20.9. Orthogonality of Sine Functions

Try these first

1. 1.

   What is orthogonality in the context of sine functions?

   Hint

   Think about how perpendicular functions behave.

2. 2.

   What is the value of the integral of sin(mx) * sin(mx) over [0,a]?

   Hint

   Refer to the self-product of sine functions.

3. 3.

   What does the integral of sin(mx) * sin(nx) equal over [0,a] if m ≠ n?

   • 0
   • a/2
   • a

   Hint

   Think about the property of orthogonal functions.

4. 4.

   True or False: The integral of sin(mx) * sin(mx) over [0,a] equals a.

   Hint

   Review how squares of functions behave under integration.

5. 5.

   Given the sine functions sin(2πx) and sin(3πx), calculate their integral over the interval [0,1] and explain the significance.

   Hint

   Remember to use the integral properties of sine functions.

6. 6.

   Demonstrate how the orthogonality of sine functions aids in constructing a solution for a vibrating rectangular membrane.

   Hint

   Consider how modes interact during vibration and their independence.