One Dimensional Waves

1.138J/2.062J, WAVE PROPAGATION CHAPTER TWO ONE DIMENSIONAL WAVES 1 General solution to wave equation It is easy to verify by direct substitution that the most general solution of the one dimensional wave equation:   [\frac{\partial^{2}\phi}{\partial t^{2}}=c^{2}\frac{\partial^{2}\phi}{\partial x^{2}}] (1.1) can be solved by [\phi(x,t)=f(x-ct)+g(x+ct)] (1.2) where (f(\xi)) and (g(\xi)) are arbitrary functions of ξ. In the x, t(space, time) plane (f(x-ct)) is constant along the straight line x - ct = constant. Thus to the observer (x, t) who moves at the steady speed c along the positive x-axis, the function f is stationary. Thus to an observer moving from left to right at the speed c, the signal described initially by (f(x)) at t = 0 remains unchanged in form as t increases, i.e., f propagates to the right at the speed c. Similarly, g propagates to the left at the speed c. The lines x - ct = constant and x + ct = constant are called the characteristic curves (lines) along which signals propagate. Note that another way of writing (1.2) is   [\phi(x,t)=F(t-x/c)+G(t+x/c)] (1.3)   Let us illustrate an application. 2 Branching of arteries References: Y C Fung: Biomechanics, Circulation. Springer 1997 M. J. Lighthill: Waves in Fluids, Cambridge 1978. Recall the governing equations for pressure and velocity [\frac{\partial^{2}p}{\partial t^{2}}=c^{2}\frac{\partial^{2}p}{\partial x^{2}}] (2.1) [\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\frac{\partial^{2}u}{\partial x^{2}}] (2.2) The two are related by the momentum equation [\frac{\partial u}{\partial t}=-\frac{\partial p}{\partial x}] (2.3) The general solutions are : [p=p_{+}(x-ct)+p_{-}(x+ct)] (2.4) [u=u_{+}(x-ct)+u_{-}(x+ct)] (2.5) Since [\frac{\partial p}{\partial x}=p_{+}^{\prime}+p_{-}^{\prime}] and [\frac{\partial u}{\partial t}=-cu_{+}^{\prime}+cu_{-}^{\prime}] where primes indicated ordinary differentiation with respect to the argument. Equation (2.3) can be satisfied if [p_{+}=ρcu_{+}] [p_{-}=-ρcu_{-}] (2.6) Denote the discharge by (Q=uA) then [ZQ_{\pm}=u_{\pm}A=\pm p_{\pm}] [Z=\pm\frac{p_{\pm}}{Q_{\pm}}=\frac{ρc}{A}] (2.7) where [Z=\frac{ρc}{A}] (2.8) is the ratio of pressure to flux rate and is call the impedance. It is the property of the tube. Now we examine the effects of branching; Referencing 1 to figure x, the parent tubes branches into two characterized by wave speeds (c_{1}) and (c_{2}) and impedances (Z_{1}) and (Z_{2}). An incident wave approaching the junction will cause reflection in the same tube   [p=p_{i}(t-x/c)+p_{r}(t+x/c)] (2.9) 6 String embedded in an elastic surrounding Reference: Graff: Waves in Elastic Solids If the lateral motion of the string is restrained by elastic springs along the entire length, the governing equation can be found from §1.1 by replacing the external pressure by the elastic restoring force -KVe per unit length,   [\rho\frac{\partial^{2}V}{\partial t^{2}}=Te\frac{\partial^{2}V}{\partial x^{2}}-KV] (6.9) [\frac{1}{c_{o}^{2}}\frac{\partial^{2}V}{\partial t^{2}}=\frac{\partial^{2}V}{\partial x^{2}}-\frac{K}{T}V] (6.10) [c_{a}=\sqrt{\frac{T}{\rho}}] (6.11) which can be written as where 6.1 Monochromatic waves For any linearized wave problem, if the range of the spatial coordinate x is (-∞,∞), and all coefficients are independent of x, t, then the first task is to examine the physics of sinusoidal wave train of the form: [V(x,t)=|A|cos(kx-\omega t-\phi)=\Re(Ae^{ikx-i\omega t})] (6.12)   where (A=|A|e^{i\phi_{A}}) is a complex number with magnitude |A| and phase angle (\phi_{A}). After examining the physical meaning of this special type of waves, it is possible to use the principle of superposition to construct more general solutions. It is customary to omit the symbol (\Re=) "the real part of" for the sake of brevity, i.e.,   [V(x,t)=Ae^{ikx-i\omega t}] (6.13) First of all a few definitions about sinusoidal waves in general. We shall call [\theta(x,t)=kx-\omega t] (6.14) the wave phase. Clearly the trigonometric function is periodic in phase with the period (2\pi). In the x, t plane, V has a constant value along a line of constant phase. In particular, (\theta=2n\pi (n=0,1,2,...)) correspond to the wave crests where (V=|A|) is the greatest. On the other hand, (\theta=(2n+1)\pi, (n=0,1,2,...)) correspond to the wave troughs where (V=-|A|) is the smallest. |A| is half of the separation between adjacent crests and troughs and is called the wave amplitude; we also call A the complex amplitude. Clearly (\frac{\partial\theta}{\partial x}) represents the number of phase lines per unit distance, i.e., the density of phase lines, at a given instant; it is called the wavenumber,   wavenumber = k = (\frac{\partial\theta}{\partial x}) (6.15) On the other hand (-\frac{\partial\theta}{\partial t}) represents the number of phase lines passing across a fixed x per unit time; it is called the wave frequency. wave frequency = (\omega = -\frac{\partial\theta}{\partial t}) (6.16) To stay with a particular line of constant phase, say a crest, one must have d(\theta) = kdx - (\omega)dt = 0 namely one must move at the phase velocity, (c_{e} = \frac{dx}{dt}|_{\theta=constant} = \frac{\omega}{k}) (6.17) Now back to the string. Substituting (6.18) in (6.9) we get ((-k^{2}-\frac{K}{T} + \frac{\omega^{2}}{c_{0}^{2}})V = 0) (6.18) or (\omega = c_{0}\sqrt{k^{2} + \frac{K}{T}}) (6.19) The phase speed is (c_{e} = \frac{\omega}{|k|} = c_{0}(1 + \frac{K}{Tk^{2}})^{\frac{1}{2}} > c_{0}) (6.20) Note that, due the stiffening by the lateral support, the phase speed is always greater than (c_{0}) and decreases monotonically with the wave number. Longer waves (small |k|) are faster, while shorter waves (larger |k|) are slower. As k increases, (c_{e}) approaches the finite limit (c_{0}) for the shortest waves. In general a sinusoidal wave whose phase velocity depends on the wavelength, i.e., (\omega) is a nonlinear function of k, is called a dispersive wave, and (6.19) or its equivalent (6.28) is called the dispersion relation.