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

Question:

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