Lecture 14: Speed of Waves

Wave speed

• Case 1: Wave on an infinite rope.
  
  
  • Rope is infinitely thin and under tension. What describes such rope ?
    • Tension F : units of tension are N = kg m / s^2
    • mass of 1m of the rope - i.e linear density μ . Units of linear density are kg/m
  • The only speed one can construct is v ~ ( F/μ )^½
  • Can we make something of the units of pure length ? No
  • Can we make something of the units of pure time ? No
  • Hence, the wavelength λ and the frequency f cannot be determined solely by the properties of the rope, only the speed can
  
  
• Case 2: Wave on a rope of the fixed length L
  • Rope is infinitely thin and under tension. What describes such rope ?
    • Tension F : units of force are N = kg m / s^2
    • Length L : units of length are m
    • Total mass M of the rope: Units of M are kg
  • The only velocity one can construct is again v ~ ( F L / M )^½=( F/μ )^½
  • Can we make something of the units of pure length ? Yes ! λ ~ L ?
  • Can we make something of the units of pure time ? Yes ! f ~ ( F/(M L) )^½ ?
  • Hence, on the fixed length rope, not only velocity is determined by the rope properties, but, since we have the mass and the length separately, the rope also has natural wavelength λ and frequency f associated with its properties
  
  
• Speed is a local phenomena (make small perturbation, look at it over short period of time), so should not feel what the ends of the rope do. Both infinite length and fixed length result should coincide. Indeed v = ( F/μ )^½ is the exact result.

Energy in a wave

Intensity of a spherical Wave

Intensity of a wave in a volume is the energy transported by the wave per second (i.e power) through an area of 1 m2 (i.e power per unit area) Intensity of the spherical wave is I = P /(4 π r^2 ) , where P is the power of the central source. Between two radii intensity drops according to inverse square law I (r1) / I (r2) = ( r2 /r1 )2