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					                   Wave Equations and Functions
Transverse Waves          Summarized      Longitudinal Waves
(eg. stretched string)                    (eg. vibrating rod)

          2 y 1 2 y                                            2 1  2
                                                                   
          x 2 v 2 t 2                                         x 2 v 2 t 2
                    T                                                     Y
               v                     wave speed                  v
                                                                         
                        normal mode                                           normal mode
                        solutions                                             solutions
                        (standing                                             (standing
                        waves)                                                waves)


      y  x, t   f  x cos t          general       x, t   f x  cost
                                          solution

 Boundary           both ends fixed                   Boundary                one end fixed
 Conditions:                                          Conditions:             one end free



                       n x                                               n x
 y  x, t   An sin          cos  n t               x, t   An sin             cos nt
                        v                                                     v
                                          where
                        1                                                 1

        n      T     2                              n  1   Y 
                                                                        2n  11
                                                                          2
   n             n1                     n           2
                                                                  
         L                                               L        
                                   Also (in Hz)
                               1                                                1

   fn 
        
          
            n T 
                
                               2
                                                      fn   
                                                             n  1   Y 
                                                                  2
                                                                       
                                                                                2     1

        2 2 L   
                                                              2L   
                                                                    
Week 8 Lecture 1: Problems 47, 48, F2000Q3a
 Travelling wave solutions to the wave
               equation
Recall the Plan…..
Earlier we derived the wave equation for both longitudinal
and transverse waves. There are lots of variations on this
but here is an example:

                   y  y
                    2            2
                       
                  x 2
                         T t 2
      Remember that this describes the general
     behaviour for a wave on a string. However, if
    we specifically want to know where (in y) some
     point x is along a string at a given time t, we
      have to get a solution to the wave equation


    We said there were two types of solutions

        Normal modes                 Travelling waves
              aka                           aka
      (stationary waves)             Progressive waves
       (standing waves)
  We have been working                Now need to look
 on these for the last few             at this type of
                                                     2
          lectures                        solution
Traveling Wave (Progressive Wave) Solutions to the
                   Wave Equation
                     (French pg 202-209)
 So far we have considered only the stationary wave
  (normal mode) solution to the wave equation.
 Now we will look at the traveling wave solution.
 In fact – a stationary wave is just a traveling wave
  which gets reflected from one end of a medium.
  The incident and reflected waves superimpose, and
  if conditions are right (i.e. appropriate frequencies
  and wavelengths then a standing wave ( stationary
  wave or normal mode) develops.

Traveling Wave Solution to the Wave Equation

              y  y0 sinkx  t          For a
                                           transverse wave

Note:
1. There is no phase constant in this equation as
   written (but you could add one).
2. k is not the spring constant but the wave number
   (rad/m).
3. Since a stationary vibration is actually created from
   a travelling wave, you should be able to show that
   these 2 solutions are the same and you can (see
   French pg 203-204).                                3
What exactly is the Wave Number?

“k” in a y vs x plot is the same as “” in a y vs t plot!

Explanation:
 If you plot the y displacement of a single point x on a
   string as a function of time you get:

        y
                                   T = time for 1 complete
                                   oscillation on a y vs t plot
                               t
                                              and
                   T
                                         = angular
                                        frequency
                                           = 2/T (rad/s)


 Now, if we look at a single snapshot in time you are
  essentially plotting y as a function of x:
    y                           = wavelength = distance for
                               1 complete oscillation on a y
                               vs x plot
                          x

               
                              k = wave number = 2/ (rad/m)
                                                         4
    Note here: on pg 214 French the wave number is
    defined as k = 1/, but then French corrects for the
    missing 2 by multiplying it through later. We will
    use k = 2/ but be aware of the little catch (note:
    most other books use 2/)

                  2                           2
    so                      and         k
                  T                            


traveling wave solution is    y  y0 sinkx  t 


subbing in  and k gives
                                        x t 
                        y  y0 sin 2    
                                            
                                        T 

                what fraction of a             what fraction of
                wavelength is this position?   a period has
                                               passed?




                                                           5
Wave Speed relationships
 The wave speed is the speed at which the wave
   travels along the medium (in m/s).



     put the traveling wave solution back into the wave
      equation

                            into
                                       2 y     2 y
  y  y0 sin kx  t                       v2 2
                                       t 2     x
  gives

   y 0 2 sinkx  t    v 2 k 2 y 0 sinkx  t 


                                              
       v k
          2      2 2       or          v
                                               k
                                   units are  = rad/s
                                   and k = rad/m

                    So there IS a relationship between 
                    and k – they are related by the wave
                    velocity                           6
How everything fits in…


                             22
                 v , k   , 
                   k           T

                  2    
                              f
                                                  f is
              v    
                  T  2  T                frequency
                                              in Hz



 Keep handy -                     
 will use this     v                  f
 over and over            k       T




                                                 7
One more thing to clean up…
    Recall that earlier we stated that               We stated this when
                                            T        we wrote down the
         Wave speed =            v                  wave equation but
                                                    we didn’t derive it

