IB Physics Topic C.4.2 Standing Wave Patterns in Systems Revision Notes (SL/HL)

IB Physics Topic C.4.2 Standing Wave Patterns in Systems Revision Notes (SL/HL) | RevisionDojo

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Standing Waves on Strings and in Pipes

Note

This section serves as a complement to C.4.1.

Boundary Conditions for Strings

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

Note

For a string with both ends fixed, like a guitar string, the boundary conditions are node-node.
This means standing waves can only form if the wave pattern fits these conditions.

Harmonics on Strings

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

First Harmonic

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.
The frequency of the first harmonic ($f_1$) is given by: $$
f_1 = \frac{v}{\lambda_1} = \frac{v}{2L}
$$

Example

A string of length $1.2 \, \text{m}$ with wave speed $140 \, \text{m s}^{-1}$ has a first-harmonic frequency of:

$$ f_1 = \frac{140}{2 \times 1.2} = 58.3 \, \text{Hz} $$

Higher Harmonics

Higher harmonics have more nodes and antinodes:

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Standing Waves on Strings and in Pipes

Note

This section serves as a complement to C.4.1.

Boundary Conditions for Strings

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

Note

For a string with both ends fixed, like a guitar string, the boundary conditions are node-node.
This means standing waves can only form if the wave pattern fits these conditions.

Harmonics on Strings

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

First Harmonic

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.
The frequency of the first harmonic ($f_1$) is given by: $$
f_1 = \frac{v}{\lambda_1} = \frac{v}{2L}
$$

Example

A string of length $1.2 \, \text{m}$ with wave speed $140 \, \text{m s}^{-1}$ has a first-harmonic frequency of:

$$ f_1 = \frac{140}{2 \times 1.2} = 58.3 \, \text{Hz} $$

Higher Harmonics

Higher harmonics have more nodes and antinodes:

Definition

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Topic A - Space, time and motion5 subtopics

Topic B - The particulate nature of matter5 subtopics

Topic C - Wave behaviour5 subtopics

Topic D - Fields4 subtopics

Topic E - Nuclear and quantum physics5 subtopics

IB Physics C.4.2 Standing Wave Patterns in Systems Notes

Topic A - Space, time and motion5 subtopics

Topic B - The particulate nature of matter5 subtopics

Topic C - Wave behaviour5 subtopics

Topic D - Fields4 subtopics

Topic E - Nuclear and quantum physics5 subtopics

Standing Waves on Strings and in Pipes

Boundary Conditions for Strings

Harmonics on Strings

First Harmonic

Higher Harmonics

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Standing Waves on Strings and in Pipes

Boundary Conditions for Strings

Harmonics on Strings

First Harmonic

Higher Harmonics

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