USU PHYS 3750 - Traveling Waves, Standing Waves and the Dispersion Relation

Unformatted text preview:

Lecture 6 Phys 3750 D M Riffe -1- 2/1/2013 Traveling Waves, Standing Waves, and the Dispersion Relation Overview and Motivation: We review the relationship between traveling and standing waves. We then discuss a general relationship that is important in all of wave physics – the relationship between oscillation frequency and wave vector – which is known as the dispersion relation. Key Mathematics: We get some more practice with trig identities and eigenvalue problems. I. Traveling and Standing Waves A. Basic Definitions The simplest definition of a 1D traveling wave is a function of the form () ( )ctxgtxq −=,1 (1a) or () ( )ctxftxq +=,2, (1b) where c is some positive constant.1 The constant c is the speed of propagation of the wave. The wave in Eq. (1a) propagates in the positive xdirection, while the wave in Eq. (1b) propagates in the negative xdirection. Now the functions g and f in Eq. (1) can essentially be any (well behaved) function, but often we are interested in harmonic waves. In this case the functions g and f in Eq. (1) take on the form () ()+−=−φλπctxActxg2sin (2a) and () ()++=+ψλπctxBctxf2sin, (2b) where A and B are the amplitudes, φ and ψ are the phases, and λ is the wavelength of the wave. Now the x- and t -dependent parts of the sine-function arguments are often written as txkω±, and so we can identify the wave vector k as 1 As we shall see, the functions in Eq. (1) are the general solutions to the wave equation, which we will study in short order. However, we shall also see, when we study the Schrödinger equation, that not all waves have these functional forms.Lecture 6 Phys 3750 D M Riffe -2- 2/1/2013 λπ2=k (3) and the angular frequency ω as2 λπωc2=. (4) You should recall from freshman physics that the speed c, frequency ()πων2= and wavelength λ are related by λνc= . So what is a standing wave? Simply put, it is the superposition (i.e., sum) of two equal-amplitude, equal-wavelength (and thus equal-frequency) harmonic waves that are propagating in opposite directions. Using Eqs. (2), (3), and (4) (and for simplicity setting 0==ψφ) we can write such a wave as () ( ) ( )[]tkxtkxAtxqSωω++−= sinsin, . (5) With a bit of trigonometry, specifically the angle-addition formula for the sine function, Eq. (5) can be rewritten as () () ()tkxAtxqSωcossin2, = (6) So instead of being a function of tkxω±, a standing wave is a product of a function of x and a function of t . Equation (6) also show us how to identify the wave vector k and angular frequency ω in the case of an harmonic standing wave: whatever multiplies x is the wave vector and whatever multiplies t is the angular frequency. In fact, for nonharmonic standing waves it is probably safe to define a standing wave as a wave where all parts of the system oscillate in phase, as is the case of the harmonic standing wave defined by Eq. (6). B. Connection to the Coupled Oscillator Problem Let's now go back to the coupled oscillator problem, and reconsider the n th normal-mode solutions to that problem, which we previously wrote as 2 Do not confuse this definition of ω (the angular frequency of the wave) with our earlier definition of ω~ (=mk) that arises when discussing single or coupled harmonic oscillators. It is easy to confuse the two definitions because for a single harmonic oscillator ω~ is also an angular frequency.Lecture 6 Phys 3750 D M Riffe -3- 2/1/2013 ()()()()()()()()()tintinNnNnNnNnnNnneBeANtqtqtqtqΩ−Ω+++++=1111321sin3sin2sin1sinππππMM, (7) where ()[]12sin~2 +=Ω Nnnπω. As we mentioned last time, these modes are essentially standing waves. Let's see that this is the case by writing Eq. (7) in the form of Eq. (6). After we do this, let's also identify the wave vector k and frequency ω for the normal modes. As written, Eq. (7) explicitly lists the motion of each individual oscillator. But the n th normal mode can also be written as a function of object index j and time t as ()()()tintinnnneBeAjNntjqΩ−Ω++=1sin,π, (8) where j labels the oscillator. Although j is a discrete index, we have included it in the argument of the normal-mode function because it is the variable that labels position along the chain. Equation (8) is almost in the form of Eq. (6). In fact, if we take the specialized case of ABAnn== , where A is real, then Eq. (8) can be written as ()() ()tjNnAtjqnnΩ+= cos1sin2,π. (9) This is very close, except in Eq. (6) the position variable is the standard distance variable x, while in Eq. (9) we are still using the object index j to denote position. However, if we define the equilibrium distance between nearest-neighbor objects as d , then we can connect x and j via jdx=, and so we can rewrite Eq. (9) as ()() ()tdxNnAtxqnnΩ+= cos1sin2,π. (10) Now, remembering that whatever multiplies x is the wave vector k and that whatever multiplies t is the angular frequency ω, we have dNnk11+=π (11)Lecture 6 Phys 3750 D M Riffe -4- 2/1/2013 and ()+=Ω=12sin~2Nnnπωω (12) for the coupled-oscillator standing waves. III. Dispersion Relations A. Definition and Some Simple Examples Simply stated, a dispersion relation is the function ()kω for an harmonic wave. For the simplest of waves, where the speed of propagation c is a constant, we see from Eqs. (3) and (4) that their dispersion relation is simply ()ckk =ω. (13) That is, the frequency ω is a linear function of the wave vector k. We also see from Eq.(13) that the ratio kω is simply the propagation speed c . As we will discuss in more detail in a later lecture, the ratio kω is technically known as the phase velocity.3 Now you may be thinking, what is the big deal here? – Eq. (13) is so simple, what could be interesting about it? Now it is simple if the phase velocity c is independent of k . But this is usually not the case. Recall, for example, the propagation of light in a dielectric medium (such as glass) where the index of refraction ccn0= (where 0c is the speed of light in a vacuum) depends upon the wavelength (and thus the wave vector).4 In this case Eq. (13) becomes ()()kknck0=ω. (14) The dispersion relation now has the


View Full Document

USU PHYS 3750 - Traveling Waves, Standing Waves and the Dispersion Relation

Download Traveling Waves, Standing Waves and the Dispersion Relation
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Traveling Waves, Standing Waves and the Dispersion Relation and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Traveling Waves, Standing Waves and the Dispersion Relation 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?