Optics Series Lecture, Lecture – XII and – XIII.
“Traveling waves, Differential wave equations, Particle and wave velocities.” These lectures were delivered on 17th and 20th February 2017, in two lecture sessions of 1 and 1/2 hours each. The web version has been named “Waves.” and the lectures were delivered to Physics honors students.
In one of our earlier optics session lecture I had hinted at having waves defined by their pulse shape called as wave profile — or alternatively wave shape or wave form, and transcribing them into forms that represent actual wave motion. The later are then called as traveling or progressive waves. The former, the so called wave shape or wave profile are then time-snapshots of the full fledged time varying waves that we just called traveling waves. Remember that stationary or standing waves are not wave profiles or any snapshots of a single traveling wave, they are rather the superposition of an advanced and a retarded wave — that is one traveling wave moving forward and another exactly shaped traveling wave moving in the reverse direction. We studied advanced and retarded waves, here. We have also already dealt with traveling waves in much detail, eg, here and here. This lecture will justify what we have been espousing all along. Also in complex waves that are found in quantum mechanical theories, we have what are called as stationary states, these are like the time-snapshots of the quantum mechanical waves, represented through the energy of the system. Since the full energy or wave cycle is not necessarily contained in a given amount of time called as a time window, we have a corresponding uncertainty relation called an energy-time uncertainty relation. But talking about an instant of time, a stationary state which represents the energy of the wave in that instant, are well defined states of energy and called as eigen-states. But what would happen if one takes a picture of a dynamic system? The fuzzed out region or so called “motion blur” might show up, because these time instants are not well defined eigen states rather superposition of random number of any of them, may be.
One dimensional Traveling Waves.
A traveling wave is a self sustaining oscillation of particles of a medium or oscillations of any physical quantity at different space-time points so that energy is transported across the medium when the oscillation propagates in the medium. There is no motion of the relevant medium in the ideal description of the wave. The oscillating particles move periodically about their equilibrium locations and in the case of physical quantities they take values around their equilibrium values. Examples of waves are mechanical waves: waves on strings, surface waves such as water waves, sound waves, compression waves in solids and liquids.
Depending on the relation between oscillation direction and wave propagation direction there are two basic types of waves, the longitudinal waves and the transverse waves .
- Longitudinal waves:
- The oscillating particles in the medium move in a direction along the direction of wave motion. Examples of such waves are sound waves and longitudinal wave motion of springs.
- Transverse waves:
- The oscillating particles move in a direction perpendicular to the direction of wave propagation. In case physical quantities define the wave, their oscillation direction is perpendicular to the wave direction. The examples of such waves are electromagnetic waves and transverse waves on strings, eg when the strings of a violin are plucked they produce transverse waves.
Waves move at great speed because there is no associated movement of the medium, there is only advancement of energy.
Let us consider a wave ψ which is moving in +x direction at speed v. So ψ is a function of space coordinate x and time t. ψ (x, t) = f (x, t) — eqn (i). f (x, t) represents a particular wave-shape at the instant of time t. Thus wave-profile is the shape of the wave at a given instant of time. Thus ψ (x, t)|t=0 = f (x, 0) = f(x) ≡ wave profile — eqn (ii). Waves can take any possible wave forms. Examples of such wave forms are Gaussian or bell shape and sinusoidal or harmonic functions. Gaussian: f(x) = e-ax2 , Harmonic: f(x) = Aeikx — eqn (iii). Let us consider waves which do not change shape during propagation. Then the pulse of such waves moves a distance of vt in time t. Consider a particular wave-pulse in frame of reference S and also in frame of reference S’. The wave pulse moves at speed v in frame S so that in frame S’ it is at rest. Evidently the frame S’ moves at speed v wrt frame S. The situation is depicted in the following diagrams. In frame S’ ψ = f (x’) — eqn (iv). We want to transform eqn (iv) for coordinates of S. For this we need to recall the Galilean Transformation, according to which: x’ = x – vt — eqn (v), so as a result we have ⇒ ψ (x, t) = f (x – vt) — eqn (vi).
So ψ (x, t) in eqn (vi) is a general form of wave function where the wave-disturbance ψ (x, t) is moving at speed v in +x direction. ψ (x, t) is called a traveling or progressive wave in 1-dimension. — we can easily change this wave function and any of its related equations to its 3-dimensional form. We learned that no matter what shape the wave-form has, we can get the corresponding progressive wave by substituting x with x – vt. A harmonic wave-profile gives rise to a harmonic traveling wave and a Gaussian wave-profile that we mentioned a little while ago gives rise to a Gaussian traveling wave.
If a time Δt elapses and the wave pulse has advanced a distance x+vΔt then ⇒ f[(x+vΔt) – v (t+Δt)] = f (x – vt). Thus the wave shape or profile is unchanged. Similarly ψ (x, t) = f (x + vt) for v > 0 gives a wave moving in the -x direction. We can easily see the following alternative form to be a valid transformation of the advanced and retarded wave we just discussed.
Differential wave equation.
