10.7.2 String Resonance

In the set-up shown below, a vibrator is used to produce waves running along a string. When the vibrator is set to certain “special” frequencies, the string becomes “possessed” and does a vigorous standing wave. Is this black magic or what?

Let’s try to visualize what’s happening in the string. The vibrator produces a sinusoidal wave. This incident wave propagates rightward along the string, undergoes reflection when it reaches the right end, and returns as a reflected wave. When this reflected wave reaches the left end, it undergoes reflection and sets off again as a new incident wave. That’s right. The wave is eternally trapped between the two fixed ends of the string as it undergoes reflection repeatedly.

Since the vibrator keeps the wave train continuous, it means that that are numerous incident and reflected waves in the string at the same time. And all these waves superpose with one another. At most frequencies, the interference is mostly destructive, and the resultant wave on the string has negligible amplitude. At some special frequencies, however, constructive interference occurs, and the resultant wave is a standing wave whose amplitude is many times that of the vibrator.

The string is now in resonance[1]. And the frequencies at which resonance occurs are called the resonant frequencies.

The H2 syllabus does not require you to work out the resonant frequencies of a given string based on first principle (you should read the appendix if you’re interested). All the H2 syllabus requires is that you are able to work out the resonant frequencies of given string by drawing. Yes, we are going to solve by doodling. How fun!

The key idea is this: a standing wave on a string must have nodes at both (fixed) ends. After some thought, we realize that in order to have two nodes (N and N) at both ends, we can only go with NAN, then NANAN, then NANANAN and so on.

Basically, we start out with one single loop, and keep squeezing in another loop to progress to the next harmonic.

Since each loop corresponds to half-a-wavelength, we are basically fitting integer number of half-wavelengths onto the string. So

\displaystyle n\frac{{{{\lambda }_{n}}}}{2}=L,\text{ }n=1,2,3,...\text{ }

This means that the resonant wavelengths are

\displaystyle {{\lambda }_{n}}=\frac{{2L}}{n},\text{ }n=1,2,3,...\text{ }

Since v=f\lambda , the resonant frequencies are

\displaystyle {{f}_{n}}=n\frac{v}{{2L}},\text{ }n=1,2,3,...\text{ }

Some nomenclature:

A resonant frequency is also called a harmonic. So a given string is associated with many harmonics.

The lowest harmonic, f1, is also called the fundamental frequency. The next harmonic is called the 2nd harmonic, since its frequency is {{f}_{2}}=2{{f}_{1}}. Basically, a harmonic with frequency {{f}_{n}}=n{{f}_{1}} is called the nth harmonic.

Harmonics above the fundamental are also called overtones. For example, the 2nd harmonic is called the 1st overtone. Basically, the mth resonant frequency after the fundamental is called the mth overtone.

The first 5 harmonics of a string of length L are tabulated below. Notice how the wavelengths of the overtones are \displaystyle \frac{1}{2},\text{ }\frac{1}{3},\text{ }\frac{1}{4},\text{ }\frac{1}{5}... of the fundamental wavelength, and the frequencies of the overtones are 2, 3, 4, 5,… times the fundamental frequency.

Demonstration

String and Vibrator

Slinky and Hand

Concept Test

2060


[1] Because this resonance has many similarities with the resonance phenomenon in SHM, they are called by the same name. But bear in mind there are differences between them too.

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