Wave Notes

Wave A wave is a disturbance or oscillation that travels through spacetime, accompanied by a transfer of energy. Wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass transport. They consist, instead, of oscillations or vibrations around almost fixed locations. Waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this equation varies depending on the type of wave. There are two main types of waves. Mechanical waves propagate through a medium, and the substance of this medium is deformed. The deformation reverses itself owing to restoring forces resulting from its deformation. For example, sound waves propagate via air molecules colliding with their neighbors. When air molecules collide, they also bounce away from each other (a restoring force). This keeps the molecules from continuing to travel in the direction of the wave. The second main type of wave, electromagnetic waves, do not require a medium. Instead, they consist of periodic oscillations in electrical and magnetic fields generated by charged particles, and can therefore travel through a vacuum. These types of waves vary in wavelength, and include radio waves, microwave es, infrared d radiation, visible light, ultra aviolet rad diation, X-rrays, and gam mma rays. Wave equatio on Conside er a traveliing transve erse wave (which ma ay be a pullse) on a sttring (the medium m). Conside er the strin ng to have a single sp patial dime ension. Co onsider this s wave as s traveling Waveleng gth λ, can be e measured between b any two correspo onding pointss on a wavefform ƒ in th he directtion in space. E.g., le et the posittive and the negatiive direc ction be to the left. ƒ with h constant amplitude ƒ with h constant velocity ƒ , where dire ection be to o the right,, is ƒ independen nt of wavelength (no o dispersio on) ƒ independen nt of ampliitude (linea ar media, not n nonline ear). with h constant waveform,, or shape This wa ave can the en be desc cribed by th he two-dim mensional functions f (wavefo orm trav veling to th he right) (wavefo orm trav veling to th he left) o more generally, by d'Alembertt's formula: or, represe enting two componen nt waveforrms and travelin ng through h the medium m in opposite directio ons. This wave w can also a be rep presented by b the parttial differen ntial equattion Wave forms Sine, squ uare, triangle and sawtootth waveformss. The forrm or sha ape of F in n d'Alembe ert's formu ula involve es the arg gument x − vt. Constan nt values of o this arg gument corrrespond to t constan nt values of o F, and th hese constan nt values occur o if x increases at the same rate th hat vt incre eases. Tha at is, the wav ve shaped d like the e function F will mo ove in the e positive x-direction at velocity yv (and G will w propag gate at the same spee ed in the negative n x-direction). In the case c of a periodic p fu unction F with w period d λ, that is, F(x + λ − vt) = F(x − vt), the perriodicity of o F in space means s that a snapshot of o the wav ve at a given g time tfin nds the e wave e varying perriodically in s space with period λ (the wave elength of the wav ve). In a similar fa ashion, th his period dicity of F imp plies a perriodicity in time as well:F(x w − v(t v + T)) = F(x F − vt) prrovided vT T = λ, so an observatio o on of the wave w at a fixed location x find ds the wa ave undula ating periodic cally in tim me with perriod T = λ/v v. Ampliitude an nd modu ulation Illustration of the enve elope (the slo owly varying red curve) off an amplitud de-modulated d wave. The fast varying blue curve is the t carrier wa ave, which iss being modu ulated. The am mplitude of o a wave e may be e constan nt (in wh hich case the wave e is a c.w. or o continuo ous wave),, or may be b modulatted so as to t vary with time an nd/or position n. The outtline of the e variation n in amplittude is ca alled the en nvelope off the wave. Mathematic M cally, the modulated m wavecan be b written in the form m: where is the e amplittude en nvelope of the e wave, is the wavenumb w ber and is thephas se. If the group ve elocity (s see below w) is wavelength-independentt, this equa ation can be b simplified as: showing th s hat the env velope mov ves with th he group ve elocity and d retains its s shape. Oth herwise, in cases whe ere the gro oup velocitty varies with w w wavelength h, the pulse shape ch hanges in a manner often o desc cribed usin ng a envelop an pe equation n