One of the following statements is inccorect. Select it. A standing wave on a st

Question

One of the following statements is inccorect. Select it.

A standing wave on a string is the superposition of two waves of equal amplitude and frequency but traveling in opposite directions.

If two or more waves are passing through a point at the same time, the net displacement of the medium is equal to the sum of the displacements induced by the individual waves.

The longest possible wavelength of a standing wave on a string fixed at both ends is equal to the twice the length of the string.

Doubling the length of a string fixed at both ends halves its fundamental harmonic frequency. A pulse propagating down a taut string is reflected from the end of the string. The end of the string is fixed. The reflected pulse has the same sign as the incident pulse.

Doubling the amplitude of a wave quadruples its intensity.

Explanation / Answer

Statement 1: - A standing wave on a string is the superposition of two waves of equal amplitude and frequency but traveling in opposite directions.

This statement is true because this is the very definition of standing waves.

Statement 2: If two or more waves are passing through a point at the same time, the net displacement of the medium is equal to the sum of the displacements induced by the individual waves.

The principle of superposition of waves says that "when two or more waves pass through a point at the same time, the net disturbance at the point is equal to the sum of individual disturbances which would have been produced by the individual waves in absence of other waves".

Thus, as per this principle the Statement 2 is correct.

Statement 3: The longest possible wavelength of a standing wave on a string fixed at both ends is equal to the twice the length of the string.

For a string of length L fixed at both ends, we have

L = n/2 where = wavelength and n = 1,2,3,4.......

or = 2L/n

so the longest wavelength would be found when n=1, then:

= 2L/(1) = 2L

So we see that longest possible wavelength of a standing wave on a string fixed at both ends is equal to the twice the length of the string. Thus Statement 3 is correct.

Statement 4: Doubling the length of a string fixed at both ends halves its fundamental harmonic frequency.

for a string of length L fixed at both ends, we have

= nV/2L, where = frequency of standing wave on the string for a particular value of n.

fundamental frequency is given by the above formula when n=1. so, the fundamental frequency is

0 = V/2L, Now, after doubling the length of the string the fundamental frequency becomes

0 = V/4L (new length of string is 2L)

So the fundamental frequency is halved.

So, the Statement 4 is correct.

Statement 5: A pulse propagating down a taut string is reflected from the end of the string. The end of the string is fixed. The reflected pulse has the same sign as the incident pulse.

If a wave enters a region where wave velocity is smaller, the reflected wave is inverted. The end of the string is fixed, so obviously the speed will be zero in that region which is smaller than the speed of the incident wave. So the reflected wave in this case will be inverted and will have opposite sign copared to the incident wave/pulse.

So Statement 5 is incorrect.

Statement 6: Doubling the amplitude of a wave quadruples its intensity.

Intensity of a wave is directly proportional to the square of amplitude of the wave. So it's clear that doubling the amplitude would quadruples the intensity

So, Statement 6 is correct.

So, this concludes the answer. If you find anything lacking, please let me know. I will provide further clarification needed by you.

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