• The speed of a wave travelling along a string with two fixed ends is given by:

• Where:
  • T?= tension in the string (N)
  • μ = mass per unit length of the string (kg m–1)

   

• At the fundamental frequency,?f0?of a stationary wave of length?L, the wavelength,?λ?=?2L
• Therefore, according to the wave equation, the speed of the stationary wave is:

v?=?fλ?= f × 2L

• Combining these two equations leads to the equation for the?fundamental frequency?(sometimes referred to as the first harmonic):

• Where:
  • f?= frequency (Hz)
  • L?= the length of the string (m)
  • T?= the tension in the string (N)
  • μ?= mass per unit length (kg m-1)

   

• Mass per unit length,?μ?can be calculated by dividing the mass of the string by the length of the string

Diagram showing the first three modes of vibration of a stretched string with corresponding frequencies

Part (a)

Step 1: Write the known quantities in S.I. units

• • Tension,?T?= 65 N
  • Mass,?m?= 3.2 g = 3.2 × 10?3?kg
  • Length of string,?L?= 90 cm = 0.90 m
    • Mass per unit length, ??= 3.56 × 10?3?kg m?1

Step 2: Write the equation for speed on a string and calculate

• • v?=?fλ?= f × 2L?AND ?
  • So,?v =

= 135

Step 3: Write the answer to the correct significant figures and include units

• • The speed of the wave on the string,?v?= 140 m s?1

Part (b)

Step 1: Write the known quantities in S.I. units

• • Tension,?T?= 65 N
  • Length of string under tension,?L?= 75 cm = 0.75 m
  • Mass per unit length,?μ =?3.56 × 10?3?kg m?1?(from part (a))

Step 2: Identify the length of one wavelength at the fundamental frequency,?f0

Step 3: Write the equation for fundamental frequency and calculate

= 90.1

Step 3: Write the answer to the correct significant figures and include units

• • The fundamental frequency, f0?= 90 Hz