Question

A sinusoidal wave can be described by a cosine function, which is negative just as often as positive. So why isn’t the average power delivered by this wave zero?

Short answer

The average power remains positive.

Step-by-step solution

Concept of Power of the wave

The power of a mechanical wave is equal to the square of the amplitude and the square of the frequency of the wave.

The power of the wave expression is given as:

$$\begin{gathered} \mathrm{P}\left(\mathrm{x},\mathrm{t}\right)=\mathrm{F}\left(\mathrm{x},\mathrm{t}\right).\mathrm{v}\left(\mathrm{x},\mathrm{t}\right) \\ =\mathrm{KAsin}\left(\mathrm{kx}-\mathrm{\omega t}\right).\mathrm{\omega Asin}\left(\mathrm{kx}-\mathrm{\omega t}\right) \\ ={\mathrm{K\omega A}}^2{\sin }^2\left(\mathrm{kx}-\mathrm{\omega t}\right) \end{gathered}$$

Here, is the power, P is the power, F is the force, v is the velocity, x is the displacement, and t is the time, k is the spring constant, is the force constant, A is the amplitude, $\omega$is the angular frequency.

The power is the square of the cosine function. So whole part of the function remains positive all time. So, the power also remains positive.

Thus, the average power remains positive.