A second loo at waves ravelling waves A first loo at Amplitude Modulation (AM) Stationary and reflected waves Lossy waves: dispersion & evanescence I thin this is the MOS IMPORAN of my eight lectures, and I am determined not to rush it, even if it means that we don t cover the whole of lecture 8
If you derive u Recap on the wave equation he wave equation is : u( x, u t c f ( x + c + g( x c u x for the D string, we could interpret c We studied d' Alembert' s solution Now we study a particular u tt, u, u, u u and u Given c, you can fix either x xx u( x, Ae you will find that : instance of j( t x) or but not both + Be So, not surprisingly, it follows that u( x, is a solution of c t tt xx u this solution : j( t+ x) the wave equation if and only if Alternatively, you have such a function u( x, t, ) and it satisfies the wave equation u t u x
Space & time variables For simplicity, we initially concentrate on First, fix t t Now fix x x 0 f ( x, Ae he constant A is the amplitude of the wave he term ( t x) determines the phase j( t x) Since c, we call c the phase velocity of in order to concentrate on the spatial term x the wave π Evidently, the wave cycles spatially π his equates to the wavelength λ is called the wave number and is the number of waves for a π spatial displacement 0 in order to concentrate on the temporal aspect. As above, the period variables associated with time, t variables associated with space, x of the wave is π is called the circular frequency of, circular frequency, period, wave number λ, wavelength the wave
Circular frequency ψ Period, t Low Circular Frequency, High Circular Frequency,
ψ Wave number λ Wavelength, λ z Low Wave Number, High Wave Number,
ravelling wave
ravelling wave We continue to concentrate on f ( t x) Ae j( t x) Note that f ( t x) corresponds to a shift of f ( to the right by x* f( f( x Suppose that time increases by δt but the shape stays the same ( t x) ( ( t + δ ( x + δx)) 0 δt δx δx c δt hat is, f ( t x) is a wave that moves to the right with speed c * Change coordinates or read Kreysig page 59 t
Forwards and bacwards ψ f ( x, Aexp j ( t x) Corresponds to a wave that propagates, or travels, in the forward x direction Similarly: ( t x) ψ b ( x, B exp j + Corresponds to a wave that propagates, or travels, in the bacward x direction General solution ψ ( x, ψ f ( x, + ψ ( x, b Consists of a forward and bacward moving wave
Wave Velocity: Phase Velocity he phase of a wave was defined to be the term φ ( t x) (Recall that phase φ wraps around every π) Differentiating phase: dφ dt dx If we are moving with the wave, at the wave velocity, then d( φ) 0 It follows that: dx dt dx 0 or dt c his is why we called c the Phase Velocity v p c
he string wave: a non-dispersive wave Earlier, we saw that: v p c In this case, v p does not depend on wave properties (such as wave number,, or angular frequency, ) A wave for which the phase velocity is independent of wave properties is called non-dispersive We will return to a discussion of dispersive waves after a brief discussion of standing waves
Dispersive vs non-dispersive waves Non-dispersive wave (lef and dispersive wave (righ
Standing Waves Standing waves occur when two travelling waves of equal amplitude and speed, but opposite direction, superpose We found that, in general: ψ ( x, Ae j( t x) + Be j( t+ x) If the amplitudes are equal, A B ψ ( x, ( j( t x) j( t+ x) + e ) [ ] jt jx jx e e A e Ae + Ae j t [ cos x + j sin x + cos x j sin x] Ae j t cos x
ψ ( x, Ae jt cos x Nodes occur for Antinodes occur for π cos x 0 x (n + ) cos x x (n + ) π cos x > 0 All points for which vibrate in phase with each other, and in antiphase with those for which cos x < 0 ψ Antinodes z Nodes A standing wave t t
Nodes and Antinodes of a pluced string
Example : pluced string* 0 L f ( x) L L g( x) 0 x ( L x) if if L 0 < x < L < x < L Since g( x) 0, Bn * 0, for all n o solve for Bn either apply orthogonality relations or consult HL *Kreysig, 8 th Edn, page 593
Fourier coefficients We need to extend the string to a periodic function We now that the solution is a sum of sin terms, so choose an odd function periodic extension: Using HL, orthogonality, or Kreysig,p545-6 B n ( ) n π n 8 8 π πc 3π 3πc u( x, sin x cos t sin x cos t + π L L 3 L L 5 5π 5πc sin x cos L L t...
