Two waves traveling on a string in the same direction both have a frequency of 92 Hz, a...

Question:

Two waves traveling on a string in the same direction both have a frequency of 92 Hz, a wavelength of 0.19 m, and an amplitude of 0.32 m.

a. What is the amplitude of the resultant wave if the original waves differ in phase by {eq}\pi {/eq}/3 rad?

b. What is the phase difference between the two waves if the amplitude of the resultant wave is 0.25 m?

Two-Wave Superposition:

Let us consider two identical waves with a phase difference traveling through a tensile string.

Let the equations of the waves:

{eq}\displaystyle{\begin{align*} y_1 &= y_0 \ \sin(\omega t -kx)\\ y_2 &= y_0 \ \sin(\omega t - kx + \phi)\\ \end{align*}} {/eq}

Therefore, the equation of the resultant wave can be given by:

{eq}\displaystyle{\begin{align*} y &= y_1 + y_2\\ y &= y_0 \ \sin(\omega t - kx) + y_0 \ \sin(\omega t - kx + \phi)\\ y &= 2y_0 \ \cos\left(\dfrac{\phi}{2}\right) \ \sin\left(\omega t - kx + \dfrac{\phi}{2}\right)\\ \end{align*}} {/eq}

The amplitude of the resultant wave can thus be given by:

{eq}\displaystyle{Y_0 = 2y_0 \ \cos\left(\dfrac{\phi}{2}\right)} {/eq}

The phase difference can be expressed as:

{eq}\displaystyle{\phi = 2 \ \cos^{-1}\left(\dfrac{Y_0}{2y_0}\right)} {/eq}.

Answer and Explanation: 1

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a.

Given

  • The amplitude of each of the original waves: {eq}y_0 = 0.32 \ \rm m {/eq}.
  • The phase difference between the original waves: {eq}\phi =...

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Law of Superposition: Definition & Concept

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Chapter 1 / Lesson 6
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Learn the definition of the law of superposition, what the law of superposition states, and how the law of superposition is applied with examples.


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