Wave Motion and String Waves Concept Page - 9

Definition

Understand and use the relation between fundamental frequency and higher overtones for string fixed at one end

Vibration of a string fixed at one end standing waves can be produced on a string which is fixed at one end and whose other end is free to move in a transverse direction. Such a free end can be nearly achieved by connecting the string to a very light thread. If the vibrations are produced by a source of correct frequency, standing waves are produced. If the end x=0$x=0$ is fixed and x=L$x=L$ is free.
y=2A sin kxcos$y=2Asinkxcos$ ωt$\omega t$
with the boundary condition that x=L$x=L$ is an antinode. The boundary condition that x=0$x=0$ is a node is automatically satisfied by the above equation. For x=L$x=L$ to be an antinode,

sinkL=±1$sinkL=\pm 1$

Or, kL=(n+12)π$kL=(n+\frac{1}{2})\pi$
d
Or, 2πLλ=(n+12)π${\displaystyle \frac{2\pi L}{\lambda }}=(n+{\displaystyle \frac{1}{2}})\pi$

Or, 2Lνv=n+12${\displaystyle \frac{2L\nu }{v}}=n+{\displaystyle \frac{1}{2}}$

Or, ν=(n+12)v2L=n+122LF/μ−−−−√$\nu =(n+{\displaystyle \frac{1}{2}}){\displaystyle \frac{v}{2L}}={\displaystyle \frac{n+{\displaystyle \frac{1}{2}}}{2L}}\sqrt{F/\mu }$.

These are the normal frequencies of vibration. The fundamental frequency is obtained when n=0$n=0$, ie,

ν0=v/4L${\nu }_0=v/4L$.

The overtone frequencies are

ν1=3v4L=3v0${\nu }_1={\displaystyle \frac{3v}{4L}}=3v_0$,
ν3=5v4L=5v0${\nu }_3={\displaystyle \frac{5v}{4L}}=5v_0$,

ν3=7v4L=7v0${\nu }_3={\displaystyle \frac{7v}{4L}}=7v_0$, etc.

We see that all the harmonics of the fundamental are not the allowed frequencies for the standing waves. Only the odd harmonics are the overtones. Figure shows shapes of the string for some of the normal modes.

Example

Solve problems on standing waves with given information about its possible harmonics

Example: A rope, under a tension of 200 N and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by : y=0.1sin(πx2)sin12πt$y=0.1sin(\frac{\pi x}{2})sin12\pi t$. Where x = 0 at one end of the rope, x is in meters and t is in seconds. If the rope oscillates in a third-harmonic standing wave pattern, the period of oscillation is 1/x$1/x$ sec. Find x$x$.

Solution:
In the second harmonic, ω=12π$\omega =12\pi$.Hence the time period T2=2πω=16$T_2={\displaystyle \frac{2\pi }{\omega }}={\displaystyle \frac{1}{6}}$⟹$⟹$ Frequency=1T2=6=ν2${\displaystyle \frac{1}{T_2}}=6={\nu }_2$For a standing wave pattern, νn=n2lTμ−−√∝n${\nu }_n={\displaystyle \frac{n}{2l}}\sqrt{{\displaystyle \frac{T}{\mu }}}\propto n$⟹ν3=32ν2=9$⟹{\nu }_3={\displaystyle \frac{3}{2}}{\nu }_2=9$⟹T3=19$⟹T_3={\displaystyle \frac{1}{9}}$

Diagram

Draw diagrams for standing waves on strings for different configurations

Example

The nature of standing waves for rods clamped at different points along the length

Example: A massless rod of length l$l$  is hung from the ceiling with the help of
two identical wires attached at its ends. A block is hung on the rod at a distance x$x$ from the left end. In the case, the frequency of the 1st harmonic of the wire on the left end is equal to the frequency of the 2nd harmonic of the wire on the right. Find the value of x$x$?

Solution:
Since, the frequency of the first harmonic from the left is equal to that of second harmonic from right,
ν1=2ν2${\nu }_1=2{\nu }_2$
⟹T1−−√=2T2−−√$⟹\sqrt{T_1}=2\sqrt{T_2}$ (T1$T_1$ and T2$T_2$ are the tensions developed in the two wires).
⟹T1=4T2$⟹T_1=4T_2$
Thus, according to the question,
T1(x)=T2(l−x)$T_1(x)=T_2(l-x)$
⟹4T2(x)=T2(l−x)$⟹4T_2(x)=T_2(l-x)$
Solving this equation for T1$T_1$ and T2$T_2$ we get the value of x=l5$x={\displaystyle \frac{l}{5}}$

Example

Infer reflected wave on a string on reflection from a rigid boundary

Example: A pulse in a rope, as shown in the given figure, approaches a solid wall and it gets reflected from it. What will be the nature wave pulse after reflection?

Solution:
Wave pulse on a string moving from left to right towards the end which is rigidly clamped. As the wave pulse approaches the fixed end, the internal restoring forces which allow the wave to propagate exert an upward force on the end of the string. But, since the end is clamped, it cannot move. According to Newton's third law, the wall must be exerting an equal downward force on the end of the string. This new force creates a wave pulse that propagates from right to left, with the same speed and amplitude as the incident wave, but with opposite polarity (upside down)At a fixed (hard) boundary, the displacement remains zero and the reflected wave changes its polarity (undergoes a 1800${180}^0$  phase change).

Definition

Formation of Standing Wave

The result of the interference of the two waves gives a new wave pattern known as a standing wave pattern. Standing waves are produced whenever two waves of identical frequency and amplitude interfere with one another while traveling opposite directions along the same medium.

Definition

Node and Antinodes

A node is a point along a standing wave where the wave has minimum amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the note played. The opposite of a node is an anti-node, a point where the amplitude of the standing wave is a maximum. These occur midway between the nodes.

Formula

Equation of standing wave in a string

The general equation of a standing wave on a string is:
y=Asin[kx]cos[wt]$y=A\sin [kx]\cos [wt]$

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