                                                         Mass/length

 Can we derive this? (of course, or I wouldn’t be asking..)
   Consider the following travelling wave pulse on a rope:
If the pulse is small enough then we
can assume the tension T in the rope
doesn’t change, and we can consider                        Speed v
the wave a circular arc.        s
                           
                                  R
Sum forces (tension):

Horizontal:     SFh=0
                                                           s
Vertical (also the net radial force):
                                                    ½
                                                T                   T
SFv=2Tsin½  2T ½ T                                         R



Mass of segment         m = s=R

Apply Newton’s 2nd law (radial)
               v2
    T    R
               R
                                        T
                                v                                       8
                   Voilla!!
                                        
            Remember this slide??

 General Wave Equation (1D):
                                           2 y     2 y
                                                 v2 2
                                           t 2     x
                                  where v is the wave speed

Can write this for all sorts of situations!
 stretched string      transverse displacement

  (transverse             2 y T 2 y               T      tension
                                              v
   oscillations):         t 2  x 2                     mass/
                                                           length

                        longitudinal displacement
 longitudinal waves       Y
                            2         2
                                                    Y
                                                           Young’s
                                              v          modulus
  in a solid:             t 2  x 2                     density




 transverse (shear)      2 y Y 2 y               n      shear
                                              v          modulus
                          t 2
                                  x 2             
  waves in a solid:

                           2 B  2               B      bulk
 longitudinal waves                          v          modulus
                          t 2
                                  x 2             
  in a gas or liquid:
                                          The velocity terms are
                                          here. We derived the
                                          top one and could9
                                          derive the others if we
                                          wanted…. Naw…
           Particle Velocity and Pressure
 So far the only parameter we have looked at is
  particle displacement
             y  y0 sinkx  t       (transverse wave)
               0 sin kx  t      (longitudinal wave)

      However, other parameters such as particle velocity
       and pressure (stress) are also important.


 Considering longitudinal waves in a thin rod (so Y is
  the appropriate modulus) (you could also do for transverse)

 particle displacement:             0 sin kx  t 

 particle velocity:                
                             
                                       0 coskx  t 
  (NOT same as wave speed)
                                    t
                                
 pressure:             strain      0 k coskx  t 
                                x
                                  
                       stress  Y      Y0 k coskx  t 
                                  x
                     pressure  P  compressiv stress
                                                e
                                Y0 k coskx  t 

                                                              10
           OR if you have a gas modulus B:
                             P   B0 k coskx  t 
                                                                     Note this is
                                                                     analogous to
                      Specific Acoustic Impedance                    refractive index in
                                (Courseware pg 72-73)                E/M waves
       Very important property of materials that are transmitting
        sound waves (but “impedance” generally is applicable to
        any wave in any medium).
                                  pressure       P
           Defined as                            z
                               particle velocity 
                                                 

  (Effectively a resistance term – high zs mean that it takes a large pressure to
        induce a given particle velocity  high “stiffness”).
                                                                        (since Y is a
                              Y0 k coskx  t  Yk
plug in equations
for pressure and                                                      stiffness this
                       z                         
                              0 coskx  t                        fits!)



   Now we really need z in a form that we can associate
     with material properties, so recall some previous
     relationships…                 then z 
                                               Y
                      since v 
                                k               v
             v
                  Y                        v 2
   also
                  
                       or   Y  v  z 
                                   2
                                                  v
                                            v

                                 z  v               units  kg  m 
                                                               3  
                                                                          kg
                                                                         2
                                                               m  s  m s

   So materials with higher zs (stiffer) have higher wave
    velocities.
                            P                         So particle speed 
         also note             z  v
                            
                                                 is inversely proportional
                                                      to wave speed v
                                                                         11

               See table of z and v values 2 pages along!
             Traveling Waves: Summary Page
  Wave Velocities
                               T                                              Y
   transverse           v                   longitudinal oscillation v 
   oscillation (string)                     (solid rod)                       

   transverse                  n             longitudinal oscillation          B
                       v                                             v
   oscillation (solid)                      (gas or liquid)                   

                                       2                  2
       wave number  k                             
                                                          T    All of these
                                           
                        v                          f
                                                                equations appear
                                                                on the formula
                                   k        T                   sheet of my exams
Longitudinal Wave: (solid)
 particle displacement:   0 sinkx  t                               But
                                                                           Not these
 particle velocity:                          0 coskx  t 
                                            
                                                
 pressure:                        P  Y           Y0k coskx  t 
                                                x

Specific Acoustic Impedance:
                               P Yk Y
                    z  v       
                                 v
                               
Intensity:                                      2                  Note we haven’t
               I  zm 2  m  2 
                   1        1P     watts
                                                                  done intensity yet –
                   2        2 z  m 
                                                                 next lecture
                             I
     decibels dB  10 log10 12 dB                                            12
                           10
Calculated from
long. velocity




        13

				
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