Question

Two traveling sine waves, identical except for a phase difference \(\phi,\) add so that their superposition produces another traveling wave with the same amplitude as the two component waves. What is the phase difference between the two waves?

Short answer

The phase difference between two identical sine waves for maintaining the same amplitude when superposed is given by: $$\phi = 2n\pi$$ where \(n\) is an integer.

Step-by-step solution

Write the equation for the two sine waves

Let the two traveling sine waves be given by: $$y_1 = A\sin(kx - \omega t)$$ $$y_2 = A\sin(kx - \omega t + \phi)$$ where \(A\) is the amplitude, \(k\) is the wave number, \(\omega\) is the angular frequency, and \(\phi\) is the phase difference between the two waves.

Write the equation for their superposition

The superposition of the two waves is the sum of the two waves: $$y_{total} = y_1 + y_2 = A\sin(kx - \omega t) + A\sin(kx - \omega t + \phi)$$

Use the sine addition formula to simplify the superposition equation

To simplify the equation for the total wave, we can use the sine addition formula: $$\sin(a + b) = \sin a \cos b + \cos a \sin b$$ Applying this formula to our superposition equation, we get: $$y_{total} = A[\sin(kx - \omega t)\cos\phi + \cos(kx - \omega t)\sin\phi]$$

Find the condition for the amplitude to remain the same

We want to find the value of \(\phi\) for which the amplitude of the resulting wave \(y_{total}\) remains \(A\). Notice that \(\sin(kx - \omega t)\) and \(\cos(kx - \omega t)\) are functions of \(x\) and \(t\), but \(\phi\) is a constant. So, we want to find a value of \(\phi\) such that the amplitude of the sum remains constant at \(A\) for all values of \(x\) and \(t\). To do this, let's rewrite the total wave equation: $$y_{total} = Y(x,t)\sin(kx - \omega t) + Z(x,t)\cos(kx - \omega t)$$ where \(Y(x,t) = A\cos\phi\) and \(Z(x,t) = A\sin\phi\). We want \(Y(x,t)^2 + Z(x,t)^2\) to be constant for all \(x\) and \(t\). This is only possible if: $$Y(x,t)^2 + Z(x,t)^2 = A^2$$

Solve for the phase difference \(\phi\)

Using the expressions for \(Y(x,t)\) and \(Z(x,t)\), we can rewrite the equation as: $$[A\cos\phi]^2 + [A\sin\phi]^2 = A^2$$ Dividing both sides by \(A^2\), we get: $$\cos^2\phi + \sin^2\phi = 1$$ This equation holds true for all values of \(\phi\). However, to find the specific value of \(\phi\) that causes the amplitude to remain the same, we must consider the nature of the sine and cosine functions. Recall that \(\cos\phi\) and \(\sin\phi\) will have their maximum values of 1 when \(\phi\) is an integer multiple of \(2\pi\). So, the phase difference \(\phi\) that will result in the superposition maintaining the same amplitude as the component waves is given by: $$\phi = 2n\pi$$ where \(n\) is an integer.