Sinusoidal Waves

1
Sinusoidal Waves

• y(x,t)ym sin( kx- ? t) describes a wave moving
  right at constant speed v ?/k
• ? 2?f 2?/T k 2?/?
• v ?/k f ? ?/T
• wave speed one wavelength per period
• y(x,t)ym sin( kx ?t) is a wave moving left

2
Transverse Velocity

• y(x,t)ym sin( kx- ?t)
• uy(x,t) ? y/? t partial derivative with
  respect to t
• derivative of y with respect to t keeping x
  fixed
• -ym ? cos( kx- ?t)
• maximum transverse speed is ym ?
• A more general form is y(x,t)ym sin( kx- ?t-?)
• (kx- ?t-?) is the phase of the wave
• two waves with the same phase or phases differing
  by 2?n are said to be in phase

3
Phase and Phase Constant

• y(x,t)ym sin( kx- ?t-?) ym sin k(x -?/k) - ?t
  ym sin kx- ?(t?/?)

4
Wave speed of a stretched string

• Actual value of v ?/k is determined by the
  medium
• as wave passes, the particles in the medium
  oscillate
• medium has both inertia (KE) and elasticity (PE)
• dimensional argument v length/time LT-1
• inertia is the mass of an element ?mass/length
  ML-1
• tension F is the elastic character (a force)
  MLT-2
• how can we combine tension and mass density to
  get units of speed?

5
Wave speed of a stretched string

• v C (F/?)1/2 (MLT-2/ML-1)1/2 L/T
• detailed calculation using 2nd law yields
  C1 v (F/?)1/2
• speed depends only on characteristics of string
• independent of the frequency of the wave f due
  to source that produced it
• once f is determined by the generator, then
• ? v/f vT

6
(a) 2,3,1
(b) 3,(1,2)
7
Summary

• ? 2?f 2?/T k 2?/?
• v ?/k f ? ?/T
• wave speed one wavelength per period
• y(x,t)ym sin( kx- ?t-?) describes a wave moving
  right at constant speed v ? /k
• y(x,t)ym sin( kx ?t-?) is a wave moving left
• v (F/?)1/2
• F tension ? mass per unit length

8
Waves
F
F
F/2
F/2
v(F/?)1/2
9
Wave Equation

• How are derivatives of y(x,t) with respect to
  both x and t related gt wave equation
• length of segment is ?x and its mass is m? ?x
• net force in vertical direction is Fsin?2 - Fsin
  ?1
• but sin? ?tan ? when ? is small
• net vertical force on segment is F(tan?2 - tan
  ?1 )
• but slope S of string is Stan ? ?y/?x
• net force is F(S2 - S1) F ?S ma ??x?2y/?t2

10
Wave Equation

• F ?S ??x?2y/?t2 force
  ma
• ?S/?x (?/F)?2y/?t2
• as ?x gt 0, ?S/?x ?S/ ?x ?/ ?x (?y/ ?x)
  ?2y/?x2
• any function yf(x-vt) or yg(xvt) satisfies
  this equation with
• v (F/?)1/2
• y(x,t) A sin(kx-?t) is a harmonic wave

11
Energy and Power

• it takes energy to set up a wave on a stretched
  string y(x,t)ym sin( kx- ?t)
• the wave transports the energy both as kinetic
  energy and elastic potential energy
• an element of length dx of the string has mass dm
  ?dx
• this element (at some pt x) moves up and down
  with varying velocity u dy/dt (keep x fixed!)
• this element has kinetic energy dK(1/2)(dm)u2
• u is maximum as element moves through y0
• u is zero when yym

12
Energy and Power

• y(x,t) ym sin( kx- ?t)
• uydy/dt -ym ? cos( kx- ?t) (keep x fixed!)
• dK(1/2)dm uy2 (1/2) ?dx ?2 ym2cos2(kx- ?t)
• kinetic energy of element dx
• potential energy of a segment is work done in
  stretching string and depends on the slope dy/dx
• when yA the element has its normal length dx
• when y0, the slope dy/dx is largest and the
  stretching is maximum
• dU F( dl -dx) force times change in
  length
• both KE and PE are maximum when y0

13
Potential Energy

• Length
• hence dl-dx (1/2) (dy/dx)2 dx
• dU (1/2) F (dy/dx)2 dx potential energy of
  element dx
• y(x,t) ym sin( kx- ?t)
• dy/dx ym k cos(kx - ? t) keeping t
  fixed!
• Since F?v2 ??2/k2 we find
• dU(1/2) ?dx ?2ym 2cos2(kx- ? t)
• dK(1/2) ?dx ? 2ym 2cos2(kx- ? t)
• dE ??2ym 2cos2(kx- ?t) dx
• average of cos2 over one period is 1/2
• dEav (1/2) ? ? 2ym 2 dx

14
cos(x)
0.
cos2(x)
.5