### Waves and oscillations in nature : an introduction

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Advertisement Hide. An Introduction to Waves and Oscillations in the Sun. Authors view affiliations A. Front Matter Pages i-xii. Pages Electromagneto Statics. MHD Equations and Concepts. Waves in Uniform Media. Waves in Nonuniform Media.

Waves in the Sun. Back Matter Pages The phase of a point tells you how far through the cycle you are like the phases of the Moon. From above, velocity and displacement are related by: This can be written as: Displacement and time are related by: These two results can be repesented graphically, as shown on the right.

The velocity, as a function of time, can be found by differentiating displacement as a function of time. Putting these relationships together as functions of both time and displacement.

## An Introduction to Waves and Oscillations in the Sun | A. Satya Narayanan | Springer

Energy relationships in SHM. It has a mass of 30 kg and a force constant of N.

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The "free" vibrations on a real physical system will decrease in amplitude with time because there is always some friction or viscous drag on the moving object. The damping of the piston provides a typical resistive force, proportional to the velocity, i.

The solution is probably a sine curve that decays exponentially, so this solution is assumed and checked by substitution. Trial solution:. For small damping the damped frequency is only slightly smaller than the free vibration, i.

### Lecture 18

For large damping , the frequency is also zero Hz, but it takes a longer time to return to zero displacement. The total mechanical energy also decays exponentially with time. Example O2 A damped simple harmonic oscillator has a mass of 0. Find the number of periods it oscillates before the energy drops to half the initial value. When a system is subjected to an external sinusoidal force it oscillates at the external frequency and not at its natural frequency.

This is called forced oscillation. For the spring-plus-mass system, the natural frequency is:. If G goes to a minimum, then the resultant amplitude will go to a maximum whose size is limited by the friction coefficient r.

## Oscillations and Waves: An Introduction

Larger friction causes a smaller maximum amplitude. As shown in the graph on the right, there is an increase in the amplitude of the oscillation when the forcing frequency approaches the natural frequency of the system. This kind of situation is called a resonance condition. As also indicated, the sharpness of the peak depends on on the friction coefficient r. Example O3 A cosine oscillator with amplitude mm drives a mass of 0.

Find a the resonant frequency, and b an expression for the displacement at frequency 2. Example O3 Vibrations and built structures.

Vibrations in built structures can just be a nuisance, or cause illness, or even structural damage. Vibrations come from two main sources, internal or external. Inside buildings, vibrations come from machines such as air conditioning units, pumps, elevators, fans, front end loaders, presses, etc, or from people walking, jumping, or running around. External vibrations come from road and rail traffic, subways, construction, earthquakes and winds. The turbulence generated by obstacles may last up to times the height of the structure.

The famous failure of the Tacoma Narrows Bridge is an example of how winds can cause dangerous vibrations. It had shallow plate girders instead of the normal deep stiffening trusses.

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## Sound Waves

When it twisted along its length Newton's law for angular motion applied i. The bridge failed because of torsional "flutter" where one mode at 0.

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It is not properly "resonance" because there is no forcing frequency.