Tuesday, 4 January 2011

Physics: Waves I

I assume readers have basic optics knownledge (F3 level), and the optic topics will be disclosed later.

Wave motion
Wave transmits energy without transferring matter.
Transverse VS longitudinal wave – when transverse wave passes through a medium, it oscillate in a direction perpendicular to the direction of wave motion, while the motion of particle is parallel to direction of wave motion in longitudinal wave. Examples of transverse wave include water wave, EM wave, while longitudinal wave include sound wave. Long spring and slinky spring are used to perform transverse and longitudinal waves respectively.
Travelling waves carries the energy away while energy is stored up in stationery waves.
Description of waves
-          Crest and trough: In longitudinal waves, the most compressed position is called center of compression while the most rarefacted position is called the center of rarefaction.
-          Displacement (s): displacement from the center (equilibrium position – s=0)
-          Amplitude (A): max. displacement from equilibrium position.
-          Wavelength (λ): length between two neighbouring crest/trough/centers.
-          Frequency (f): Number of waves produced in 1s, in Hz, period (T) = 1/f.
-          v = s/t = λ/T = fλ, where v is the speed of waves, only depending on the medium.
-          Phase: stages of oscillation, when two particles in the waves has the same motion all the time then they’re in phase. If not, they’re out of phase. If they always have opposite motion, they’re out of phase exactly.
-          In transverse waves, all particles oscillate at the same f and A.
-          Waves can be described as s=cos (t+θ0), then consider s’ = v = sin (t+θ0), speed of particles maximized at equilibrium position, and momentarily at rest at crest and trough.
-          Displacement-distance graph (s-x) is the appearance of wave at an instant of time.
-          Displacement-time graph (s-t) is the motion of given particle in different time.
In longitudinal waves, s-x graphs can be plotted on a straight line rather than a wave. In other way using a curve shows the displacement from equilibrium position for each particle.
Speed of transverse wave in a taut string: v = (t/μ)0.5, μ is the linear density (kgm-1).
Speed of compressional waves in a long and thin solid: v = (E/ρ)0.5, E is the young modulus.
Observing waves: using ripple tank, water with detergents is used (to reduce surface tension), sponge in the boundary prevents the reflection of waves. In the waves crest are bright and trough is dark since crest acts as convex lens to converge lights. Straight bar is used to produce straight waves while dipper is used to produce circular waves.
In a diagram, wavefronts show the neighbouring particles which are in phase (and usually crest). Rays show the direction of travel of the wave (dotted arrow).
Reflection of waves
It obeys angle of incidence = angle of reflection. For circular waves, we can reflect the position of dipper through the barrier, and after the wave is reflected, it’s like the image of reflected dipper producing the circular waves (having center at reflected dipper).
When a string connected to a fixed end (or from a less dense to a more dense medium), the reflected wave has a π phase change (half a period), while there’s no change in free end (from a more dense to a less dense medium).
Refraction of waves
When a waves entering a shallower region, v decreases and f unchanged, therefore λ decreases. Consider straight waves, when it enters shallower region, v decreases, then the ray bends towards normal. When it enters a deeper region, v increases and it bends away from the normal. Convex and Concave shallow region act as convex and concave lens and converge / diverge rays respectively.
Diffraction of waves
When it passes through a barrier, the boundary of waves diffracts into straight line + curve or nearly a semi-circle (pass through a slit). Note that when λ increases or width of gap decreases, the degree of diffraction increases (a large curved wavefront produced.
Interference of waves
Coherent waves must be used to perform interference, or we can say same f and similar A. Also, there must be two sources, or straight waves passing through a double slit.
Superposition: when two waves meet, they add each other (i.e. cos(t+θ0)+ (t+θ1)), when they separate, they back to original state as if nothing has happened.
Consider two source, S1 and S2, consider a point X. Denote path difference Δλ = S1X ~ S2X (physics notation), = |S1X - S2X|, in terms of λ.
Now when Δλ = nλ where n is an integer, then it has a constructive interference since when Δλ is an integer, then the motion produced by the two waves are in phase, that is, the waves added up and reinforce to give a motion of doubled amplitude at that point.
If Δλ=(n+0.5) λ, where n is an integer, then it’s a destructive interference since motion produced by the two waves are out of phase exactly, then the two waves eliminated each other. Then that particle will have no motion at all (A=0).
In a diagram we can draw a line connecting all particles with the same path difference. Those who joins particles with constructive interference is called antinodal lines(maxima) while those who joins particles with destructive interference is called nodal lines(minima).
When the two sources are close together, the maxima and minima are closely packed while they spread widely when the two sources are farther apart.
When Δλ = nλ or = (n+0.5)λ, the antinodal/nodal lines related are the nth-order maximum/minimum. The line joining particles that Δλ=0 is called central maximum.
Stationery wave – waves that stay at fixed points. They’re usually with a fixed end. When the waves are reflected (note that it has a π phase change), it has superposition with the original wave. Different particle has different A. Those points which have no motions are called node. All particles inside two successive nodes are in phase (s:A is the same at any time) While particles in two sides of nodes are all out of phase exactly. λ of a stationery waves is given by 2*length between two successive nodes.

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