91Ó°ÊÓ

The tension is increased to ${\mathrm{Ï„}}_f=4{\mathrm{Ï„}}_i$.

By using the formulas for resonant frequency fand the speed v of the wave, we can find the frequency of oscillation in terms of and by using the formula for the new wavelength we can find the wavelength of the waves in terms of .

Formula:

The resonant frequency of a wave,$f=\frac{v}{\mathrm{λ}}......\left(1\right)$

The speed of wave, $v=\sqrt{\frac{\mathrm{τ}}{\mathrm{μ}}}.........\left(2\right)$

The nth frequency of a wave, $f_n=\frac{nv}{2L}.....\left(3\right)$

Substituting the equation (2) in equation (3), we get the nth frequency as:

$f_n=\frac{n}{2L}×\sqrt{\frac{\mathrm{τ}}{\mathrm{μ}}}$

Hence, the third frequency is given as:

$f_3=\frac{3}{2L}×\sqrt{\frac{{\mathrm{τ}}_i}{\mathrm{μ}}}......\left(4\right)$

When,${\mathrm{Ï„}}_f=4{\mathrm{Ï„}}_i$, using this in equation (4),we get the new frequency is given as:

$$\begin{gathered} f{\text{'}}_3=\frac{3}{2L}×\sqrt{\frac{{\mathrm{Ï„}}_i}{\mathrm{μ}}} \\ =2\left[\frac{3}{2L}×\sqrt{\frac{{\mathrm{Ï„}}_i}{\mathrm{μ}}}\right](âˆ\mu {\mathrm{Ï„}}_f=4{\mathrm{Ï„}}_i) \\ =2f_3 \end{gathered}$$

Hence, the value of the new frequency is $2f_3$

Using equation (1), we can get the new wavelength as given:

$$\begin{gathered} \mathrm{λ}{\text{'}}_3=\frac{v\text{'}}{f{\text{'}}_3} \\ =\frac{\sqrt{{\displaystyle \frac{{\mathrm{Ï„}}_i}{\mathrm{μ}}}}}{2f_3}(âˆ\mu u\sin gequation(2\left)\right) \\ =2\left[\begin{array}{c}\frac{\sqrt{{\displaystyle \frac{{\mathrm{Ï„}}_i}{\mathrm{μ}}}}}{2f_3}\\ \end{array}\right](âˆ\mu f_3\text{'}=f_{3}and{\mathrm{Ï„}}_f=4{\mathrm{Ï„}}_i) \\ =\frac{v}{f_3} \\ ={\mathrm{λ}}_3 \end{gathered}$$

Hence, the value of new wavelength is ${\mathrm{λ}}_3$