Question

How is the equation of a standing wave in a string fixed at both ends?

Answer 1

Hint: First let us see what standing wave is-
A standing wave sometimes referred to as a stationary wave, is a wave in physics that oscillates in time but whose overall amplitude profile does not shift in space.
In any point in space, the peak amplitude of the wave oscillations is consistent with time and the oscillations are in phase in various points in the motion.

Complete step by step solution:
A standing wave pattern is a vibration pattern produced within a medium where the source’s vibrational frequency induces reflection waves from one end of the medium to interact with the source’s incident waves. Only at particular frequencies of vibration are certain patterns produced within the medium.
The equation for standing wave fixed at both ends of a string is given by-
$y=2A\sin kx\cos \omega t$

We can also write it as
$$\begin{gathered} y=A\left[\sin (kx+\omega t)+\sin \left(kx-\omega t\right)\right] \\ y=A\left[\sin (\omega t+kx)-\sin (\omega t-kx)\right] \end{gathered}$$

The standard wave equation is given by –
$y\left(x.t\right)=A\sin \left(kx-\omega t+\varphi \right)$

Now comparing the equations for standing wave with the standard wave equation we get-
Amplitude value $=A$
Direction of wave $=x$
Frequency $=\omega$
Phase constant $=k$

The wavelengths of the normal modes for a string of length $L$ fixed at both ends are ${\lambda }_n={\displaystyle \frac{2L}{n}}n=1,2,3.......$

Here nodes occur when $kx=0=kL$

$$\begin{gathered} kL=n\pi \\ k={\displaystyle \frac{2\pi }{\lambda }}=n{\displaystyle \frac{\pi }{L}} \\ L=n{\displaystyle \frac{\lambda }{2}} \end{gathered}$$

$n$ is the $n^{th}$ normal mode of oscillation
The natural frequencies are
$f_n=n{\displaystyle \frac{v}{2L}}={\displaystyle \frac{n}{2L}}\sqrt{{\displaystyle \frac{T}{\mu }}}$

Here $f$ is also called the fundamental frequency.

Additional information:
Standing waves are synonymous with resonance all the time. The resonance of the resulting vibrations can be identified by a drastic rise in amplitude. Compared to moving waves of the same amplitude, it is relatively effortless to produce standing waves.

Note: For the string fixed at both ends $x=0$ and $x=L$. The amplitude must be zero at the ends and the argument of sin must be zero. Also the antinodes have maximum amplitude at $2A$ and the nodes have zero amplitude.