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Following points are the summary of these Lecture Slides : Waves, Oscillating Disturbances, Move, Quantum Waves, Space, Mechanical Waves, Disturbance, Medium, Travel, Vacuum
Typology: Slides
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All waves are oscillating disturbances that move through space over time. Some types of waves (electromagnetic and quantum waves of matter) can travel in a vacuum. Others, known as mechanical waves, (water waves, sound) require a medium that supports the disturbance.
CC: BY Amagill (flickr) http://creativecommons.org/licenses/by/2.0/deed.en
Mechanical waves produce a displacement of particles within a medium that both
There are two basic types of mechanical waves:
1. longitudinal: The particle displacement is parallel to the direction of the traveling wave. sound is a longitudinal wave 2. transverse: The particle displacement is perpendicular to the direction of the traveling wave. waves on a pond’s surface are transverse waves
A simple and useful example is a sinusoidal wave
which describes SIMPLE HARMONIC MOTION IN BOTH SPACE AND TIME. Such a wave is characterized by A : maximum displacement amplitude (or, simply, the amplitude ) ω : angular frequency ( ω = 2π/ T = 2π f ) k : angular wavenumber ( k = 2π/λ)
The combination kx – ω t defines the phase of a rightward traveling wave. A point of fixed phase (e.g. a “peak” or a “trough”) will move in the +x direction with a velocity given by
Alternate forms for the wave speed are v = f λ = λ / Τ.
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y ( x , t ) = A cos( kx ± ω t )
v = ω / k , the wave speed.
y
x
v
v particle
The wave disturbance travels through the medium with a speed
At the same time, a small piece (a particle) of the medium oscillates about its equilibrium location, implying a
The direction of the particle velocity is either perpendicular to (for transverse waves, as shown above) or parallel to (for longitudinal waves) the direction of the wave velocity.
A string with mass per unit length μ stretched under tension F (we use F here to avoid confusion with the period T ) supports transverse waves that travel with speed
Waves travel faster when the tension is higher or when the medium is less dense.
This expression implies an energy equation for a small length of string Δ x of the form
meaning tension supplies the work needed to support wave motion in the string.
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v =
F μ
( μΔ x ) v^2 = F Δ x
When two traveling waves y 1 ( x , t ) and y 2 ( x , t ) intersect, their wave displacements add
A wave that travel to a boundary is reflected. Upon reflection, the wave’s velocity reverses and its phase is either
changed by π radians if the boundary is fixed (closed) or unchanged if the boundary is free (open).
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y ( x , t ) = y 1 ( x , t ) + y 2 ( x , t )
For animated examples of wave superposition and reflection, see http://www.kettering.edu/~drussell/Demos/superposition/superposition.html
For a taut string tied at both ends (as in any stringed musical instrument), the existence of nodes at at x =0 and x = L requires that
If the string supports wave velocity v , this condition restricts the frequencies that can be expressed by the string to a discrete set of values
f n = n ( v /2 L ) ; n =1, 2, 3, …
termed the fundamental mode ( n =1) and higher harmonics ( n >1).
CC: BY-NC headspacej (flickr) http://creativecommons.org/licenses/by-nc/2.0/deed.en
