Spectral Analysis of Wave Motion Dr. Chih-Peng Yu

Dr. C. P. Yu2 Elastic wave propagation Unbounded solids –P-wave, S-wave Half space –Surface (Rayleigh) wave Double bounded media –Lamb waves Slender member

Dr. C. P. Yu Fall Waves in Slender members –longitudinal wave –flexural wave –torsional wave Waves by different approximate theories –Elementary member –Deep member

Dr. C. P. Yu Spring General derivation of waves in solids –P-wave, S-wave, Surface (Rayleigh) wave Modification due to bounded media –Lamb waves

Dr. C. P. Yu5 General Function of Space and Time At a specific point in space, the spectral relationship can be expressed as In general, at arbitrary position

Dr. C. P. Yu6 Imply (discrete) Fourier Transform pairs Or, in a simpler form as

Dr. C. P. Yu7 Spectral representation of time derivatives Assuming linear functions Or, for simplicity, discretecontinuous

Dr. C. P. Yu8 Derivatives of general order It is clear to see the advantage of using spectral approach –time derivatives replaced by algebraic expressions in Fourier coefficients => simpler

Dr. C. P. Yu9 Spectral representation of spatial derivatives Nothing special Or, for simplicity, discretecontinuous

Dr. C. P. Yu10 It is clear to see another advantage of using spectral approach –partial differential equation becomes ordinary differential equation in Frequency domain => solution form is solvable or at least easier to be solved –This is also true for using other transform integrals, such as Laplace Transform, Bessel-Laplace Transform

Dr. C. P. Yu11 Spectral relation Consider a general, linear, homogeneous differential equation for u(r,t) –with all coefficients independent of time –Assume one dimensional problem

Dr. C. P. Yu12 The spectral representation of the general differential equation becomes

Dr. C. P. Yu13 e i  n t is independent for all n. Thus the spectral representation results in n simultaneous equations as Or, in a general form as A j depend on frequency and are complex.

Dr. C. P. Yu14 When A j (x,  ) independent of position, the original partial differential equation has been transformed into n simultaneous ordinary linear differential equation. The solution form is e t, the transformed ODE becomes then The equation in the ( ) is called characteristic equation, which can be solved to give values for can be complex, so the solution form is in a form as with =  + ik

Dr. C. P. Yu15  is referred to as the attenuation factor of the wave motion. It represents the non-propagating and the attenuated components of the wave. k is the wave number. It represents the propagating parts of the wave. So, for a propagating component of the wave, the solution can be expressed as with =  ± ik ± stands for the traveling direction (to the right or left)

Dr. C. P. Yu16 Propagating speeds Consider the propagating component The time response is then in the form as j represents the number of characteristic constant

Dr. C. P. Yu17 The time response is then in the form as For each j, we can see the response corresponds to (infinite) sinusoids traveling with a speed of c j is called the phase speed corresponding to j

Dr. C. P. Yu18 So, for a specific j with only components traveling towards one direction (say -kx), we have the wave response expressed as Consider the interaction between two propagating wave components, the resultant response is thus

Dr. C. P. Yu19 The first sinusoid is the average response called carrier wave. It travels at the average speed of the two interacting wave components, c * =  * / k *. The second term represents the modulated effect between the interacting components. This is called group wave traveling at a speed

Dr. C. P. Yu20 It can be expected when there are many waves interacting together, the overall effect would be a carrier wave modulated by a group wave. In reality, the individual sinusoids is hard to be observed unless through an FFT scheme. The wave energy and varying amplitude of the wave envelope travel at group speed.

Dr. C. P. Yu21 Transfer function Let’s exam again the displacement function In a displacement – force relationship, transfer function is then the inverse of dynamic stiffness function.

Dr. C. P. Yu22 Summary of wave terms Angular frequency (rad/s) =  Cyclic frequency (Hz) = f =  / 2  Period (sec)= T = 1/f = 2  /  Wave number (1/length) = k = 2  / =  / c Wave length (length) = = 2  c /  = 2  / k Phase (rad) =  = (kx -  t) = Phase velocity (length/s) = c =  / k =  / 2  Group speed (length/s) = c g = d  / dk

Dr. C. P. Yu23 Specific terms Spectrum relation :  vs k Dispersion relation :  vs c Non-dispersion : phase velocity is constant for all frequency Evanescent wave : the attenuated non-propagating components of waves Carrier wave : main zero-crossing sinusoid waves Group wave : modulation of wave groups

Dr. C. P. Yu24 Simple wave examples Wave equation of the 1-D axial member –Non-dispersion Flexural wave in a beam –dispersion