7.5 Stationary Waves on a Stretched String

7.5 Stationary Waves on a Stretched String

Consider a string of length $\ell$ which is kept stretched by clamping its ends so that the tension in the string is $F$.

(a) String Plucked at its Middle Point

If the string is plucked at its middle point, two transverse waves will originate from this point. One of them will move towards the left end of the string and the other towards the right end. When these waves reach the two clamped ends, they are reflected back thus giving rise to stationary waves. As the two ends of the string are clamped, no motion will take place there. So, nodes will be formed at the two ends and one mode of vibration of the string will be as shown in Fig. 7.7(a) with the two ends as nodes with one antinode in between. Visually, the string seems to vibrate in one loop. As the distance between two consecutive nodes is one half of the wavelength of the waves set up in the string, so in this mode of vibration, the length $\ell$ of the string is:

where $\lambda_{1}$ is the wavelength of the waves set up in this mode. The speed $v$ of the waves in the string depends upon the tension $F$ of the string and $m$, the mass per unit length of the string. It is independent of the point from where string is plucked to generate wave. It is given by:

Knowing the speed $v$ and wavelength $\lambda_{1}$, the frequency $f_{1}$ of the waves is:

Substituting the value of $v$:

Thus, in the first mode of vibration shown in Fig. 7.7 (a), waves of frequency $f_{1}$ only will be set up in the given string.

(b) String Plucked at Quarter Length

If the same string is plucked from one quarter of its length, again stationary waves will be set up with nodes and antinodes as shown in Fig. 7.7 (b). Note that now the string vibrates in two loops. This particular configuration of nodes and antinodes has developed because the string was plucked from the position of an antinode. As the distance between two consecutive nodes is half the wavelength, so the length $\ell$ of string is equal to the wavelength of the waves set up in this mode. If $\lambda_{2}$ is the measure of wavelength of these waves, then:

Comparison of Eq. (7.5) with Eq. (7.1) shows that the wavelength in this case is half of that in the first case. Equation (7.2) shows that the speed of the waves depends on tension and mass per unit length of the string, it is independent of the point where the string is plucked. So, speed remains the same in both cases.

If $f_{2}$ is frequency of vibration of string in its second mode, then $f_{2} = v/\lambda_{2}$

Since $\lambda_{2} = \ell$, therefore $f_{2} = v/\ell$

Multiplying and dividing by $2$, we have $f_{2} = 2v/2\ell$

We know that: $f_{1} = v/2\ell$, so $f_{2} = 2f_{1}$

Thus, when the string vibrates in two loops, its frequency becomes double than when it vibrates in one loop.

(c) String Plucked at an Arbitrary Point

Let the string resonates in $n$ number of loops with $(n + 1)$ nodes and $n$ antinodes. Thus, we can say that if the string is made to vibrate in $n$ loops, the frequency of stationary waves set up on the string will be:

and the corresponding wavelength is:

where $n = 1, 2, 3, \ldots$

It is clear that as the string vibrates in more than one loop, its frequency $f$ goes on increasing and the wavelength $\lambda$ gets correspondingly shorter. However, the product of the frequency and wavelength is always equal to $v$, the speed of waves.

The above discussion clearly establishes that:

1. The stationary waves have a discrete set of frequencies $f_{1}$, $2f_{1}$, $3f_{1}$, $\ldots$, $nf_{1}$, which is known as harmonic series. The lowest characteristic frequency of vibration is the fundamental frequency $f_{1}$, corresponds to the first harmonic. The frequency $f_{2} = 2f_{1}$ corresponds to the second harmonic and so on.

2. In other words, quantum jumps in frequency exist between the resonance frequencies. This phenomenon is known as the Quantization of frequencies. It means $f_{n} = nf_{1}$, where $n = 1, 2, 3, \ldots$ (Integral multiples). The stationary waves can be set up on the string only with the frequencies of harmonic series determined by the tension, length and mass per unit length of the string. Waves which are not in harmonic series are quickly damped out.

3. The frequency of a string on a musical instrument can be changed either by varying the tension or by changing the length. For example, the tension in guitar and violin strings is varied by tightening the pegs on the neck of the instrument. Once the instrument is tuned, the musicians vary the frequency by moving their fingers along the neck, thereby changing the length of the vibrating portion of the string.

Example 7.3

A stationary wave is established in a string which is $120\,\mathrm{cm}$ long and fixed at both ends. The string vibrates in four segments at a frequency of $120\,\mathrm{Hz}$. Determine its wavelength and the fundamental frequency?

Solution

$\ell = 120\,\mathrm{cm} = 120/100\,\mathrm{m} = 1.2\,\mathrm{m}$

$n = 4$

$f_{4} = 120\,\mathrm{Hz}$

$f_{1} = ?$

$\lambda = ?$

(i) As the string vibrates in four segments and the distance between two consecutive nodes is $\lambda/2$, so the wavelength of the string is:

$\ell = n(\lambda_{n}/2)$ or $\lambda_{n} = 2\ell/n$

(ii) If string vibrates in $n$ loops, then frequency of stationary wave will be:

$f_{n} = nf_{1}$

$120\,\mathrm{Hz} = 4f_{1}$