17. Modelling – Vibrating String, Wave Equation

Try these first

1. 1.

   What is the wave equation for a vibrating string?

   Hint

   Think of how wave motion in strings is mathematically represented.

2. 2.

   List two assumptions made when modeling a vibrating string.

   Hint

   Consider the physical constraints of the vibrating string.

3. 3.

   What is the general form of the wave equation for a vibrating string?

   • $$ \\frac{\\partial^2 u}{\\partial t^2} = c^2 \\frac{\\partial^2 u}{\\partial x^2} $$
   • $$ \\frac{\\partial u}{\\partial t} = c \\frac{\\partial^2 u}{\\partial x^2} $$
   • $$ \\frac{\\partial^2 u}{\\partial x^2} = c^2 \\frac{\\partial^2 u}{\\partial t^2} $$

   Hint

   Recall the standard mathematical formulation for wave motion.

4. 4.

   True or False: The wave speed is inversely proportional to the tension in the string.

   • True
   • False

   Hint

   Consider how increasing tension affects the wave speed.

5. 5.

   Consider a string fixed at both ends, initially at rest. If displaced at the center and released, derive the equations governing its motion over time.

   Hint

   What boundary conditions apply at the fixed ends?

6. 6.

   Suppose the string's tension is modified. Explain how this change impacts the wave speed and the normal modes.

   Hint

   Think about how frequency relates to wave speed and string length.