Vibrations and Waves


Simple Harmonic Motion

What is it?

The motion of an object under the influence of either

a force characterized by Hook's law:

F = -kx 

or equivalently 

a potential given by

PEsho = (1/2)kx2 

where sho stands for simple harmonic oscillator.




Force and potential (energy) are two ways to describe a conservative interaction.

Force is equal to the negative of the slope of the corresponding potential.




Examples to be studied: The motion of an object attached to a spring and the motion of a pendulum.


Why is it important?

Almost all types of motion that have an equilibrium position can be approximated by an simple harmonic oscillator if their amplitudes are sufficiently small. 

The understanding of simple harmonic motion is also the basis for understanding the wave motion.


What happens in a simple harmonic motion?

Understanding based on Newton's second law:

a = F/m = -(k/m)x 




Understanding based on the conservation of energy:

a) No friction (no damping)

E = KE + PE = const 

(1/2)mv2 + (1/2)kx2 = E = const 

from which it is clear that energy can be written in terms of the amplitude as

E = (1/2)kA2 

It is also clear that the maximum velocity is achieved at the equilibrium position (x=0) and

E = (1/2)mvmax2 

(1/2)mvmax2 = E = (1/2)kA2 

In a simple harmonic motion, kinetic energy and potential energy are converting back and forth into each other. The energy is purely kinetic at the equilibrium position, and it is pure potential energy at the maxima positions.

b) With friction

DE = Wfriction = -fs 

Example 13.4 Motion with and without friction


Parameters that characterize a simple harmonic motion

Amplitude A and frequency f.

Amplitude: the maximum distance traveled by an object away from its equilibrium position.

Amplitude depends on the initial condition, i.e. how you get the motion started; how much energy you put into the system.

Frequency: the number of cycles per unit time.

Frequency is an intrinsic property of the oscillator and is independent of the initial conditions (as long as you do not stretch it too hard).

f = (1/2p)(k/m)1/2 

This equation is derived by solving the Newton's equation of motion. The solution gives us position as a function of time:

x = A cos w t 

where

w = 2pf = (k/m)1/2 

is called the angular frequency (rad/s).

There is another parameter that is related to frequency. It is the period T.

Period T: time it takes to complete one cycle.

T = 1/f = 2p(m/k)1/2 

(f = 1/T may be even easier to see).

Example 13.7 The vibrating mass-spring system




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There is in fact a third parameter that characterizes a simple harmonic motion. It is called phase and is determined by initial conditions. The concept of phase is very important in understanding the interference between two waves. Briefly, the more general description of displacement as a function of time is

x = A cos(w t+f ) 

in which f is called the phase.

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Pendulum

A pendulum is a good example of the motion near an equilibrium being well approximated by a simple harmonic oscillator. For small angles, the restoring force for a pendulum is

F = -mg sin q -mg q = -(mg/L)s 

which corresponds to a simple harmonic oscillator with an effective spring constant

k = mg/L 

The period of a pendulum is therefore given by

T = 2p(L/g)1/2 

Example 13.8 What is the height of that tower?


Wave

What is it?

Propagation of energy or disturbance.

A spatially correlated oscillations.


Parameters that characterize a (monochromatic) wave

Amplitude and two out of the following three parameters:

which are related by

v = f l 

Using light (an electromagnetic wave) as an example, the amplitude is the parameter that determines the intensity of the light and frequency (or wavelength) is the parameter that determines the color of the light.

Example 13.9 A traveling wave


Transverse and longitudinal waves

Transverse wave: a wave in which the displacement is perpendicular to the direction of propagation.

Longitudinal wave: a wave in which the displacement is parallel to the direction of propagation.

Figure 13.19 is a illustration of two types of waves.




Examples

a) Light (electromagnetic) wave

Light waves of all frequency travel at the same speed in vacuum. The speed of light in vacuum:

c =  2.99792458 108 m/s

The speed of light in a medium is generally frequency-dependent. This phenomenon is called (or gives rise to what is called) dispersion.

Light wave is transverse in nature.

b) Transverse wave on a string

The speed of a transverse wave on a string is determined by

v = (F/m)1/2 

in which F is the tension in the string and m is mass per unit length.

Example 13.12 A pulse traveling on a string

c) Sound wave

A sound wave in air is a longitudinal wave.

Audible sound has a frequency range of 20 Hz to 20,000 Hz.

Doppler effect


Superposition and interference of waves

The displacement at a point which is under the influence of multiple waves is equal to the sum of displacements induced by each individual wave at that point.

Constructive interference: The influences of two waves add to each other.

Destructive interference: The influences of two waves cancel each other.

Figure 13.25 illustrates constructive interference and Figure 13.26 illustrates destructive interference.

The two waves in Figure 13.25 are said to be in phase (oscillations induced by both waves at a specific point reach their maxima at the same time).

The two waves in Figure 13.26 are said to be 180 degrees out of phase (oscillation induced by one of waves is at its minimum when the oscillation induced by the other is at its maximum).




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Waves are described mathematically by so called wave equations. Waves satisfying the superposition principle are described by equations which are linear. Maxwell's equation for an electromagnetic wave in vacuum and the Schrodinger equation in quantum physics are examples of wave equations, both of which are linear.

A medium may become nonlinear if the wave traveling through it is too intense (amplitude too big).

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Copyright 1997 by Bo Gao. All rights reserved.