his is a standing wave he n th term in the expression is 8 (n + ) πx (n + ) πct sin cos π (n + ) L L Changing coordinates to the midpoint we see that of cosines in x this is indeed a sum
Impedance boundary Consider again the wave y ( x, A e j( wt x) We have seen that this corresponds to a travelling wave, it travels to the right with phase velocity v in the case of a D string wave, c this tells us that the denser the material, the slower the wave you already now this from Snell' s law in optics p c Now we study what happens when there is a sudden change in mass per unit length, m, at x0: an impedance boundary his situation arises in many important practical cases: - sound waves moving from one medium to another - waves in a fluid with suddenly changing properties - ultrasound waves crossing a tissue boundary
Impedance boundary j( t x) Suppose that the wave y ( x, A e approaches an impedance boundary at x 0 from the left ψ y y x Suppose that the density of Recall that the string is to the left and to the right c and We assume (reasonably) that, don' t change c
What happens at the impedance boundary? Some of the energy is absorbed (attenuated) Some of the energy is transmitted in to the different material x > 0 and some of the energy is reflected in the direction -x c c c We assume initially that all the waves have the same frequency, that is, that the waves propagate in a non-dispersive medium ) ( ) ( ), ( x t j x t j R e Ie t x y + + ) ( ), ( x t j e t x y Put y y incident + y reflected y y transmitted I - amplitude of incident wave - amplitude of transmitted wave R - amplitude of reflected wave where I R
Boundary Conditions Evidently, the wave is continuous, and moves continuously at x0 At x 0 y 0, y ( (0, t () y ( t y t ) 0, ) (0, x x () So () gives Ie jt + R e jt e jt I + R his is a statement of conservation of energy I R
Applying boundary condition y(0, x y(0, x From which we find: j j Ie e jt jt + j R. e jt ji + jr j I R so ( I R) Remembering that I + R we get I + and R I +
I + I R + he reflected wave is only in phase with the incident wave if > hat is, the reflected wave is only in phase if the incident wave is in a denser medium he transmitted wave is always in phase with the incident wave c > < < > c c Recall that: so that Interpreting these two equations
Coefficient of ransmission his is defined to be the ratio of the magnitudes of the transmitted and the incident waves I + τ so c τ + We now that Given the Characteristic Impedance for a string Z Z Z Z + We have τ the coefficient of transmission for a wave
Coefficient of Reflection his is defined to be the ratio of the magnitudes of the reflected and the incident waves I R + ρ so c ρ + We now that We define the Characteristic Impedance for a string Z Z Z Z Z + ρ - the coefficient of reflection for a wave So that
Reflection of waves transmission from less dense to more dense Suppose that 4 times denser the material to the right than the material 4 of to the left the impedance boundary is so that and so Z Z hat is : Z Z 0.5
ψ Z y y Z Z x Recall: transmitted incident Z Z + Z and reflected incident Z Z + Z Z Previous Example Z /Z / transmitted incident + 3 reflected incident + 3
Reflection of waves transmission from denser to far less dense Suppose that the material to the right of the impedance boundary is only /4 times as dense as the material to the left 4 so that and so Z Z hat is : Z Z
ψ Z y y Z Z x Recall: transmitted incident Z Z + Z and reflected incident Z Z + Z Z he second example Z /Z transmitted incident 4 + 4 3 reflected incident + 3
Impedance boundaries Left ventricle and myocardium Cyst in the breast
Dispersive or Lossy (damped) wave What s a reasonable model for the decay in the amplitude A of the wave (as a function of x)? Exponential decay A( x) Ae λx
Forward moving wave (again) Recall that a forward moving wave can be described by V Ae j ( t x) where π c λ In the cases we have considered previously, A was a constant. Now we let A(x) decay exponentially.