A differential wave equation is a linear, homogeneous, second order partial differential equation whose solution is the progressive or traveling wave represented by the wave-form: ψ = f(x’). A linear equation is one where two different solution when added together linearly, is another solution. This results from the linearity of the terms, that is, each term has a degree either 0 or 1, containing only the functions or constants multiplied by the derivatives and they are grouped in one side and equated to 0. The square and cubes and further orders are not allowed for functions or derivatives, but higher order differentiation is allowed. They would be in-homogeneous if instead of equating the grouped terms to 0 one equates them to another function (which is not multiplied to any derivative). Thus having a constant term (a special function) independent of derivatives is the sign of a non-homogeneous or in-homogeneous equation. Equations are called as partial DE (diff eqns) if there are more than 1 independent variables but ordinary DE if there is only one dependent variable. When a 2nd order differential equation is in the offing, one has two arbitrary constants, in case of wave equation this is supplied by amplitude and frequency (or amplitude and wavelength). From rules of partial differentiation and Galilean Relativity that we have used earlier in this lecture: Similarly: . ⇒ By applying our method once again: andBut ⇒ but we have seen already: ⇒ So finally we have what we call the differential wave equation:
Particle and wave velocity.
Harmonic progressive wave. Wave parameters and their relations.
Lets discuss a wave whose wave-form or wave-profile is given by a harmonic function, that is; a sine or cosine function. Accordingly such waves are called as sinusoidal, harmonic or simple harmonic waves. Its important to study such waves even if they are ideal, because any complicated wave can be considered as a combination of large number of simple harmonic waves. The wave profile is given by ψ (x) = f (x). So ψ (x, t)|t=0 = ψ (x) = A sin kx = f(x), where k is a positive constant known as propagation number or wave number and A is is known as the amplitude which gives the maximum possible value of the wave function: ψ (x).
Let us create a traveling wave corresponding to the sinusoidal wave profile ψ (x). To achieve this we let x through the Galilean transformation. So x → x – vt, where v > 0 and wave moves in the +x direction. So we have: ψ (x, t) = A sin k (x – vt) = f(x – vt). We have set the differential wave equation in the last section. This harmonic wave-function ψ (x, t) = A sin k (x – vt) should be a solution to that differential wave equation, namely: .Take this as a home work and show that this is indeed the case. Let us define: λ ≡ the spatial period, known as wave-length as the distance traversing which the wave changes by ± 2π and the wave-function returns to its previous value. That means when x → x ± λ there is a change in the argument of the harmonic wave function of ± 2π and ψ is unaltered. So: ψ (x, t) = ψ (x ± λ, t). Since the traveling wave is based upon a harmonic or sinusoidal variation this implies: sin k (x – vt) = sin k [(x ± λ) – vt] = sin k [(x – vt) ± 2π ] ⇒ kλ = 2π or k = 2π/λ. Similarly when t → t ± τ, ψ is not changed. We have: ψ (x, t) = ψ (x, t ± τ), τ is called as the time period or temporal period of the wave or wave-function. Its the time during which one cycle of wave is completed. This gives us a condition in terms of the sine variation. We have: sin k (x – vt) = sin k [x – v (t ± τ)] and sin k (x – vt) = sin k [(x – v t) ± 2π]. From these two we obtain kvτ = 2π or (2π/λ)vτ = 2π or τ = λ/v. But υ = 1/ τ is called as the linear frequency or linear temporal frequency — remember that we can now in analogy with this define the linear spatial frequency, but always remember by default we do not mention the temporal part, so anytime we must use the analogy privilege in nomenclature we must explicitly specify any spatial variable eg by terminology such as spatial frequency and spatial period and so on.
With all the analysis in the section above we see that: v = υλ, this is the velocity of the wave disturbance and was derived by Newton, really. Remember that all sorts of waves are always blessed with an amicable relation between their linear frequency and wavelength which multiply to give the velocity. But it is the right time for us to define two other parameters so that we do not commit any mistakes of confusion. We define corresponding parameters and relate them to their linear or angular counterpart as it befits. Namely; angular (temporal) frequency; ω = 2π/τ = 2πυ and linear wave number or spatial frequency: κ = 1/λ. So there are 4 in total, linear frequency for time, linear frequency for length and their angular counterparts, in the respective order: υ = 1/τ, κ = 1/λ, ω = 2πυ, k = 2πκ.
Phase and phase velocity.
Now we are in a situation where we can define the phase of a wave. We have taken as an example of our wave the harmonic wave-function. Here the total angular argument of the sine or cosine function is called as the phase of the wave. Thus simply saying phase is the angular state — a function of space coordinate as well as time coordinate, of the wave. Its given by: φ = kx ± ωt for the wave-function given by: ψ (x, t) = A sin k (x ± vt). If at the initial instant t = 0, and x = 0 the wave-function takes the form, ψ (x, t)|t = 0, x = 0 = ψ (0, 0) = 0. But in general: ψ (x, t) = A sin (kx ± ωt + ε) where ε is called as the initial phase, phase constant or epoch. Let us evaluate the rate of change of phase with respect to time, at a constant location. As we can see this gives the angular frequency of the wave. Similarly the rate of change of phase at a particular instant of time gives us the (angular) wave number that we have defined earlier, k. We have: According to rules of partial differentiation: . So we have: . The factor in parenthesis is called as the phase velocity, it is the distance covered by points of constant phase in an unit time. We see that its numerically equal to wave velocity v and has the same direction as that of wave velocity if wave is harmonic.
Particle velocity of wave.
The particle motion in a medium is described by a velocity which is different from the wave or phase velocity. then Thus maximum particle velocity is given by: . There is yet another velocity for waves and its called as group velocity. When there are multiple waves spanning a space-time the phase of different υ or λ move at different phase velocity. Overall we have vg = ∂ω/∂k instead of vp = ω/k.