一根绳中的波

Waves in a string can have many interesting properties. Some of the most notable wave occurrences on a string are related to standing waves. This means that the ends on either sides are fixed, and that both the peaks and troughs of the wave occur at the same points along the string.

Let us analyze some of the properties of standing waves in a string. To start off, for any standing wave with fixed endpoints, the length of the string will be $1/2$ $\lambda$, where $\lambda$ is the wavelength corresponding to the fundamental frequency of the string. Thus, for a string with a fixed length of $l$, the fundamental frequency, $f_1$, of its vibration can be mathematically represented by manipulating the relationship between wave frequency, velocity, and wavelength:

$$v=f\lambda$$ $$f=\dfrac{v}{\lambda}$$

Since the wavelength is equal to $2l$ in this case, we obtain the equation for the fundamental frequency of a fixed string:

$$f_1=\dfrac{v}{2l}$$

In the equation above, $v$ is the velocity with which the wave is traveling. The above equation can also be extrapolated upon such that if there are an integer number of half wavelengths between the two fixed ends of the string, a standing wave will also form.

Mathematically noted, that is, if:

$$l=\dfrac{n}{2}\lambda$$

The velocity of a wave in such a string can be calculated utilizing the mass ($m$), length ($l$), and tension ($T$) of the string:

$$v=\sqrt{\dfrac{T}{{m}{l}}}$$

Using this formula, we can develop a more comprehensive formula for the fundamental frequency of a string:

$$f_1=\dfrac{\sqrt{\dfrac{T}{{m}{l}}}}{2l}$$

This type of formula can have applications in determining frequencies to tune musical instruments at. It is a rather basic introduction into the field of mathematical harmonics.