Oscillations and Waves - Examples & Applications

Engineering Applications

$Millennium Bridge: Driven Lateral Vibration$

When the Millennium Bridge in London opened in 2000, pedestrian motion and the bridge's lateral response became coupled: people tended to synchronize their steps with the moving deck, reinforcing lateral excitation. Explain why added damping reduces the response without treating the event as a simple undamped oscillator.

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$P- and S-Wave Arrival-Time Interpretation$

An earthquake generates P-waves and S-waves. Explain what a station learns from their different particle motions and arrival times, and distinguish source distance from epicenter location.

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Simple Harmonic Motion (SHM) Examples

$Mass-Spring Period and Frequency$

A $2.0 \text{ kg}$ mass is attached to a horizontal spring with a spring constant $k = 50 \text{ N/m}$ . The mass is displaced $0.1 \text{ m}$ from equilibrium and released from rest. What is the period and frequency of the oscillation?

Horizontal spring-mass system labeled m equals 2.0 kilograms, k equals 50 newtons per meter, equilibrium position, and initial displacement 0.10 meter. — Mass-Spring Period Diagram

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$Simple Pendulum on the Moon$

An astronaut on the Moon wants to create a simple pendulum that has a period of exactly $2.0 \text{ seconds}$ . If the acceleration due to gravity on the Moon is $1.62 \text{ m/s}^2$ , how long must the pendulum string be?

Simple pendulum geometry showing the unknown string length L, small swing angle, and lunar gravitational acceleration 1.62 meters per second squared. — Lunar Pendulum Diagram

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$Energy Conservation in a Mass-Spring Oscillator$

A $0.5 \text{ kg}$ block attached to a spring ( $k = 200 \text{ N/m}$ ) oscillates on a frictionless horizontal surface. If its maximum speed (velocity at equilibrium) is $3.0 \text{ m/s}$ , what is the amplitude of the oscillation?

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Wave Properties and Speed Examples

$Radio-Wave Wavelength from Frequency$

A radio station broadcasts an FM radio wave (an electromagnetic wave traveling at the speed of light, $c = 3.0 \times 10^8 \text{ m/s}$ ) at a frequency of $95.5 \text{ MHz}$ . What is the wavelength of the radio wave?

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$Wave Speed on a Tensioned String$

A $2.0 \text{ m}$ long guitar string has a mass of $0.005 \text{ kg}$ . It is pulled tight with a tension of $400 \text{ N}$ . If the string is plucked, how fast does the transverse wave pulse travel along the string?

Two-metre tensioned string with total mass 0.005 kilogram, equal 400-newton end tensions, and a transverse pulse propagating along the string. — Tensioned-String Wave-Speed Diagram

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$Open-Pipe Fundamental and Second Harmonic$

An organ pipe that is open at both ends has a fundamental frequency (first harmonic) of $250 \text{ Hz}$ . If the speed of sound in air is $340 \text{ m/s}$ , how long is the pipe? What is the frequency of its second harmonic?

Open-open pipe fundamental standing-wave pattern with antinodes at both ends, a node at midspan, and pipe length equal to one-half wavelength. — Open-Pipe Standing-Wave Diagram

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$Temporal Beat Frequency from Two Nearby Tones$

Two coherent sound waves at the same observation point have frequencies $f_A=256\text{ Hz}$ and $f_B=260\text{ Hz}$ with comparable amplitudes. Determine the beat frequency and the time between successive amplitude maxima.