Lossy wave V x Now consider a damped, or lossy, wave, whose amplitude A decays as the wave propagates, say according to: A( x) Ae lx where l is the "loss factor"
Complex wave number V A( x) e Ae lx j e ( t x) j ( t x) Ae j[ t ( jl) x] hat is, for a lossy wave, with loss factor l, the wave number is complex : ( jl)
D string in a viscous fluid We introduced the wave equation by studying the D string. We now revisit the string but this time place it in a viscous medium so that it loses energy to the medium. his causes damped vibrations, familiar to you from the first year. β u( x, α x δx Re-doing the analysis that led to the wave equation, the equilibrium equation becomes ψ ψ ψ ψ x x β x, where the term β corresponds to viscous damping t x t t
Damped wave equation x t t + ψ ψ β ψ x c t t + Γ ψ ψ ψ Straightforwardly, the damped wave equation for the D string becomes: Slightly more generally, for lossy wave problems Substituting into this equation the general solution for a forward travelling wave (recalling that the wave number κ(-jl) is from now on generally complex) ( ) x t j Ae V κ V c V j V j ) ( κ + Γ We find that: κ c j Γ is called the Dispersion Relation
Analysing the dispersion relation We substitute the (complex) wave number κ Equating real and imaginary parts ( jl) in to the dispersion relation: ( l j l) j Γ c ( l ) c Γ c from which Γ l c c l ± Γ + c ( Γ ) (the loss factor)
Extreme case : Light Damping Γ 0 l Γ ± c ± c so that is, the loss factor is small c Not surprisingly, the situation is almost that of an undamped wave. ψ ( x, Ae Ae j ( t κx ) lx e j ( t x)
Extreme case. Heavy Damping c l Γ ( ) Γ + ± c >> Γ he dispersion relationships: Heavy damping implies that ( ) l c c Γ Γ + ± so that
Phase & group velocity for a vibrating string in a viscous fluid he damped equation was: ψ + Γ t and the complex wave number was Recall that the dispersion relation was Phase velocity Group velocity By definition, ψ c t κ jl ψ x j Γ c equating real parts : v p c d By definition, vg : but d c and so vg l d d c l κ ( l ) c c Note that v p vg c
Chec what happens when we put the damping to zero Γ 0 First: l 0 also c Phase velocity: Group velocity v p v g c c As expected, these do not depend on the wave properties, so the medium is now non-dispersive
Sum of two waves Consider two (rightwards travelling) waves, (a) and (b), which have different frequencies and wavelengths. he sum of the two waves at the same instant is shown in (c) v pa a a ψ ( x, a Ae j ( t x) a a v pb b b ψ ( x, b B e j ( t x) b b ψ ψ + ψ c a b
he non-dispersive case If the two component waves (a) and (b) have the same velocity, v, their sum (c) maintains the same shape and simply travels to the right at velocity v. If the wave is non-dispersive, then the sinusoidal component waves of different frequencies that mae up the wave, travel with the same velocity. he dispersive case If the velocities of waves (a) and (b) are different, they would then slide passed one another as they travelled, and their sum, (c) would change shape as the waves travelled along.
Amplitude modulated (AM) waves he important thing in any communications system is to be able to send information from one place to another. his means we have to find a way to impress that information on the radio wave in such a way that it can be recovered at the other end. his process is nown as modulation. In order to modulate a radio wave, we have to change either or both of the two basic characteristics of the wave: the amplitude or the frequency. Here we just consider the amplitude.
Amplitude Modulated (AM) waves We start, as usual, from a forward moving wave, which defines the carrier of the signal V Ae j( t x) From this, we create two waves, one by decrementing both and ; the other by incrementing them by the same amount, then consider the difference between these two waves: ψ ( x, ψ ( x, ψ ( x, Ae Ae ψ ψ j [( + ) t ( + ) x] j [( ) t ( ) x] where <<, and <<
Carrier & signal Re( ψ ) A cos[( t x) + ( t x)] cos[( t x) ( t x)] sin ( t x) sin( t x) Carrier Signal Ψ( x, t
When the signal and carrier are transmitted through a dispersive medium, they travel at different speeds AM in a dispersive medium v p constant (non - dispersive) v p v p () For the dispersive wave, we can write d + d ( )! d + d For the amplitude modified wave ψ Asin Asin (dispersive)... ( t x) sin( t x) ( t ) x sin t x d d d d
Evidently, the carrier has velocity /, which is equal to the phase velocity v p Similarly, the signal has velocity v g d d Phase Velocity his is called the Group Velocity For a non-dispersive medium, for which / is constant, For a dispersive medium, v v g v p p d v d dv p λ dλ p + dv d p v p + d d dv p dλ dλ d v p