# Simple Harmonic Motion and Waves

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Simple Harmonic Motion and Waves

Wave Motion 4.4.2 State that progressive (travelling) waves transfer energy. Students should understand that there is no net motion of the medium through which the wave travels. A wave is a disturbance travelling through a medium. In a wave, ther is a transmission of ENERGY, but no transmission of MATTER.

This illustrates that there is no net transfer of the medium through which the wave travels, only energy moves from place to place. In many examples, the wave carrying medium will oscillate with simple harmonic motion (i.e. a  -x).

Wave Motion 4.4.2 State that progressive (travelling) waves transfer energy. Students should understand that there is no net motion of the medium through which the wave travels. A wave is a means by which energy is transferred between two points in a medium without any net transfer of the medium itself. Medium - The substance or object in which the wave is travelling. When a wave travels in a medium parts of the medium do not end up at different places The energy of the source of the wave is carried to different parts of the medium by the wave.

Wave Motion A wave travels along its medium, but the individual particles just move up and down.

Wave Motion A pulse is a short burst of wave energy.
4.4.1 Describe a wave pulse and a continuous progressive (travelling) wave. Students should be able to distinguish between oscillations and wave motion, and appreciate that, in many examples, the oscillations of the particles are simple harmonic. A pulse is a short burst of wave energy. In a continuous wave, energy is being continuously transmitted by the wave, and the wave profile travels onward with characteristic wave velocity v.

Wave Motion All types of traveling waves transport energy.
4.4.1 Describe a wave pulse and a continuous progressive (travelling) wave. Students should be able to distinguish between oscillations and wave motion, and appreciate that, in many examples, the oscillations of the particles are simple harmonic. All types of traveling waves transport energy. Study of a single wave pulse shows that it is begun with a vibration and transmitted through internal forces in the medium. Continuous waves start with vibrations too. If the vibration is SHM, then the wave will be sinusoidal.

Types of Waves Waves can be described as transverse or longitudinal. In a transverse wave the vibration (oscillation) of particles is at right angles to the direction of energy transfer. Transverse waves cannot move through a gas. Examples In a longitudinal wave the vibration of particles is in the same plane as the direction of energy transfer. Examples:

(Label a crest (peak), trough, rarefaction, compression)
Transverse: Longitudinal: Energy Transfer Oscillation Energy transfer (Label a crest (peak), trough, rarefaction, compression)

Demos

Types of Waves: Transverse and Longitudinal
The motion of particles in a wave can either be perpendicular to the wave direction (transverse) or parallel to it (longitudinal). Sound waves are longitudinal waves. Light waves are transverse waves.

Types of Waves: Transverse and Longitudinal
Sound waves are longitudinal waves:

4.4.4 Describe waves in two dimensions, including the concepts of wavefronts and of rays.
Think about the wave that are constantly hitting a beach. Each wave has a crest and trough. Imagine you are in a helicopter. If you look down you will see wavefronts. Lines perpendicular to the wave fronts are called rays; they point in the direction of propagation of the wave.

4.4.4 Describe waves in two dimensions, including the concepts of wavefronts and of rays.
Two- or three-dimensional waves can be represented by wave fronts, which are curves of surfaces where all the waves have the same phase. Lines perpendicular to the wave fronts are called rays; they point in the direction of propagation of the wave and the direction energy is traveling.

Energy Transported by Waves
If a wave is able to spread out three-dimensionally from its source, and the medium is uniform, the wave is spherical.

Wave Motion Wave characteristics: Amplitude, A Wavelength, λ
4.4.5 Describe the terms crest, trough, compression and rarefaction. Wave characteristics: Amplitude, A Wavelength, λ Frequency f and period T Wave velocity

Describing Waves Crest and trough: Points on the wave where particles are oscillating at maximum positive and negative displacement. Rarefaction and compression: Areas in a longitudinal wave where particles are far apart (lower than normal density) and close together (higher than normal density).

Wave Motion rarefactions wavelength Many times longitudinal wave are graphed to better analyze them. Notice that the high pressure corresponds to a compression and low pressure corresponds to a rarefaction.

Wave Motion 4.4.6 Define the terms displacement, amplitude, frequency, period, wavelength, wave speed and intensity. Students should know that intensity ∝ amplitude2. Displacement (s) is the distance that any particle is away from its equilibrium position at an instance. Measured in meters. Amplitude (A, a) This is the maximum displacement of a particle from its equilibrium position. Measured in meters. Period (T) This is the time that it takes a particle to make one complete oscillation. Measured in seconds.

