Question

the following figure from wikipedia.org shows a standing wave pattern:

1. which harmonic is this?
2. how many nodes and how many antinodes are in this figure?
3. if the distance from the left end of the figure to the right end is 75 cm, what is the wavelength of the standing wave?
4. suppose this was a wave form on a string held under tension. if the tension was decreased in the string without changing the driving frequency until a new standing wave pattern formed, would there be more of fewer loops between the two ends? explain the reasoning behind your answer.

Explanation:

Step1: Identify harmonic number

The number of half - wavelengths in a standing wave gives the harmonic number. Here, there are 5 half - wavelengths, so it is the 5th harmonic.

Step2: Count nodes and antinodes

Nodes are points of zero displacement. Antinodes are points of maximum displacement. In a standing wave with \(n\) half - wavelengths (\(n\)th harmonic), the number of nodes \(N=n + 1\) and the number of antinodes \(A=n\). For \(n = 5\), nodes \(N=6\) and antinodes \(A = 5\).

Step3: Calculate wavelength

For a standing wave on a string of length \(L\) with \(n\) half - wavelengths, \(L=\frac{n\lambda}{2}\). Given \(L = 75\space cm\) and \(n = 5\), we solve for \(\lambda\):

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Step4: Analyze effect of tension decrease

The speed of a wave on a string \(v=\sqrt{\frac{T}{\mu}}\) (where \(T\) is tension and \(\mu\) is linear mass density). The frequency \(f=\frac{v}{\lambda}\). A decrease in \(T\) decreases \(v\). If the driving frequency is changed until a new standing wave forms, using \(f=\frac{n}{2L}\sqrt{\frac{T}{\mu}}\), a lower \(T\) for the same \(f\) (assuming driving frequency is adjusted) would lead to a longer \(\lambda\) (\(\lambda=\frac{2L}{n}\) for a standing wave). Since \(\lambda\) is longer (more distance per half - wavelength), there would be fewer loops (because \(n=\frac{2L}{\lambda}\), smaller \(n\) for larger \(\lambda\)).

Answer:

1. 5th harmonic.
2. 6 nodes and 5 antinodes.
3. \(\lambda = 30\space cm\).
4. Fewer loops.