Honors Physics

Standing Waves on a String

Here we are looking at the first through the third harmonicof a standing wave on a string

Discussion of a Standing Waves

Standing Wave Description
Calculations Decsription Picture
Nodes Destructive Interferenc Here the incoming crest of the the wave is meeting with the trough of the reflection of the previous wave, thereby creating a Node. Anti- Nodes Constructive Interference Here the incoming crest of the the wave is meeting with the crest of the reflection of the previous wave, thereby creating an Anti-Node Note that in all Cases the Length of the String is Constant
The speed of a wave in a given medium is also Constant and is given by:
$V_{wave}=f\lambda$
In General Frequency is given by:
$f=\frac{V_{wave}}{\lambda}$
First Harmonic or Fundemental Frequency

Wavelength Calculations:

$\lambda _{1}=2L$
or
$L=\frac{1}{2}\lambda_{1}$

Lowest possible frequency. This is usually the frequency that is played on stringed instruments. We are seeing half a wave, with an anti-node in the center and nods at either end Frequency Calculations:

$f_{1}=\frac{V_{wave}}{2L}$

Second Harmonic Wavelength Calculations:
$\lambda _{2}=L$
or
$L=\lambda_{2}$
Here we have the second possible frequency and we are seeing a full wavelength. Notice that we now have two anti-nodes and we have added a node in the centre for a total of three nodes. Frequency Calculations:

$f_{2}=\frac{V_{wave}}{L}$

Third Harmonic Wavelength Calculations:
$\lambda_{3}=\frac{2L}{3}$ or

$L=\frac{3}{2}\lambda_{3}$

This is the third possilble frequency and we are seeing a wavelength and a half. There are now three anti-nodes and we have nodes at either end plus two in the middle Frequency Calculations:

$f_{3}=\frac{3V_{wave}}{2L}$

General Equations for Frequency $f_{n}=\frac{nV_{wave}}{2L}$
or $f_{n}=nf_{1}$
n = 1, 2, 3, 4, etc this is the harmonic number.
General Equations for Wavelength $\lambda_{n}=\frac{2L}{n}$ n = 1, 2, 3, 4, etc this is the harmonic number.