Wave Motion Frequency (f) This is the number of oscillations made per second by a particle. The SI unit of frequency is the hertz -Hz. (1 Hz is 1 oscillation per second). Clearly then, f = 1/T Wavelength () This is the distance along the medium between two successive particles that have the same displacement and the same phase of motion. Measured in meters. Wave Speed (v, c) This is the speed with which energy is carried in the medium by the wave. Measured in ms-1

Example Problem 1 If the waves in the ocean are timed so that they come to shore every 1.74 seconds, what is the frequency of these wave? Answer: 0.575Hz

Example Problem 2 A radio wave has a frequency of 2 MHz. Calculate the time period between successive waves. Answer: 5x10-7s

Wave Characteristics Amplitude.
Height of a wave from origin to a peak/crest. Affects brightness and intensity. Wavelength. Distance from crest to crest. Distance for one full cycle. Visible light: nm.

Wave Characteristics Period
time that it takes to complete a full cycle. Measured in seconds Frequency. number of cycles per second. Measured in hertz(Hz) High frequency = high energy

Wave Characteristics Speed of light Speed of light a constant:
3.00 X 108 m/s. Frequency and Wavelength related by the equation:  = c / 

Draw one wave with wavelength 2 cm
4.4.7 Draw and explain displacement – time graphs and displacement-position graphs for transvers and for longitudinal waves. Split up into groups of 2. Draw one wave with wavelength 2 cm Draw second wave with wavelength 4 cm. Draw a third wave with wavelength 8cme. Draw 3 more waves with the same wavelengths as the first set but with an amplitude of 6 cm.

The Wave Equation 4.4.8 Derive and apply the relationship between wave speed, wavelength and frequency. There is a very simple relationship that links wave speed, wavelength, and frequency. It applies to all waves. The time taken for one complete oscillation is the period of the wave, T. In this time, the wave pattern will have moved on by one wavelength, λ. This means that the speed of the wave must be given by V=d =λ since T= 1 t T f Then we get

Example 3 A radio station broadcasts with a wavelength of 160m. If the velocity of the radio signal is 3 x 108m/s, calculate the frequency of the wave. Answer: 1.88MHz

Example 4 A water wave reaches the beach every 2.5 seconds. If it’s wavelength is 5m, calculate the speed of the wave. Answer: 2m/s

Example 5 A stone is thrown into a still water surface and creates a wave. A small floating cork 2m away from the impact point has the following displacement-time graph (time is measured from the instant the stone hits the water).

Example 5 Determine: The amplitude The velocity The period of the wave
The frequency of the wave The wavelength of the wave The time when the reaches its maximum vertical height

Example 5 Answers a) 4cm b) .77m/s c) 2.9 – 2.6 = .3s d) 3.33Hz
e) 0.231M f) 2.675s

The Electromagnetic Spectrum
Electromagnetic waves are waves have a magnetic field oscillating in one dimension and an electric field oscillating in another direction. They are transverse waves, however they are waves of fields and not matter. Because of this they can propagate (travel) in a vacuum.

EM properties All EM waves share similar properties.
They all propagate by means of changing electric and magnetic fields. They are all transverse in nature. They all travel at the same speed in a vacuum. (3x108m/s = c)

Electromagnetic waves
Light is a form of Electromagnetic Radiation. EM Radiation has waves in the electric and magnetic fields Electromagnetic waves have two basic parts. electric field magnetic field The fields are perpendicular to each other.

Electromagnetic Spectrum
Many parts including: Gamma Rays (10-11 m) X-Rays (10-9 m) Ultra-violet (10-8 m) Visible (10-7 m) Infared (10-6 m) Microwave (10-2 m) TV/Radio (10-1 m)

Electromagnetic Spectrum
Visible Spectrum: ROY G BIV Red Orange Yellow Green Blue Indigo Violet The frequency of an electromagnetic wave is related to its wavelength:

Notice that the speed of the light is a constant for any point of the spectrum. Ex Radio

The Electromagnetic Spectrum
4.4.9 State that all electromagnetic waves travel with the same speed in free space, and recall the orders of magnitude of the wavelengths of the principal radiations in the electromagnetic spectrum. Long  Short  Low f High f Radio Waves Micro Waves Infra red VISIBLE Ultra Violet X rays Gamma rays

The Electromagnetic Spectrum
Electromagnetic waves can have any wavelength; we have given different names to different parts of the wavelength spectrum. Wave Type Order of Magnitude of the wavelength (m) Radio Microwave Infrared Visible 7x10-7-3x10-7 Ultraviolet 3x X-ray Gamma

EM Background James Maxwell helped to develop our understanding of EM waves. Used four equations to describe EM waves Showed EM waves travel at 2.998x108 m/s Showed that speed of EM waves is dependent on two constants, the permittivity of free space, ε0 and the permeability of free space, µ0.

The permittivity of free space, ε0 describes the electric characteristic of the wave.
The permeability of free space, µ0 describes the magnetic characteristic of the wave. ε0 = 8.85 x C2/Nm2 µ0 = 4π x 10-7 m/A So…. c = 1/(√ε0µ0) c = 1/(√8.85 x x 4π x 10-7 = x 108 m/s = 3 x 108 m/s

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