Tacoma Narrows Bridge (1940)

Saturn

Chapter 20: Sound

Sound waves are compression (longitudinal) waves and may be excited in gases, solids, and liquids. They need media (materials) in which to propagate unlike Electromagnetic Waves which can propagate in the absence of material, in vacuo. We have already covered much of the material in Chapter 20 in our discussion of waves and their properties. In this lecture we describe:

  • Resonant Processes
  • refraction and reflection of sound waves (and waves in general).

Total Lunar Eclipse (Why can we see the Moon and why does the Moon appear reddish?)

Resonances and Forced Oscillations and Vibrations

Pendula and Vibrations: To the right is shown a pendulum. The period of swing for a pendulum depends only on its length (for small swings)!!! The period of swing is
P = 2 (L/1 meter)1/2 seconds

where P stands for the period of swing and L is the length of the pendulum.

Simple Harmonic Oscillators: A mass attached to a spring is pulled downward by gravity. It assume as an equilibrium stretch such that its restoring spring force (a Hooke's Law force, F = -Kx) balances the downward pull of gravity. If the mass is disturbed it oscillates about this equilibrium position with period

P = 6.3 (M/K)1/2

All mechanical systems made of elastic materials when struck, ring like a bell. The exact frequency with which they ring, depnds on their structure. Generally small objects ring faster than large objects and dense objects ring faster than less dense objects.

Even stars ring; our Sun is observed to oscillate in over 1 million different modes, four types are shown below:

Resonant effects arise when a forcing frequency excites resonance frequencies in say a bridge, pendulum, spring, or any mechanical system. Driving at these special frequencies allows large amounts of energy to be transferred compared to driving off resonance.

Free Oscillators

Simple harmonic oscillator oscillates with frequency f = (K/M)1/2, where K is the spring constant defined by the Hooke's Law force = -Kx, and M is the mass of the suspended weight

Simple Pendulum oscillates with natural period P = 2 (L/1 meter)1/2 seconds.

Tacoma Narrows Bridge (1940)

Tacoma Narrows Bridge

The Tacoma Narrows Bridge in Washington measured a length 1.9 km and was one of the largest suspension bridges of its time. It spanned the Tacoma Narrows channel and collapsed in a dramatic manner on Thursday November 7, 1940. Moderate winds of 65 km/hr to 75 km/hr led to aerodynamic effects (lift) which produced an oscillation which eventually broke the bridge, an effect known as flutter . The bridge first vibrated torsionally, giving it a twisting motion. Later, the vibrations excited a natural resonance of the bridge and the amplitude of the oscillations greatly increased leading to the break-up.

Saturn's Rings and Resonances

Saturn's rings exhibit effects of resonance with the motions of Saturn's moons. There are gaps and edges at locations of resonance. The outer edge of the A-ring is rougnly a six-petaled flower lies at a 7:6 resonance with the moon Janus. At this resonance, ring materials circle Saturn 7 times while Janus orbits 6 times. The boundary between the B-ring and the Cassini Gap is located at the 2:1 resonance with Mimas. The Encke Gap is at a 5:3 resonance with Mimas.

Besides gaps, resonant effects appear in the A-Ring as a 5:3 resonance with Saturn's moon Mimas. Two ave patterns are formed from the Mimas resonance. The left is an outward propagating spiral density wave. The right is an inward propagating spiral bending wave that is made up of vertical oscillations of the ring material. The features are separated by about 400 kilometers.

Refraction Idea

Refraction of Sound Waves

The speed of sound is given by v = (1.4 P/density)1/2 for the atmospheric gases on Earth. Near the surface of the Earth where the temperature is 20 Celsius (293 Kelvin), the speed of sound is 343 meters per second. The sound is determined by how fast the compression is carried through the air; it is reasoanble that the faster the particles move (the higher the temperature), the faster the information propagates. The higher the temperature, the faster sound travels. The lower the temperature, the slower sound travels.

In the atmosphere (in the troposhere), the temperature generally falls with altitude. However, this can change locally depending on whether it is night or day and other conditions (such as in Los Angeles). The temperature variation causes sound waves to refract (to bend) as they move through the atmosphere. This refraction affects our ability to hear distant events: