Waves can carry energy from one place to another without transporting the medium itself over large distances. In a vibrating string, the disturbance moves along the string while each element of the string oscillates up and down. In sound, the disturbance travels through a fluid as alternating compressions and rarefactions. The important measurable quantity is often the average power, which tells us how much energy passes a point each second.
For a sinusoidal transverse wave on a string, the average power is
\[
P=\frac12 \mu \omega^2 A^2 v,
\]
where \(\mu\) is the linear mass density of the string, \(\omega\) is the angular frequency, \(A\) is the amplitude, and \(v\) is the wave speed. This expression comes from averaging the instantaneous energy flow over one full cycle. The factor of \(\tfrac12\) appears because the squared sine and cosine terms contribute an average value of one half over a period.
The corresponding average energy per unit length on the string is
\[
u=\frac12\mu\omega^2A^2.
\]
Since the energy density \(u\) moves with the wave at speed \(v\), the average power is simply \(P=uv\). This is a useful way to interpret the formula physically: energy density times transport speed gives power flow.
For sound in a fluid, the same basic idea applies, but now the medium is three-dimensional. The average energy density is written as
\[
u=\frac12\rho\omega^2A^2,
\]
where \(\rho\) is the mass density of the fluid. Multiplying by the sound speed gives the average intensity,
\[
I=\frac12\rho\omega^2A^2v.
\]
Intensity is power per unit area, so if the wave passes through an area \(S\), the total power is
\[
P=IS.
\]
These formulas show several important proportionalities. First, power is proportional to the square of amplitude. That means doubling the amplitude multiplies the power by four. Second, power also increases with the square of angular frequency. Faster oscillations transfer energy more rapidly. Third, denser media and larger wave speeds tend to support larger power flow for the same amplitude and frequency.
Consider the sample string values \(\mu=0.01\ \text{kg/m}\), \(\omega=10\ \text{rad/s}\), \(A=0.1\ \text{m}\), and \(v=20\ \text{m/s}\). Substituting gives
\[
P=\frac12(0.01)(10^2)(0.1^2)(20)=0.1\ \text{W}.
\]
So the correct result for that sample is \(0.1\ \text{W}\), not \(1\ \text{W}\). This is a good reminder to check the powers of ten and the amplitude-squared term carefully.
In real applications, wave power can vary enormously. A plucked guitar string may carry a tiny fraction of a watt, while seismic waves or large water waves can transport enormous amounts of energy. In electromagnetism, an analogous idea appears in the Poynting vector, which describes the rate and direction of energy transport in electromagnetic waves. Although the formulas differ, the central idea is the same: waves transmit energy, and their power depends on both how strongly the medium oscillates and how quickly the disturbance moves.
This calculator lets you explore both one-dimensional string waves and fluid sound waves with the same structural idea: average energy density times propagation speed gives the rate of energy transfer.
