Standing Waves

1. A stretched wire 1.0 m long has a fundamental frequency of 300 Hz.
• (A) What is the speed of the waves in the wire ?
• (B) What are the frequencies of the first three overtones ?


• (A) = 2l = 2.0 m

  v = f = 300 x 2 = 600 m/s

  
  
• (B) f_n = vn/2l = 600n/2 = 300n

  f₁ = 300 Hz - Fundamental

  f₂ = 600 Hz - 1^st overtone

  f₃ = 900 Hz - 2^nd overtone

  f₄ = 1200 Hz - 3^rd overtone

  
  
  
2. The vibrating part of the E string of a violin is 330 mm long and has a fundamental frequency of 659 Hz. What is its fundamental frequency when the string is pressed against the fingerboard at a point 60 mm from its end ? What are the first and second overtones of the string under these circumstances ?
   

   v = 2l f₁ = 2 x 0.33 x 659 = 435 m/s

   f_n = vn/2l = 435n/(2 x 0.27) = 805 n

   1^st overtone, f₂ = 805 x 2 = 1610 Hz

   2^nd overtone, f₃ = 805 x 3 = 2415 Hz

   
   
   
   
3. The vibrating part of the G string of a certain violin is 330 mm long and has a fundamental frequency of 196 Hz when under a tension of 50 N.
• (A) Find the linear density of the string.
• (B) Where should the string be pressed in order for it to vibrate at 220 Hz ?


• (A) v = 2l f₁ = 2 x 0.33 x 196 = 129.36 m/s

  v = (T/(m/l))^½

  Therefore, m/l = T/v² = 50/(129.36)² = 2.99 x 10^-3 kg/m

  
  
• (B) v = 2l f₁

  Therefore, 2l = v/f₁ = 129.36/220 = 0.588

  l = 294 mm

  String should be pressed 36 mm from its end
  




4. The vibrating part of a violin string whose linear density is 4.7 g/m is 30 cm long. What tension should the string be under if its fundamental frequency is to be 440 Hz, the musical note A ?
   
   

   v = (T/(m/l))^½

   Therefore, T = v²(m/l)

   But, v = 2l f₁

   So that T = 4l²f₁(m/l) = 4 x 0.3 x 0.3 x 440 x 4.7 x 10^-3 = 0.744 N