Animated Solution for Physics - Waves: A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation, $y(x,t)=(0.01\text{ m})[\sin (62.8{\text{ m}}^{-1})x]\cos [(628{\text{ s}}^{-1})t]$. Assuming $\pi =3.14$, the correct statement(s) is (are)

When two identical progressive waves traveling in opposite directions superpose, they form a standing wave (or stationary wave).

Unlike traveling waves, standing waves do not transfer energy through the medium. Instead, energy is localized within vibrating loops bounded by points of zero displacement called nodes and points of maximum displacement called antinodes.

In this problem, we are analyzing a stretched string of length $L$ fixed at both ends, vibrating in its fifth harmonic ($n=5$).

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We are given the standing wave equation:

Comparing this with the standard mathematical form of a standing wave:

We can directly extract the following physical parameters:

Amplitude of the standing wave (at antinodes): $A=0.01\text{ m}$ Wave number: $k=62.8{\text{ m}}^{-1}$ Angular frequency:* $\omega =628{\text{ s}}^{-1}$

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First, let's calculate the wavelength $\lambda$ of the constituent traveling waves using the relation between wave number and wavelength:

Substituting the given values (with $\pi =3.14$):

Now, for a string fixed at both ends, the boundary conditions require nodes at both ends ($x=0$ and $x=L$). This restricts the allowed wavelengths to those that can fit an integer number of half-wavelengths within the string length $L$:

Since the string is vibrating in its fifth harmonic ($n=5$):

This confirms that statement (b) is correct.

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For a string fixed at both ends vibrating in its $n$-th harmonic:

The number of vibrating loops is equal to $n=5$. The number of nodes is always $n+1=5+1=6$ (including the two fixed ends). * The number of antinodes is equal to $n=5$.

Since the total number of nodes is $6$, statement (a) is incorrect.

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The mid-point of the string is located at:

Let's determine whether this point is a node or an antinode by evaluating the spatial term $\sin (kx)$ at this position:

Using trigonometric identities:

Since the spatial term is at its maximum value of $1$, the mid-point of the string is an antinode.

At any antinode, the maximum displacement of the particles from their equilibrium position is equal to the amplitude of the standing wave:

Thus, statement (c) is correct.

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Harmonic Frequency

The frequency of the current vibration (the 5th harmonic) is:

Since the frequency of the $n$-th harmonic is an integer multiple of the fundamental frequency ($f_n=n{f}_0$):

Therefore, the fundamental frequency is $20\text{ Hz}$, which makes statement (d) incorrect.

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By systematically analyzing the wave equation, we have determined:

1. The number of nodes is 6 (Statement a is incorrect). 2. The length of the string is 0.25 m (Statement b is correct). 3. The mid-point is an antinode with a maximum displacement of 0.01 m (Statement c is correct). 4. The fundamental frequency is 20 Hz (Statement d is incorrect).

Thus, the correct options are (b) and (c).