Superposition of Waves  |  Standing Wave Resonance

When two waves with the same frequency and wavelength are traveling in opposite directions, they interfere in such a way that a characteristic Standing Wave Pattern shows up.

In the above animation the two top waves are added together through the principle of superposition (just added) and the resultant is the bottom wave. Notice in the resulting standing wave that there locations that are never displaced, these are the nodes of the standing wave. In between the nodes are locations that undergo large displacements, in fact much larger than when there was a single wave. These are called anti-nodes. The wave itself does not appear to be traveling, hence the term standing.

If the waves fit perfectly into the system that contains them, they can create a resonance. To determine what wavelengths match this system size condition you need to consider the Boundary Conditions (BC). Consider a wave sent down a taught string that reflects off the fixed end. Such a system is great for generating standing waves because the reflections at the end points ensure there are waves traveling in opposite directions, a requirement for standing waves. If the endpoints are fixed, the displacement must be zero there. So, conceptually, the longest wavelength that would fit in that system, is if the wavelength is twice the length of the string.

The fundamental mode shown above is the one with the longest wavelength. We represent the standing wave in this form with two lines weaving back and forth, but it's important to realize this is just a static represnetation of the actual wave pattern, which is animated below for the 4th harmonic. The fundamental mode is the first harmonic, folled by the next longest wavelength that match the boundary counditions, which would be the second harmonic.

If the length of the system is a value $L$, than the wavelengths that match the boundary conditions are equal to $\frac{2L}{m}$, where $m$ is any integer $m=1, 2, 3, ...$. These are the equations for systems where the ends are both fixed, or both free, so long as they are Symmetric boundary conditions.

If the ends are the opposite of each other the system is considered to be Anti-symmetric. There is then different series that determines the resonant modes. A summary of the two types of boundary conditions is below where $v$ is the speed of the wave in the medium and $L$ is the length of the system.

Symmetric Boundary Conditions:

wavelength    $\lambda_m = \frac{2L}{m}$,    frequency    $f_m=m \frac{v}{2L}=mf_1$,    where m=1,2,3,...

$\Delta f=f_{m+1}-f_m=f_1$

Antisymmetric Boundary Conditions:

wavelength    $\lambda_m = \frac{4L}{m}$,    frequency    $f_m=m \frac{v}{4L}=mf_1$,    where m=1,3,5,...

$\Delta f=f_{m+2}-f_m=2f_1$

Some important features of these solutions are that for the symmetric BC, the difference in the frequency between adjacent modes is the fundamental frequency. For the anti-symmetric BC the difference is twice the fundamental. Also notice that in the anti-symmetric case you only use odd integers, which means the harmonic mode is not the same as the m-value. For example the 2nd resonance mode has an m-value of 3. In the symmetric case the harmonic and the m-value are the same. 

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OpenStax Section 17.5  |  Sound Interference and Resonance: Standing Waves in Air Columns

1.  A $1.5 \, m$ long string that is pinned down to a wall at both ends vibrates with a standing wave pattern that has $3$ antinodes. 

      (a) Which harmonic is this? 

      (b)  What is the wavelength of this wave?

2.  A soda bottle of length $26 \, cm$ can be modeled as an open-closed tube.  While blowing air across the opening of the bottle, it is determined that the standing wave pattern inside has $3 \, antinodes. 

      (a)  Which harmonic is this? 

      (b)  What is the wavelength of this wave?

3.  Human ear canal’s are roughly $2.4 \, cm$ long and can be modeled as an open-closed tube. 

      (a)  What is the fundamental frequency and wavelength of an ear canal? 

      (b)  What is the frequency and wavelength of an ear canal’s $2^{nd}$ harmonic?

4.  An open-open tube of air supports standing waves at frequencies of $816 \, Hz$ and $1224 \, Hz$, and at no frequencies between these two.  Which of the following are also resonant frequencies of the same tube?

     (a)  $204 \, Hz$

     (b)  $408 \, Hz$

     (c)  $612 \, Hz$

     (d)  $1428 \, Hz$

     (e)  $1632 \, Hz$

Click HERE for solutions.

Short foundation building questions, often used as clicker questions, can be found in the clicker questions repository for this subject.

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• Openstax has a few practice problems for standing wave resonance, Link

For additional practice problems and worked examples, visit the link below. If you've found example problems that you've used please help us out and submit them to the student contributed content section.

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