A wire having length L is kept under tension between x=0 and x=L. In one experiment, the equatio

step 1 In this question we will learn about some basic concept of the oscillation and waves. The questi

A wire having length L is kept under tension between x=0 and...

A wire having length $\mathrm{L}$ is kept under tension between $\mathrm{x}=0$ and $\mathrm{x}=\mathrm{L}$. In one experiment, the equation of the wave and energy is given by $\mathrm{y}_{1}=\mathrm{A} \sin (\pi \mathrm{x} / \mathrm{L}) \sin \omega \mathrm{t}$ and $\mathrm{E}_{1}$ respectively. In another experiment, it is $\mathrm{y}_{2}=\mathrm{A} \sin$ $\{(2 \pi \mathrm{x}) / \mathrm{L}\} \sin 2 \omega \mathrm{t}$ and $\mathrm{E}_{2}$. Then........ (A) $E_{2}=E_{1}$ (B) $E_{2}=2 \mathrm{E}_{1}$ (C) $\mathrm{E}_{2}=4 \mathrm{E}_{1}$ (D) $E_{2}=16 \mathrm{E}_{1}$

A wire having length $\mathrm{L}$ is kept under tension between $\mathrm{x}=0$ and $\mathrm{x}=\mathrm{L}$. In one experiment, the equation of the wave and energy is given by $\mathrm{y}_{1}=\mathrm{A} \sin (\pi \mathrm{x} / \mathrm{L}) \sin \omega \mathrm{t}$ and $\mathrm{E}_{1}$
respectively. In another experiment, it is $\mathrm{y}_{2}=\mathrm{A} \sin$ $\{(2 \pi \mathrm{x}) / \mathrm{L}\} \sin 2 \omega \mathrm{t}$ and $\mathrm{E}_{2}$. Then........
(A) $E_{2}=E_{1}$
(B) $E_{2}=2 \mathrm{E}_{1}$
(C) $\mathrm{E}_{2}=4 \mathrm{E}_{1}$
(D) $E_{2}=16 \mathrm{E}_{1}$

Key Concepts

Standing Waves

Standing waves are a form of vibration in which specific points on a medium, termed nodes, appear to stay stationary while the rest of the medium vibrates with maximum amplitude at antinodes. They occur when waves of the same frequency and amplitude interfere, leading to a pattern that is fixed in space. In systems like a stretched wire or string, these standing waves represent resonant vibrational modes.

Modes of Vibration

Modes of vibration refer to the distinct patterns of standing waves that can exist on a vibrating string or in another system. Each mode is characterized by a specific number of nodes and antinodes. The fundamental mode (first harmonic) has the lowest frequency and one antinode between the fixed ends, while higher modes (overtones) have more nodes and antinodes and higher frequencies.

Wave Energy in Strings

The energy stored in a vibrating string includes both kinetic energy, due to the motion of the string elements, and potential energy, due to the deformation of the string. This energy is closely linked to the amplitude of vibration and the rate at which the string is moving or bending, making it dependent on the square of the amplitude as well as the square of the angular frequency.

Frequency Dependence of Energy

For a harmonic oscillator such as a vibrating string, the energy in the oscillation is proportional to the square of the angular frequency when the amplitude is kept constant. This means that if the frequency doubles, the overall energy of the vibration increases by a factor of four. This scaling arises because both the kinetic and potential energy components are proportional to the square of the velocity and displacement gradients, respectively.

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