C.4.2 Standing wave patterns in systems

Standing Waves on Strings and in Pipes

Standing waves are a fascinating phenomenon that occur when two identical waves traveling in opposite directions interfere with each other.

Standing Waves on Strings

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:
   • Fixed Ends: The displacement at the end is always zero, creating a node.
   • 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)

The first harmonic is the simplest standing wave pattern, with:

• Two nodes (one at each end) and one antinode in the middle.
• A wavelength of ${\lambda }_1=2L$, where $L$ is the length of the string.

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

Higher harmonics have more nodes and antinodes:

• Second Harmonic: Two antinodes and three nodes, with ${\lambda }_2=L$
• Third Harmonic: Three antinodes and four nodes, with ${\lambda }_3=\frac{2L}{3}$

In general, the wavelength of the $n$th harmonic is:

$${\lambda }_n=\frac{2L}{n}$$

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

1. Open Ends: The displacement is maximum, creating an antinode.
2. Closed Ends: The displacement is zero, creating a node.

Harmonics in Pipes

The harmonic patterns in pipes depend on whether the pipe is open at both ends, closed at both ends, or open at one end and closed at the other.

Pipes Open at Both Ends

For a pipe open at both ends, the boundary conditions are antinode-antinode.

The harmonics are similar to those on a string with both ends free:

• First Harmonic: Two antinodes and one node in the middle, with ${\lambda }_1=2L$
• Second Harmonic: Three antinodes and two nodes, with ${\lambda }_2=L$.
• Third Harmonic: Four antinodes and three nodes, with ${\lambda }_3=\frac{2L}{3}$.

The wavelength and frequency relationships are the same as for strings:

$${\lambda }_n=\frac{2L}{n}\text{and}{f}_n=n\cdot f_1$$

Pipes Closed at One End

For a pipe closed at one end, the boundary conditions are node-antinode.

This creates a unique pattern:

• First Harmonic: One node and one antinode, with ${\lambda }_1=4L$.
• Third Harmonic: Two nodes and two antinodes, with ${\lambda }_3=\frac{4L}{3}$.
• Fifth Harmonic: Three nodes and three antinodes, with ${\lambda }_5=\frac{4L}{5}$.

The frequency of the $n$th harmonic (where $n$ is odd) is:

$$f_n=n\cdot f_1$$

Comparing Strings and Pipes

The table below summarizes the wavelength and harmonic relationships for strings and pipes:

By mastering the principles of standing waves on strings and in pipes, you gain insight into the physics behind musical instruments and many other real-world phenomena.