local gravitational acceleration g. This is the isochronism of the pendulum, and it is also a way to measure g: plot a T²-L graph, whose slope is k = 4π²/g.

Increase the amplitude angle beyond 30° and the measured period will be longer than the formula predicts. Try the Moon or Mars: the period gets longer.

Energy in Simple Harmonic Motion

At the highest point the bob's speed is zero: kinetic energy is zero and potential energy is at its maximum. At the lowest point its speed is greatest: kinetic energy is at its maximum and potential energy at its minimum. Without resistance, kinetic and potential energy convert into each other and mechanical energy is conserved:

E = Eₖ + Eₚ = constant

Increase the damping and mechanical energy is gradually converted into internal energy, so the amplitude keeps shrinking. This is damped oscillation. Note that the period of a damped oscillation hardly changes.

Forced Oscillation & Resonance

A motor drives an eccentric wheel that moves the top of the spring up and down periodically, so the oscillator undergoes forced oscillation under the driving force. Once steady, the oscillator's frequency equals the frequency of the driving force, regardless of its natural frequency.

The closer the driving frequency is to the oscillator's natural frequency f₀ = (1/2π)√(k/m), the larger the amplitude; when the two are equal the amplitude is greatest. This is resonance.

Press "Sweep driving frequency" to plot the resonance curve automatically. The smaller the damping, the higher and sharper the resonance peak. Troops breaking step on a bridge and the shock absorbers in a washing machine both relate to resonance.

Formation of Mechanical Waves

A row of particles is linked by elastic forces. The leftmost particle oscillates under an external force and sets its neighbors oscillating one after another. In this way the form of the vibration and its energy propagate outward, forming a mechanical wave.

Note that the particles themselves do not travel with the wave; each only oscillates about its own equilibrium position (watch the red particle). If the particles oscillate perpendicular to the direction of propagation, it is a transverse wave; if along the same line, it is a longitudinal wave, with rarefactions and compressions.

v = λ f = λ / T

Reflection of Waves

Flick a pulse from one end of the rope. When the pulse reaches the other end, it is reflected back.

· Fixed end: the end of the rope is clamped and cannot move. The reflected wave is inverted relative to the incident wave (a crest becomes a trough), as if half a wavelength were lost. This is the half-wave loss (phase reversal).· Free end: the end of the rope is looped around a smooth vertical rod and can slide freely. The reflected wave is in phase with the incident wave.

The wave speed is the same before and after reflection.

Superposition & Standing Waves

When several waves meet, the displacement of each particle equals the sum of the displacements each wave would cause on its own. This is the principle of superposition. After meeting, each wave keeps its original shape and travels on.

The source at the left end oscillates continuously; the wave reflects at the fixed right end, and the incident and reflected waves superpose. When the rope length is exactly an integer multiple of half a wavelength, a stable standing wave forms: some points never move (nodes) and some have the largest amplitude (antinodes).

fₙ = n v / 2L (n = 1, 2, 3 …)

This is how the strings of a guitar or violin make sound.

Interference & Diffraction of Water Waves

Two sources that vibrate in exactly the same way (coherent sources) set up water waves. Where the path difference from a point to the two sources is an integer multiple of the wavelength, the vibration is reinforced; where it is an odd multiple of half a wavelength, the vibration is weakened. Stable regions of reinforcement and cancellation alternate across the surface. This is wave interference.

Δr = kλ constructive Δr = (2k+1)λ/2 destructive

Switch to single slit: a plane wave passing through a slit in a barrier spreads out in all directions. This is diffraction. Diffraction is most noticeable when the slit is narrower than, or about the same as, the wavelength.

Doppler Effect

When a sound source approaches an observer, the wavefronts ahead of it are "squeezed" together and the wavelength gets shorter, so the observer hears a higher frequency. When the source moves away, the wavelength gets longer and the frequency drops. This is the Doppler effect:

f′ = f · v / (v ∓ vₛ)

Turn on the sound and switch the "Listener position" to hear the pitch change. When the source moves faster than sound, the wavefronts pile up into a cone, the Mach cone, producing a sonic boom.

Properties of Sound & Resonance

Sound is produced by vibrating objects and travels as sound waves. The oscilloscope shows the waveform picked up by the microphone:

· Loudness is set by the amplitude· Pitch is set by the frequency· Timbre is set by the waveform (its mix of harmonics). A tuning fork gives an almost pure sine wave; a piano or violin is rich in harmonics.

Strike a tuning fork and hold it over the mouth of the resonance tube. Adjust the water level to change the length L of the air column. When

L = (2n − 1) λ / 4

the air column resonates and the sound becomes clearly louder. The difference λ/2 between two resonance lengths lets you measure the speed of sound.

Sandbox

Every parameter is open: how the left end oscillates, the boundary at the right end, wave speed, frequency, amplitude and damping.

Try these:

· Left end driven, right end absorbing: a continuous traveling wave· Left end driven, right end fixed: a standing wave· Left end pulse, right end free: in-phase reflection· Turn up the damping: the wave decays as it travels