[Physics] Expression describing standing waves

waves

In my textbook, it says that a stationary wave is generated upon interference with the incident and the reflected wave. Well, if there is a reflected wave, I suppose that there would be a phase change of 180 degrees. When that happens won't we express the reflected wave as the following?

$$y = A\sin (kx + \omega t + \pi)$$

Assuming that the reflected wave is coming from the opposite end and having a negative amplitude.

Best Answer

A standing wave is formed when you have a continuous source of disturbance on one end. Consider a string whose one end is fixed and the other end is attached to a source of disturbance.

Let the equation of the wave created by the source of disturbance be:

$$y = A\sin (kx + \omega t)$$

This wave travels through the string and reflects from the fixed end of the string. The reflected wave is traveling in the opposite direction and is also 180 degrees out of phase (reflections from fixed ends of a string are 180 degrees out of phase). Therefore, the equation of the wave is given by:

$$y = A\sin (kx - \omega t + \pi)$$

The reflected wave is NOT the equation of the standing wave. As you might already know, two waves can interfere to form a new wave. The new wave is given by the sum of the two waves.

The equation of the standing wave is given by (for a general case where the phase difference is $\phi$):

$$y = A\sin (kx - \omega t - \phi) + A\sin (kx + \omega t)$$ $$ = 2A \cos(kx + \frac{\phi}{2})\sin(\omega t - \frac{\phi}{2})$$

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