"The English may not like music, but they absolutely love the noise it makes"
Thomas Beecham

Principle of Superposition

"When two or more waves of the same nature travel past a point at the same time, the displacement at that point is the sum of the instantaneous displacements of the individual waves"
 
• Consider a traveling wave on a string moving from left to right. If the far right end of the string is fixed the wave will be reflected (moving right to left). The incident and reflected waves will undergo "superposition", interfering constructively as indicated below.

  

• Fixing the far left of the string also will result in a standing wave pattern.  The whole string will look as if it is vibrating in Simple Harmonic Motion.  Points on the string which remain stationary are called nodes, maximum displacement positions are called antinodes.

  

• For a string of fixed length L, the frequency of the standing wave is given by  .  The wavelength () is given by 2L/n, where n can take on any positive integer value.  In the above diagram the upper standing wave has n=1 and the lower wave has n=2.
• The general expression for the frequency is given by

                   (n = 1,2,3,4....)

  f₁ is called the fundamental frequency or first harmonic.

  f₂  f₃  f₄  ... are called the 2nd, 3rd, 4th  ... harmonics.

  The standing wave patterns for n=1-6 are shown in the diagram at right.

• These standing wave patterns are exactly the modes of vibration of the strings of stringed musical instruments, e.g. guitar, violin , cello . When such a string is set vibrating the sound produced is a combination of the fundamental and all harmonics.

  

• The velocity v, of a wave on a string is given by

  
  

  where T is the tension in the string and (m/L) is the mass per unit length of the string (kg/m). Tuning of a stringed instrument is achieved by adjusting the tension, which in turn changes the wave velocity and therefore the frequency.

• A vibrating column of air (constrained in a tube) will produce similar standing wave patterns.  By adjusting the length of the column different fundamental frequencies are obtained.  This is the operating principle of woodwind and brass instruments.
  

• Resonance:
  Normally the vibrations in the stretched string or column of air die away naturally due to various energy losses, e.g. sound, heat, air resistance. However, if it is possible to apply a periodic force which inputs enough energy to overcome these losses the vibrations will continue. If the energy input exceeds the losses the amplitude of vibrations will increase and it is possible the string could break. This phenomenon is known as resonance.
  Resonance occurs when the "driving frequency" is equal to the "natural frequency" of the oscillating system.

   Supplementary Problems 

"Following the dispute with the domestic servants' union at Buckingham Palace today, the Queen, a radiant figure in a white silk gown and crimson robe, swept down the main staircase and through the hall. She then dusted the cloakroom and vacuumed the lounge."

Ronnie Barker


 

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu