Question

A string of length$L$fixed at both ends vibrates in its fundamental mode at a frequency $v$, and maximum amplitude $A$.

$\left(a\right)$ Find the wavelength and the wave number$k$.

$\left(b\right)$ Take the origin at one end of the string and the $X-$axis along the string. Take the $Y-$axis along the direction of the displacement. Take$t=0$ at the instant when the middle point of the string passes through its mean position and is going in the positive $y-$direction. Write the equation describing the standing wave.

Answer 1

Step 1: Given data

Length of the string$=L$

Frequency of vibration$=v$

Maximum amplitude of vibrations$=A$

Step 2: Find wavelength of the wave

Velocity of wave, $V=\surd \left(\frac{T}{m}\right)$

wavelength, $\lambda =$Velocity$/$Frequency

$=\frac{\surd \left({\displaystyle \frac{T}{m}}\right)}{\left[\right(\frac{1}{2L})\surd (\frac{T}{m}\left)\right]}$ [As, frequency$=\left(\frac{1}{2L}\right)\times V$]

$=2L$

Hence, wavelength is $2L$.

Step 3: Find wavenumber of the wave

wave number, $k=\frac{2\pi }{\lambda }$

$$\begin{gathered} \Rightarrow k=\frac{2\pi }{2L} \\ \Rightarrow k=\frac{\pi }{L} \end{gathered}$$

Hence, the wavenumber of the wave is $\frac{\mathrm{\pi }}{L}$

Step 4: Write equation for the stationary wave

Equation of the stationary wave, $y=A\cos \left(\frac{2\pi x}{\lambda }\right)\sin \left(\frac{2\pi Vt}{\lambda }\right)$

As, frequency of vibration

$$\begin{gathered} v=\frac{V}{\lambda } \\ \Rightarrow v=\frac{V}{2L} \end{gathered}$$

$$\begin{gathered} y=A\cos \left(\frac{2\pi x}{\lambda }\right)\sin \left(\frac{2\pi vt}{\lambda }\right) \\ \Rightarrow y=A\cos \left(\frac{2\pi x}{2L}\right)\sin \left(\frac{2\pi vt}{2L}\right) \\ \Rightarrow y=A\cos \left(\frac{\pi x}{L}\right)\sin \left(\frac{\pi vt}{L}\right) \end{gathered}$$

Equation for the stationary wave,

$\Rightarrow y=A\cos \left(\frac{\pi x}{L}\right)\sin \left(\frac{\pi vt}{L}\right)$

Hence, wavelength is $2L$.

Hence, the wavenumber of the wave is $\frac{\mathrm{\pi }}{L}$.

Hence, equation for the stationary wave is $y=A\cos \left(\frac{\pi x}{L}\right)\sin \left(\frac{\pi vt}{L}\right)$.