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AQA A Level Physics復(fù)習(xí)筆記3.2.3 Harmonics

Category: A-level課程, 教材筆記, 福利干貨 Date: 2022年7月7日下午12:18

Harmonics

Stationary waves can have different wave patterns, known as?harmonics
- These depend on the?frequency?of the vibration and the?situation?in which they are created
These harmonics can be observed on a string with two fixed ends
As the frequency is increased, more harmonics begin to appear

Harmonics on a String

When a stationary wave, such as a vibrating string, is fixed at both ends, the simplest wave pattern is a single loop made up of two nodes and an antinode
- This is called the?first harmonic
The particular frequencies (i.e. resonant frequencies) of stationary waves formed depend on the length of the string?L?and the wave speed?v
The frequencies can be calculated from the string length and wave equation
For a string of length?L, the wavelength of the lowest harmonic is 2L
- This is because there is only one loop of the stationary wave, which is a half wavelength
Therefore, the frequency is equal to:

3.2.3-First-Harmonic-Equation

The second harmonic has?three nodes?and?two antinodes
The wavelength is?L?and the frequency is equal to:

3.2.3-Second-Harmonic-Equation

The third harmonic has?four nodes?and?three antinodes
The wavelength is 2L / 3 and the frequency is equal to:

3.2.3-Third-Harmonic-Equation

The nth harmonic has n antinodes and n + 1 nodes
The wavelengths and frequencies of the first three harmonics can be summarised as follows:

Fixed-end-wavelengths-and-harmonics-1-e1616147493267

Diagram showing the first three modes of vibration of a stretched string with corresponding frequencies

Worked Example

A stationary wave made from a string vibrating in the third harmonic has a frequency of 150 Hz.Calculate the frequency of the fifth harmonic

Step 1: Calculate the frequency of the first harmonic

f3?= 3 f1

f1?= f3?÷ 3 = 150 ÷ 3 = 50 Hz

Step 2: Calculate the frequency of the fifth harmonic

f5?= 5 f1

f5?= 5 × 50 = 250 Hz

Exam Tip

Make sure to match the correct wavelength with the harmonic asked for in the question:

The?first?harmonic (or n = 1) is the lowest frequency with half or quarter of a wavelength
The?second?harmonic (or n = 2) is a full wavelength

Frequency of the First Harmonic

The speed of a wave travelling along a string with two fixed ends is given by:

3.2.3-Velocity-Equation

Where:
- T?= tension in the string (N)
- μ = mass per unit length of the string (kg m–1)
For the first harmonic of a stationary wave of length L, the wavelength, λ = 2L
Therefore, according to the wave equation, the speed of the stationary wave is:

v = fλ = f × 2L

Combining these two equations leads to the frequency of the first harmonic:

3.2.3-Frequency-of-First-Harmonic

Where:
- f?= frequency (Hz)
- L?= the length of the string (m)

Worked Example

A guitar string of mass 3.2 g and length 90 cm is fixed onto a guitar. The string is tightened to a tension of 65 N between two bridges at a distance of 75 cm. Guitar-String-First-Harmonic-Question Calculate the frequency of the first harmonic produced when the string is plucked.