C.4.2 Standing wave patterns in systems Notes

Standing Waves on Strings and in Pipes

Boundary Conditions for Strings

1. When a wave travels along a string, the behavior at the ends of the string determines the possible standing wave patterns.
2. There are two primary boundary conditions:
   1. Fixed Ends: The displacement at the end is always zero, creating a node.
   2. Free Ends: The displacement at the end is maximum, creating an antinode.

Harmonics on Strings

1. Harmonics are the specific standing wave patterns that can form on a string.
2. Each harmonic has a distinct wavelength and frequency, determined by the length of the string and the boundary conditions.

First Harmonic (Fundamental Frequency)

1. The first harmonic is the simplest standing wave pattern, with:
   1. Two nodes (one at each end) and one antinode in the middle.
   2. A wavelength of $\lambda_1 = 2L$, where $L$ is the length of the string.
2. The frequency of the first harmonic is called the fundamental frequency ($f_1$), given by: $$
   f_1 = \frac{v}{\lambda_1} = \frac{v}{2L}
   $$

Higher Harmonics

1. Higher harmonics have more nodes and antinodes:
   1. Second Harmonic: Two antinodes and three nodes, with $\lambda_2 = L$
   2. Third Harmonic: Three antinodes and four nodes, with $\lambda_3 = \frac{2L}{3}$
2. In general, the wavelength of the $n$th harmonic is: $$
   \lambda_n = \frac{2L}{n}
   $$
3. The frequency of the $n$th harmonic is: $$
   f_n = n \cdot f_1
   $$

Standing Waves in Pipes

1. Standing waves can also form in pipes, which are used in wind instruments like flutes and clarinets.
2. The boundary conditions for pipes depend on whether the ends are open or closed.

Boundary Conditions for Pipes